Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs
FFINITE-HORIZON PARAMETERIZING MANIFOLDS, AND APPLICATIONS TOSUBOPTIMAL CONTROL OF NONLINEAR PARABOLIC PDES
MICKA¨EL D. CHEKROUN AND HONGHU LIU
Abstract.
This article proposes a new approach for the design of low-dimensional suboptimal con-trollers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolictype. The approach fits into the long tradition of seeking for slaving relationships between the smallscales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds todo so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0 , T ] and alow-mode truncation of the PDE, a PM provides an approximate parameterization of the high modesby the controlled low ones so that the unexplained high-mode energy is reduced — in a mean-squaresense over [0 , T ] — when this parameterization is applied.Analytic formulas of such PMs are derived by application of the method of pullback approximationof the high-modes introduced in [26]. These formulas allow for an effective derivation of reduced systemsof ordinary differential equations (ODEs), aimed to model the evolution of the low-mode truncation ofthe controlled state variable, where the high-mode part is approximated by the PM function appliedto the low modes. The design of low-dimensional suboptimal controllers is then obtained by (indirect)techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs.
A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u ∗ R and the optimal controller u ∗ are derived under a second-order sufficient optimality condition. Theseestimates demonstrate that the closeness of u ∗ R to u ∗ is mainly conditioned on two factors: (i) theparameterization defect of a given PM, associated respectively with the suboptimal controller u ∗ R andthe optimal controller u ∗ ; and (ii) the energy kept in the high modes of the PDE solution either drivenby u ∗ R or u ∗ itself.The practical performances of such PM-based suboptimal controllers are numerically assessed foroptimal control problems associated with a Burgers-type equation; the locally as well as globallydistributed cases being both considered. The numerical results show that a PM-based reduced systemallows for the design of suboptimal controllers with good performances provided that the associatedparameterization defects and energy kept in the high modes are small enough, in agreement with therigorous results. Contents
1. Introduction 22. Optimal Control of Nonlinear PDEs, and Functional Framework 43. Finite-Horizon Parameterizing Manifolds: Definition, Pullback Characterization andAnalytic Formulas 64. Finite-Horizon Parameterizing Manifolds for Suboptimal Control of PDEs 115. 2D-Suboptimal Controller Synthesis Based on the Leading-Order Finite-Horizon PM:Application to a Burgers-type Equation 196. 2D-Suboptimal Controller Synthesis Based on Higher-Order Finite-Horizon PMs 317. Synthesis of m -Dimensional Locally Distributed Suboptimal Controllers 41Acknowledgments 49 Key words and phrases.
Parabolic optimal control problems, low-order models, error estimates, Burgers-type equa-tion, Backward-forward systems. a r X i v : . [ m a t h . O C ] A p r MICKAEL CHEKROUN AND HONGHU LIU
Appendix A. Suboptimal Controller Synthesis Based on Galerkin Projections and PontryaginMaximum Principle 50Appendix B. Global Well-posedness for the Two-dimensional h (1) λ -based Reduced System(5.27) 53References 541. Introduction
In this article, we propose a new approach for the synthesis of low-dimensional suboptimal controllersfor optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type.Optimal control of PDEs has been extensively studied in the past few decades due largely to itsbroad applications in both engineering and various scientific disciplines, and fruitful results have beenobtained; see e.g. the monographs [8, 10, 30, 43, 48, 55, 77, 99].Due to the complexity of most applications, optimal control problems of parabolic PDEs are oftensolved numerically. Among the commonly used methods one finds methods that solve at once theassociated optimality system using techniques such as the Newton or quasi-Newton methods [14,55, 59], or methods that use optimization algorithms involving for instance an approximation to thegradient of the cost functional; see e.g. [13, 55, 59, 99]. In this case, the gradient can be approximatedby using sensitivity methods or methods based on the adjoint equation; see e.g. [1, 15, 16, 51, 50, 61, 84,85]. Efficient (and accurate) solutions can be designed by such methods [1, 7, 15, 29, 54, 84, 85] whichmay lead however to high-dimensional problems that can turn out to be computationally expensive tosolve, especially for fluid flows applications. The task becomes even more challenging when a dynamicprogramming approach is adopted, involving typically to solve (infinite-dimensional) Hamilton-Jacobi-Bellman (HJB) equations [8, 9, 24, 34, 35, 36, 37].As an alternative, various reduction techniques have been proposed in the literature to seek insteadfor low-dimensional suboptimal controllers. The main issue related to such techniques relies howeveron the ability to design suboptimal solutions close enough to the genuine optimal one [39, 49, 56, 60,100], while keeping cheap enough the numerical efforts to do so. A general class of model reductiontechniques used extensively in this context is the so-called reduced-order modeling (ROM) approach,based on approximating the nonlinear dynamics by a Galerkin technique relying on basis functions,possibly empirical [47, 53, 54, 88]. Various ROM techniques differ in the choice of the basis functions.One popular method that falls into this category is the so-called proper orthogonal decomposition(POD); see among many others [6, 12, 56, 57, 73, 74, 82, 89], and [49, 62, 63] for other methodsin constructing the reduced basis. We refer also to [75] for suboptimal controllers designed fromthe solutions of low-dimensional HJB equations associated with POD-based Galerkin reduced-ordermodels.Such Galerkin/ROM-based techniques can lead to a synthesis of very efficient suboptimal controllersonce, at a given truncation, the disregarded high-modes do not contribute significantly to the dynamicsof the low modes. However, when this is not the case, the seeking of parameterizations of the disre-garded modes in terms of the low ones becomes central for the design of surrogate low-dimensionalmodels of good performances. The idea of seeking for slaving relationships between the unstable orleast stable modes with the more stable ones has a long tradition in control theory of large-dimensionalor distributed-parameter systems. For instance, by use of methods from singular perturbation theory,the authors in [71, 69, 70, 68] investigated the construction of such slaving functions for slow-fastsystems in terms of invariant (slow) manifolds. Such manifolds are then used to decouple the slow See also [78, Chap. 5] and [79] for the use of singular perturbation techniques for optimal control of PDEs.
INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 3 and fast parts of the dynamics and to feed back the slow component of the state only. This is espe-cially important since the fast components of the state are in general difficult to measure/estimateand consequently to feedback.Complementary to singular perturbation methods, the authors of [28] used tools of center manifoldand normal form theory to design a nonlinear controller and obtained a closed-loop center manifoldfor a truncated distributed-parameter system; in their case proximity to a bifurcation constitutes aguarantee to the separation of relevant time scales of the problem. In [31, 32], the authors have gonebeyond the finite-dimensional singular perturbation work of [68] or center-manifold-based work of [28]to exploit approximate inertial manifolds (AIMs) [45] in the infinite-dimensional case; the latter areglobal manifolds in phase space that can be thought of as generalizations of slow/center manifolds.Using AIMs, the authors of [28] designed then observer-based nonlinear feedback controllers (throughthe corresponding closed-loop AIMs) and demonstrated their performance.The potential usefulness of inertial manifolds (IMs) [33, 46, 97] or AIMs in control theory of nonlinearparabolic PDEs was actually quickly identified after IM theory started to be established [22, 31, 32, 93];see e.g. [92, 95] for a state-of-the-art of the literature at the end of the 90s. However since theseworks, IMs or AIMs have been mainly employed to derive low-dimensional vector fields for the designof feedback controllers [3, 91]. To the exception of [4, 60], the use of IMs or AIMs to design suboptimalsolutions to optimal control problems have been much less considered.The main purpose of this article is to introduce a general framework — in the continuity butdifferent from the AIM approach — for the effective derivation of suboptimal low-dimensional solutionsto optimal control problems associated with nonlinear PDE such as (1.1) given below. To be morespecific, given an ambient Hilbert space, H , the control problems of PDEs we will consider hereaftertake the following abstract form:(1.1) d y d t = Ly + F ( y ) + C u ( t ) , t ∈ (0 , T ] , where L denotes a linear operator, F some nonlinearity, and C denotes a bounded linear operator on H ; the state variable y and the controller u living both in L (0 , T ; H ) for a given horizon T >
0; seeSection 2 for more details.The underlying idea consists of seeking for manifolds M aimed to provide — over a finite horizon[0 , T ] — an approximate parameterization of the small scales of the solutions to the uncontrolled PDE associated with Eq. (1.1), namely(1.2) d y d t = Ly + F ( y ) , in terms of their large scales , so that M allows in turn to derive low-dimensional reduced models fromwhich suboptimal controllers can be efficiently designed by standard methods of finite-dimensionaloptimal control theory such as found in e.g. [18, 23, 66, 67, 94]. In that respect, the notion of finite-horizon parameterizing manifold (PM) is introduced in Definition 3.1 below. Finite-horizon PMsdistinguish from the more classical AIMs in the sense that they provide approximate parametrizationof the small scales by the large ones in the L -sense (over [0 , T ]) rather than a hard (cid:15) -approximationto be valid for each time t ∈ [0 , T ], cf. [45]. In particular, a finite-horizon PM allows to reduce the(cumulative) unexplained high-mode energy (over [0 , T ]) from the low modes to be controlled, in away different from other slaving relationships considered so far; the high-mode energy being reducedin a mean-square sense in the case of finite-horizon PMs.Obviously, the difficulty relies still on the ability of such an approach to give access to suboptimalcontrollers of good performance. A priori the task in not easy and a key feature to ensure that a“good” performance is achieved from such a suboptimal low-dimensional controller, u ∗ R , relies on theability of the manifold M derived from the uncontrolled problem to still achieve a sufficiently “small” MICKAEL CHEKROUN AND HONGHU LIU parameterization defect (over the horizon [0 , T ]) of the small scales by the large ones once a controller u ∗ R is used to drive the PDE (1.1); see (3.5) in Definition 3.1. This point is rigorously formulated asTheorem 4.1 in Section 4 (see also Corollary 4.2), which provides — under a second-order sufficientoptimality condition — error estimates on how “close” a low-dimensional suboptimal controller u ∗ R ,designed from a PM-based reduced system, is to the optimal controller u ∗ . The error estimates (4.5)and (4.10) show in particular that the closeness of u ∗ R to u ∗ is mainly conditioned on two factors: (i)the parameterization defect of a given PM, associated respectively with the suboptimal controller u ∗ R and the optimal controller u ∗ ; and (ii) the energy kept in the high modes of the PDE solution eitherdriven by u ∗ R or u ∗ itself.The article is organized as follows. The functional framework associated with optimal control prob-lems related to (1.1) is introduced in Section 2. The definition of finite-horizon PMs and a practicalprocedure to get access to such PMs are introduced in Section 3. In particular analytic formulas ofleading-order PMs are provided; the latter being subject to a cross non-resonance condition (NR) tobe satisfied between the high and the low modes; see Section 3.2. Section 4 is devoted, given an arbi-trary PM, to the derivation of rigorous a priori error estimates between a low-dimensional PM-basedsuboptimal controller and the optimal one; see Theorem 4.1 and Corollary 4.2. The performance ofthe resulting PM-based reduction approach is numerically investigated on a Burgers-type equationin the context of globally and locally distributed control laws; see Sections 5–6, and Section 7. Asa main byproduct, the numerical results strongly indicate that a PM-based reduced system allowsfor a design of suboptimal controllers with good performances provided that the aforementioned pa-rameterization defects and the energy contained in the high modes are small enough, in agreementwith the theoretical predictions of Theorem 4.1 and Corollary 4.2. This is particularly demonstratedin Section 6, where analytic formulas derived in Theorem 6.1 give access to higher-order PMs withreduced parameterization defects compared to those of the leading-order PMs introduced in Section3. In all the cases, the analytic formulas of the PMs used hereafter allows for an efficient design ofsuboptimal controllers by standard (and simple) application of the Pontryagin maximum principle[18, 19, 66, 87] to the PM-based reduced systems.2. Optimal Control of Nonlinear PDEs, and Functional Framework
The functional framework for the optimal control problem considered in this article takes place inHilbert spaces. Let us first introduce the class of partial differential equations (PDEs) to be controlled.For a given Hilbert space H , we consider H to be a subspace compactly and densely embedded in H such that A : H → H is a sectorial operator [52, Def. 1.3.1] satisfying − A is stable in the sense that its spectrum satisfies Re( σ ( − A )) < . To include in our framework PDEs for which the nonlinear terms are responsible of a loss of regularitycompared to the ambient space H , we consider standard interpolated spaces H α between H and H (with α ∈ [0 , along with perturbations of the linear operator − A given by a one-parameter family, { B λ } λ ∈ R , of bounded linear operators from H α to H , that depend continuously on a real parameter λ . By defining L λ := − A + B λ , we are thus left with a one-parameter family of sectorial operators {− L λ } λ ∈ R , each of them mapping H into H . Finally, F : H α → H will denote a continuous k -linear mapping ( k ≥
2) for some α ∈ [0 , depending on the problem at hand; see e.g. [52]. In particular, nonlinearities including a loss of regularity compared to the ambient space H , are allowed; see e.g. Section 5 below.
INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 5
The nonlinear evolution equation to be controlled takes then the following abstract form:(2.1) d y d t = L λ y + F ( y ) + C u ( t ) , t ∈ (0 , T ] , where y ∈ L (0 , T ; H ) denotes the state variable, u ∈ L (0 , T ; H ) denotes the controller; T > C : H → H denoting a bounded (and non-zero) linear control operator . In particular, we will be mainly concernedwith distributed control problems (control inside the domain) and not with problems involving acontrol on the boundary which leads typically to an unbounded control operator; see e.g. [10, Part V,Chap. 2 and 3] and [42, 43, 44]. The parameter λ governs typically the presence of (linearly) unstablemodes for (2.1). In the application considered in Sections 5–7, it will be chosen so that the linearoperator, L λ , admits large-scale unstable modes.We introduce next the cost functional J : L (0 , T ; H ) × L (0 , T ; H ) → R given by(2.3) J ( y, u ) := (cid:90) T [ G ( y ( t )) + E ( u ( t ))]d t, where G : H → R + and E : H → R + are assumed to be continuous, and to satisfy the followingconditions:(C1) G is uniformly Lipschitz on bounded sets of H , and(C2) (cid:107) u (cid:107) ≤ (cid:107) v (cid:107) = ⇒ E ( u ) ≤ E ( v ) , where (cid:107) · (cid:107) denotes the H -norm.Given such a cost functional, we will consider in this article the following type of optimal controlproblem:( P ) min J ( y, u ) s.t. ( y, u ) ∈ L (0 , T ; H ) × L (0 , T ; H ) solves Eq. (2.1)subject to y (0) = y ∈ H . To simplify the presentation, we will make the following assumptions on L λ and F throughout thisarticle: Standing Hypothesis. L λ is self-adjoint, whose eigenvalues (arranged in descending order) aredenoted by { β i ( λ ) } i ∈ N ; and the eigenvectors { e i ( λ ) } i ∈ N of L λ form a Hilbert basis of H . The eigen-vectors are regular enough such that e i ( λ ) ∈ H α for all i ∈ N . The nonlinearity F : H α → H is acontinuous k -linear mapping for some k ≥ , and for some α ∈ [0 , . In particular, F (0) = 0 . We also assume that for any initial datum y ∈ H , any T >
0, and any given u ∈ L (0 , T ; H ), theCauchy problem(2.4) d y d t = L λ y + F ( y ) + C u ( t ) , y (0) = y ∈ H , has a unique solution y ( · , y ; u ) ∈ C ([0 , T ]; H ) ∩ L (0 , T ; H α ), which lives furthermore in the space C ((0 , T ]; H ) ∩ C ([0 , T ]; H α ) ∩ L (0 , T ; H ) when y ∈ H α ; see e.g. [52, Chap. 3] and [81, Chap. 7] forconditions under which such properties are guaranteed. Section 5.1 below deals with such an example. We refer to Sections 5–7 for other type of cost functional including a terminal cost.
MICKAEL CHEKROUN AND HONGHU LIU Finite-Horizon Parameterizing Manifolds: Definition, Pullback Characterizationand Analytic Formulas
This section is devoted to the definition of finite-horizon parameterizing manifolds (PMs) for a givenPDE of type (2.4) and a general method to give access to explicit formulas of such finite-horizon PMsin practice through pullback limits associated with certain backward-forward systems built from theuncontrolled Eq. (1.2).The key idea takes its roots in the notion of (asymptotic) parameterizing manifold introducedin [26] , which reduces here of approximating — over some prescribed finite time interval [0 , T ] —the modes with “high” wave numbers as a pullback limit depending on the time-history of (someapproximation of) the dynamics of the modes with “low” wave numbers. The cut between what is“low” and what is “high” is organized in an abstract setting as follows; we refer to Section 7 for a moreconcrete specification of such a cut in the case of locally distributed controls. The subspace H c ⊂ H defined by,(3.1) H c := span { e , · · · , e m } , spanned by the m -leading modes will be considered as our subspace associated with the low modes.Its topological complements, H s and H s α , in respectively H and H α , will be considered as associatedwith the high modes, leading to the following decomposition(3.2) H = H c ⊕ H s , H α = H c ⊕ H s α . We will use P c and P s to denote the canonical projectors associated with H c and H s , respectively. Here,the usage of the eigenbasis in the decomposition of the phase space is employed for the sake of analyticformulations derived hereafter. In practice, the methodology presented below can be (numerically)adapted when the phase space H is decomposed by using other bases; see also Remark 3.1 (ii).3.1. Finite-horizon parameterizing manifolds.
Let t ∗ > V be an open set in H α ,and U an open set in L (0 , t ∗ ; H ). For a given PDE of type (2.4), a finite-horizon parameterizingmanifold M over the interval [0 , t ∗ ] is defined as the graph of a function h pm from H c to H s α , whichis aimed to provide, for any y ( t, y ; u ) solution of (2.4) with initial datum y ∈ V and control u ∈ U ,an approximate parameterization of its “high-frequency” part, y s ( t, y ; u ) = P s y ( t, y ; u ), in terms ofits “low-frequency” part, y c ( t, y ; u ) = P c y ( t, y ; u ), so that the mean-square error, (cid:82) t ∗ (cid:13)(cid:13) y s ( t, y ; u ) − h pm ( y c ( t, y ; u )) (cid:13)(cid:13) α d t , is strictly smaller than the high-mode energy of y s , (cid:82) t ∗ (cid:107) y s ( t, y ; u ) (cid:107) α d t . Herethe frequencies are understood in a spatial sense, i.e. in terms of wave numbers . In statistical terms,a finite-horizon PM function h pm can thus be thought of as a slaving relationship between the highmodes and the low ones such that the fraction of energy of y s unexplained by h pm ( y c ) ( i.e. via thisslaving relationship) is less than unity.In more precise terms, we are left with the following definition: Definition 3.1.
Let t ∗ > be fixed, V be an open set in H α , and U an open set in L (0 , t ∗ ; H ) . Amanifold M of the form (3.3) M := { ξ + h pm ( ξ ) | ξ ∈ H c } mainly in a stochastic context; see however [26, Section 8.5] for the deterministic setting. In particular, the reduction techniques developed in this article should not be confused with the reduction techniquesbased on the slow manifold theory which have been used to deal with the reduction of optimal control problems arisingin slow-fast systems , where the separation of the dynamics holds in time rather than in space; see e.g. [68, 76, 86].Furthermore, unlike slow manifolds, the finite-horizon PMs considered in this article are not invariant for the dynamics.To the contrary, they correspond to manifolds for which the dynamics wanders around, within some margin whose size(in a mean square sense) is strictly smaller than the energy unexplained by the H c -modes. over the time interval [0 , t ∗ ]. INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 7 is called a finite-horizon parameterizing manifold (PM) over the time interval [0 , t ∗ ] associated withthe PDE (2.4) if the following conditions are satisfied: (i) The function h pm : H c → H s α is continuous. (ii) The following inequality holds for any y ∈ V and any u ∈ U : (3.4) (cid:90) t ∗ (cid:13)(cid:13) y s ( t, y ; u ) − h pm ( y c ( t, y ; u )) (cid:13)(cid:13) α d t < (cid:90) t ∗ (cid:107) y s ( t, y ; u ) (cid:107) α d t, where y c ( · , y ; u ) and y s ( · , y ; u ) are the projections to respectively the subspaces H c and H s α ofthe solution y ( · , y ; u ) for the PDE (2.4) driven by u emanating from y .For a given initial datum y , if y s ( · , y ; u ) is not identically zero, the parameterization defect of M over [0 , t ∗ ] , and associated with the control u , is defined as the following ratio: (3.5) Q ( t ∗ , y ; u ) := (cid:82) t ∗ (cid:13)(cid:13) y s ( t, y ; u ) − h pm ( y c ( t, y ; u )) (cid:13)(cid:13) α d t (cid:82) t ∗ (cid:107) y s ( t, y ; u ) (cid:107) α d t . Note that in Sections 5, 6 and 7, we will illustrate numerically that finite-horizon PMs can actuallybe obtained from the uncontrolled PDE (1.2), with still possibly small parameterization defects whena controller u is applied. The procedure to build in practice such PMs from the uncontrolled PDE (1.2)is described in the next section; see also [26, Section 8.5] for the construction of PMs over arbitrarily(and sufficiently) large horizons.3.2. Finite-horizon parameterizing manifolds as pullback limits of backward-forward sys-tems: the leading-order case.
We consider now the important problem of the practical determi-nation of finite-horizon PMs for PDEs of type (2.4). As mentioned above, following [26], the pullbackapproximation of the high modes in terms of the low ones via appropriate auxiliary systems associ-ated with the uncontrolled
PDE (1.2) will constitute the key ingredient to propose a solution to thisproblem; see also [26, Section 8.5]. In that respect, we consider first the following backward-forwardsystem associated with the uncontrolled PDE (1.2):d y (1) c d s = L c λ y (1) c , s ∈ [ − τ, , y (1) c ( s ) | s =0 = ξ, (3.6a) d y (1) s d s = L s λ y (1) s + P s F ( y (1) c ) , s ∈ [ − τ, , y (1) s ( s ) | s = − τ = 0 , (3.6b)where L c λ := P c L λ , L s λ := P s L λ , and ξ ∈ H c . We refer to Section 6 for other backward-forward systems used in the construction of higher-order finite-horizon PMs.In the system above, the initial value of y (1) c is prescribed at s = 0, and the initial value of y (1) s at s = − τ . The solution of this system is obtained by using a two-step backward-forward integrationprocedure — where Eq. (3.6a) is integrated first backward and Eq. (3.6b) is then integrated forward— made possible due to the partial coupling present in (3.6) where y (1) c forces the evolution equationof y (1) s but not reciprocally. Due to this forcing introduced by y (1) c which emanates (backward) from ξ ,the solution process y (1) s depends naturally on ξ . For that reason, we will emphasize this dependenceas y (1) s [ ξ ] hereafter.It is clear that the solution to the above system is given by:(3.7) y (1) c ( s ) = e sL c λ ξ, s ∈ [ − τ, , ξ ∈ H c ,y (1) s [ ξ ]( − τ, s ) = (cid:90) s − τ e ( s − τ (cid:48) ) L s λ P s F ( e τ (cid:48) L c λ ξ )d τ (cid:48) , s ∈ [ − τ, . MICKAEL CHEKROUN AND HONGHU LIU
The dependence in τ and s in y (1) s [ ξ ] is made apparent to emphasize the two-time descriptionemployed for the description of the non-autonomous dynamics inherent to (3.6b); see e.g. [25, 27].Adopting the language of non-autonomous dynamical systems [25, 27], we then define h (1) λ ( ξ ) as thefollowing pullback limit of the y (1) s -component of the solution to the above system, i.e. ,(3.8) h (1) λ ( ξ ) := lim τ → + ∞ y (1) s [ ξ ]( − τ,
0) = (cid:90) −∞ e − τ (cid:48) L s λ P s F ( e τ (cid:48) L c λ ξ ) d τ (cid:48) , ∀ ξ ∈ H c , when the latter limit exists. We derive hereafter necessary and sufficient conditions for such a limitto exist.In that respect, first note that since L λ is self-adjoint, we have(3.9) e τ (cid:48) L c λ ξ = m (cid:88) i =1 e τ (cid:48) β i ( λ ) ξ i e i , where ξ = (cid:104) ξ, e i (cid:105) , i ∈ I := { , · · · , m } with m = dim( H c ), and (cid:104)· , ·(cid:105) denoting the inner-product in theambient Hilbert space H .Now for a fixed τ >
0, by projecting y (1) s [ ξ ]( − τ,
0) against each eigenmode e n for n > m , we obtain,by using (3.9) and the k -linear property of F ,(3.10) y (1) s [ ξ ]( − τ,
0) = (cid:88) n>m (cid:90) − τ e − τ (cid:48) β n ( λ ) (cid:68) F (cid:16) m (cid:88) i =1 e τ (cid:48) β i ( λ ) ξ i e i (cid:17) , e n (cid:69) d τ (cid:48) e n = (cid:88) n>m (cid:88) ( i , ··· ,i k ) ∈I k (cid:90) − τ e − β n ( λ ) τ (cid:48) + (cid:0) (cid:80) kj =1 β ij ( λ ) (cid:1) τ (cid:48) d τ (cid:48) (cid:68) F ( e i , · · · , e i k ) , e n (cid:69) e n . From this identity, we infer that h (1) λ is well defined if and only if each integral (cid:90) −∞ e − β n ( λ ) τ (cid:48) + (cid:0) (cid:80) kj =1 β ij ( λ ) (cid:1) τ (cid:48) d τ (cid:48) converges, whenever the corresponding nonlinear interaction F ( e i , · · · , e i k ) as projected against e n ,is non-zero. Namely, h (1) λ exists if and only if the following (weak) non-resonance condition holds:(NR) ∀ ( i , · · · , i k ) ∈ I k , n > m, it holds that (cid:16) (cid:104) F ( e i , · · · , e i k ) , e n (cid:105) (cid:54) = 0 (cid:17) = ⇒ (cid:18) k (cid:88) j =1 β i j ( λ ) − β n ( λ ) > (cid:19) ;see also [26, Sect. 7].Assuming the above (NR)-condition, it follows then from (3.8) and (3.10) that h (1) λ takes the fol-lowing form:(3.11) h (1) λ ( ξ ) = (cid:88) n>m (cid:88) ( i , ··· ,i k ) ∈I k ξ i · · · ξ i k (cid:80) kj =1 β i j ( λ ) − β n ( λ ) (cid:68) F ( e i , · · · , e i k ) , e n (cid:69) e n . In particular under the (NR)-condition, each e n -component of h (1) λ ( ξ ) is — in the ξ -variable — anhomogeneous polynomial of order k , the order of the nonlinearity F . For that reason, h (1) λ will bereferred to as the leading-order finite-horizon PM when appropriate, that is when the latter provides afinite-horizon PM. We clarify in the remaining of this section, some (idealistic) conditions under whichsuch a property is met by the manifold function h (1) λ for the PDE (2.4). In practice these conditions INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 9 can be violated, while the manifold function h (1) λ defined by (3.11) still constitutes a finite-horizonPM; see Sections 5.5 and 7 for numerical illustrations.To delineate conditions under which h (1) λ is a finite-horizon PM is still valuable for the theory. Thisis the purpose of Lemma 3.1 below which relies on another key property of h (1) λ such as defined by(3.8), that can be explained using the language of invariant manifold theory for PDEs [26, 83]. Thelatter states that the manifold function h (1) λ constitutes — for the uncontrolled PDE (1.2) — theleading-order approximation of some local invariant manifold near the trivial steady state; see [83,Appendix A] and [26, Sect. 7]. Based on this result we formulate the following lemma about theexistence of finite-horizon PMs. Lemma 3.1.
Let λ be fixed and H c be the subspace spanned by the first m eigenmodes of the linearoperator L λ . Assume that the standing hypothesis of Section 2 holds, and that (3.12) β m ( λ ) > kβ m +1 ( λ ) . Assume furthermore that the non-resonance condition (NR) holds so that the pullback limit h (1) λ definedby (3.8) exists.Assume that h (1) λ is non-degenerate in the sense that there exists C > such that (3.13) (cid:107) h (1) λ ( ξ ) (cid:107) α ≥ C (cid:107) ξ (cid:107) kα , ξ ∈ H c . Then, for any fixed t ∗ > , there exist open neighborhoods V ⊂ H s α and U ⊂ L (0 , t ∗ ; H ) containingthe origins of the respective spaces, such that h (1) λ is a finite-horizon parameterizing manifold over thetime interval [0 , t ∗ ] for the PDE (2.4) driven by any control u ∈ U and with initial data taken from V .Proof. Let us first recall some related elements from [26]. Note that the PDE (1.2) fits into theframework of [26, Cor. 7.1]. Since the nonlinearity F is assumed to be k -linear for some k ≥ M loc λ := { ξ + h loc λ ( ξ ) | ξ ∈ B } , where h loc λ : H c → H s α is the corresponding local manifold function, B ⊂ H c is an open neighborhoodof the origin in H c , and h loc λ (0) = 0. Recall that the (NR)-condition ensures the pullback limit h (1) λ given in (3.8) to be well-defined. According to [26, Cor. 7.1], the manifold function h (1) λ under its form(3.11) provides then the leading order approximation of the local invariant manifold function h loc λ , i.e. (3.15) (cid:107) h loc λ ( ξ ) − h (1) λ ( ξ ) (cid:107) α = o ( (cid:107) ξ (cid:107) kα ) . It follows from (3.15) that for all (cid:15) > B ⊂ B suchthat(3.16) (cid:107) h loc λ ( ξ ) − h (1) λ ( ξ ) (cid:107) α ≤ (cid:15) (cid:107) ξ (cid:107) k +1 α , ξ ∈ B . This together with the non-degeneracy condition on h (1) λ given by (3.13) implies that(3.17) (cid:107) h loc λ ( ξ ) (cid:107) α ≥ (cid:107) h (1) λ ( ξ ) (cid:107) α − (cid:107) h loc λ ( ξ ) − h (1) λ ( ξ ) (cid:107) α ≥ C (cid:107) ξ (cid:107) kα − (cid:15) (cid:107) ξ (cid:107) k +1 α . By possibly choosing (cid:15) smaller, and B to be a smaller neighborhood of the origin, we obtain(3.18) (cid:107) h loc λ ( ξ ) (cid:107) α ≥ C (cid:107) ξ (cid:107) kα , ξ ∈ B . Eq. (1.2) corresponds to a deterministic situation dealt with in [26] by setting the noise amplitude to zero.
We show now that the condition (3.4) required in Definition 3.1 holds for solutions of the un-controlled PDE (1.2) emanating from sufficiently small initial data on the local invariant manifold M loc λ .For this purpose, we note that for any fixed t ∗ >
0, by continuous dependence of the solutionsto (1.2) on the initial data, given any sufficiently small initial datum on the local invariant manifold M loc λ , the solution stays on M loc λ over [0 , t ∗ ]. Let B ⊂ B be a neighborhood of the origin in H c sothat each initial datum of the form y := ξ + h loc λ ( ξ ), ξ ∈ B , satisfies the aforementioned property,and the corresponding solution y ( · , y ; 0) satisfies furthermore that(3.19) y c ( t, y ; 0) := P c y ( t, y ; 0) ∈ B , ∀ t ∈ [0 , t ∗ ] , where the latter property can be guaranteed by choosing B properly thanks again to the continuousdependence of the solution on the initial data.By the local invariant property of M loc λ , we have y s ( t, y ; 0) := P s y ( t, y ; 0) = h loc λ ( y c ( t, y ; 0)) , ∀ t ∈ [0 , t ∗ ] . Now, for each such chosen initial datum, thanks to (3.16) and (3.19), we get(3.20) (cid:90) t ∗ (cid:13)(cid:13) y s ( t, y ; 0) − h (1) λ ( y c ( t, y ; 0)) (cid:13)(cid:13) α d t = (cid:90) t ∗ (cid:13)(cid:13) h loc λ ( ξ ( t )) − h (1) λ ( y c ( t, y ; 0)) (cid:13)(cid:13) α d t ≤ (cid:90) t ∗ (cid:15) (cid:107) y c ( t, y ; 0) (cid:107) k +1) α d t ≤ (cid:15) max t ∈ [0 ,t ∗ ] (cid:107) y c ( t, y ; 0) (cid:107) α (cid:90) t ∗ (cid:13)(cid:13) y c ( t, y ; 0) (cid:13)(cid:13) kα d t. Besides, by (3.18) we have(3.21) (cid:90) t ∗ (cid:107) y s ( t, y ; 0) (cid:107) α d t = (cid:90) t (cid:107) h loc λ ( y c ( t, y ; 0)) (cid:107) α d t ≥ C (cid:90) t ∗ (cid:107) y c ( t, y ; 0) (cid:107) kα d t. We obtain then for all y = ξ + h loc λ ( ξ ) with ξ ∈ B that(3.22) (cid:82) t ∗ (cid:13)(cid:13) y s ( t, y ; 0) − h (1) λ ( y c ( t, y ; 0)) (cid:13)(cid:13) α d t (cid:82) t ∗ (cid:107) y s ( t, y ; 0) (cid:107) α d t ≤ (cid:15)C max t ∈ [0 ,t ∗ ] (cid:107) y c ( t, y ; 0) (cid:107) α . The RHS can be made less than one by again the continuity argument and by possibly choosing B to be an even smaller neighborhood.By appealing to the continuous dependences on initial data y and the control u of the solution y (0 , y ; u ) to the controlled PDE (2.4), there exist an open set V in H α containing the set { y = ξ + h loc λ ( ξ ) | ξ ∈ B } , and an open set U of the origin in L (0 , t ∗ ; H ), such that the solution y (0 , y ; u )satisfies (3.22) with the RHS of (3.22) staying less than one as y various in V and the control u variesin U . The proof is complete. (cid:3) We conclude this section by some remarks regarding possible ways of constructing more elaboratedfinite-horizon PMs as well as PMs relying on decompositions of the phase space H involving otherbases than a standard eigenbasis. Remark 3.1. i) More elaborated backward-forward systems than (3.6) can be imagined in orderto design finite-horizon PMs of smaller parameterization defect than offered by h (1) λ ; see [26,Section 8.3] . The idea remains however the same, namely to parameterize the high-modes aspullback limits of some approximation of the time-history of the dynamics of low modes. We INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 11 refer to Section 6 for such a parameterization leading in particular to finite-horizon PMs whose e n -components are polynomials of higher order than for those constituting h (1) λ . As we will seein Section 6.2, such higher-order PMs can give rise to a better design of suboptimal solutionsto a given optimal control problem (including terminal payoff terms) than those accessible fromthe leading order finite-horizon PM h (1) λ ; see also Remark 6.1 below. ii) Note also that the usage of the eigenbasis in the decomposition of the phase space H is notessential for the definition of the finite-horizon PMs as well as for the construction of PMcandidates based on the backward-forward procedure presented in this section or discussed above.In practice, empirical bases such as the POD basis [57] can be adopted to decompose the phasespace into resolved low-mode part and its orthogonal complement (the high-mode part). Bydoing so, the resulting subspaces H c and H s are not invariant subspaces of the linear operator L λ anymore, and explicit formulas such as (3.11) should be revised accordingly; this importantpoint for applications will be addressed elsewhere. Finite-Horizon Parameterizing Manifolds for Suboptimal Control of PDEs
Abstract results.
Given a finite-horizon PM, we present hereafter an abstract formulation ofthe corresponding reduced equations from which we will see how suboptimal solutions to the problem( P ) can be efficiently synthesized once an analytic formulation of such reduced equations is available;see Sections 5, 6 and 7.The approach consists of reducing the PDE (2.4) governing the evolution of the state y ( t ) to anordinary differential equation (ODE) system which is aimed to model the evolution of the low modes P c y ( t ), by substituting their interactions with the high modes P s y ( t ), by means of the parameterizingfunction h associated with a given PM.For simplicity, we assume that the nonlinearity F is bilinear, denoted by B hereafter so that B : H α × H α → H , is thus a continuous bilinear mapping.For the sake of readability, the notations introduced in the previous sections are completed by thosesummarized in Table 1 below. Note also that throughout this article, B ( v ) will be sometimes used inplace of B ( v, v ) to simplify the presentation. Table 1.
Glossary of principal symbols used in Sections 4 – 6symbol terminology y c , y s the low-mode and high-mode projections of a given PDE solution y : y c := P c y and y s := P s y ( y ∗ , u ∗ ) an optimal pair for the original optimal control problem ( P ) z state variable of the PM-based reduced system (4.2a) involved in ( P sub )( z ∗ R , u ∗ R ) an optimal pair for the reduced problem ( P sub ); u ∗ R is the PM-based suboptimal control for ( P ) y ∗ R the suboptimal trajectory of the underlying PDE driven by C u ∗ R z ∗ the trajectory of the PM-based reduced system driven by P c C P c u ∗ l R the trajectory z ∗ R “lifted” onto the given parameterizing manifold: l R := z ∗ R + h ( z ∗ R ) l ∗ the trajectory z ∗ “lifted” onto the given parameterizing manifold: l ∗ := z ∗ + h ( z ∗ )Recall that the subspace H c is spanned by the first m dominant eigenmodes associated with thelinear operator L λ for some positive integer m . We denote as before its topological complements in H and H α by H s and H s α , respectively. Let h : H c → H s α be a finite-horizon PM function associated with (2.4); see Definition 3.1. The corresponding PM-based reduced optimal control problem ( P sub )below, is then built from the following m -dimensional PM-based reduced system:d z d t = L c λ z + P c B ( z + h ( z )) + P c C P c u ( t ) , t ∈ (0 , T ] , (4.1a)supplemented by z (0) = P c y ∈ H c ;(4.1b)the system (4.1a) being aimed to model the dynamics of the low modes P c y ( t ) by z ( t ), and the dynamicsof the high modes P s y ( t ) by h ( z ( t )). To avoid pathological situations, we will assume throughout thisarticle that P c C P c is non-zero.To simplify the presentation, we will assume furthermore that the PM function h has been chosenso that for any z (0) in H c , the problem (4.1) admits a well-defined global ( H c -valued) solution thatis continuous in time. Such PM functions are identified in the case of a Burgers-type equation inSections 5–7; see also Appendix B for more details on the corresponding well-posedness problem forthe associated reduced systems.Note that only the low-mode projection of the controller u , P c u , is kept in the above reducedmodel. In the following we denote by u R := P c u ∈ L (0 , T ; H c ) this m -dimensional controller. Then,the problem (4.1) can be rewritten as:d z d t = L c λ z + P c B ( z + h ( z )) + P c C u R ( t ) , t ∈ (0 , T ] , (4.2a) z (0) = P c y ∈ H c , (4.2b)and the cost functional (2.3) is substituted by(4.3) J R ( z, u R ) := (cid:90) T (cid:2) G (cid:0) z ( t ) + h ( z ( t )) (cid:1) + E ( u R ( t )) (cid:3) d t. The finite-horizon PM-based reduced optimal control problem is then given by:( P sub ) min J R ( z, u R ) s.t. ( z, u R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (4.2) . Throughout this section, we assume that the original problem ( P ) as well as its reduced form ( P sub )admit each an optimal control, denoted respectively by u ∗ and u ∗ R . Theorem 4.1 below provides thenan important a priori estimate for the theory. It gives indeed a measure on how far to the optimalcontrol u ∗ a suboptimal control u ∗ R built on a given PM is. More precisely, under a second-ordersufficient optimality condition on the cost functional J , an a priori estimate of (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) is expressed in terms of key quantities associated with a given PM on one hand, and key quantitiesassociated with the optimal control u ∗ , on the other; see (4.5) below. These quantities involve theparameterization defects associated with u ∗ and u ∗ R ; the energy contained in the high modes of theoptimal and suboptimal PDE trajectories associated with u ∗ and u ∗ R , respectively; and the high-modeenergy remainder (cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) of u ∗ . Our treatment is here inspired by [60] but differs howeverfrom the latter by the use of PMs instead of AIMs; the framework of PMs allowing for a naturalinterpretation of the error estimate (4.5) derived hereafter that as we will see in the applications, willhelp analyze the performances of a PM-based suboptimal controller; see Sections 5–6, and Section 7. Theorem 4.1.
Assume that the optimal control problem ( P ) admits an optimal controller u ∗ , wherethe cost functional J defined in (2.3) satisfies the assumptions of Section 2.Assume furthermore there exists σ > such that the following (local) second order sufficient opti-mality condition holds: (4.4) J ( y ( · ; v ) , v ) − J ( y ∗ , u ∗ ) ≥ σ (cid:107) v − u ∗ (cid:107) L (0 ,T ; H ) , INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 13 where v ∈ L (0 , T ; H ) is chosen from some neighborhood U of u ∗ , and y ( · ; v ) denotes the solution to (2.4) with v in place of the controller u .Assume finally that the corresponding PM-based reduced optimal control problem ( P sub ) admits anoptimal controller u ∗ R , which is furthermore contained in U , and that the underlying PM function h : H c → H s α is locally Lipschitz.Then, the suboptimal controller u ∗ R satisfies the following error estimate (4.5) (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) ≤ C σ (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) , where Q ( T, y ; u ∗ R ) and Q ( T, y ; u ∗ ) denote the parameterization defects of the finite-horizon PM func-tion h associated with the controllers in Eq. (2.4) taken to be respectively u ∗ R and u ∗ ; y ∗ R, s := P s y ∗ R and y ∗ s := P s y ∗ denote the high-mode projections of the suboptimal trajectory y ∗ R and the optimal trajectory y ∗ to Eq. (2.4) driven respectively by C u ∗ R and C u ∗ ; and C denotes a positive constant depending inparticular on T and the local Lipschitz constant of h ; see (4.38) below. Besides the suboptimal trajectory y ∗ R , another trajectory of theoretical interest is the “lifted” tra-jectory by the PM function h , of the (low-dimensional) optimal trajectory z ∗ R := z ( · , P c y ; u ∗ R ) of thereduced optimal control problem ( P sub ). This lifted trajectory is defined as l R ( t ) := z ∗ R ( t ) + h ( z ∗ R ( t )) , for which if z ∗ R constitutes a good approximation of the low-mode projection P c y ∗ and h has a smallparameterization defect , l R provides a good approximation of the optimal trajectory y ∗ , itself.This intuitive idea is made precise in Corollary 4.1 below that provides a general condition underwhich an error estimate regarding the distance (cid:107) y ∗ − l R (cid:107) L (0 ,T ; H ) , between the lifted trajectory l R andthe optimal trajectory y ∗ , can be deduced from the error estimate (4.5) about the distance betweenthe respective controllers; see (4.8) below. This condition concerns the L -response over the interval[0 , T ] of the PM-based reduced system (4.2a) with respect to perturbation of the control term C P c u ∗ . Corollary 4.1.
In addition to the assumptions of Theorem 4.1, assume that the PM-based reducedsystem (4.2a) satisfies the following sublinear response property:There exist κ > and a neighborhood U ⊂ L (0 , T ; H c ) of P c u ∗ , such that the following inequalityholds for all u R ∈ U : (4.6) (cid:107) z ( · , P c y ; u R ) − z ∗ ( · , P c y ; P c u ∗ ) (cid:107) L (0 ,T ; H ) ≤ κ (cid:107) u R − P c u ∗ (cid:107) L (0 ,T ; H ) , where z ( · , P c y ; u R ) denotes the solution to (4.2) emanating from P c y and driven by C u R .Then, the following error estimate between the optimal trajectory z ∗ R := z ( · , P c y ; u ∗ R ) for the reducedoptimal control problem ( P sub ) and the low-mode projection y ∗ c := P c y ∗ of the optimal trajectoryassociated with ( P ) , holds: (4.7) (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) ≤ T (cid:16) (cid:101) C Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:101) C (cid:107) C (cid:107) (cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) + 2 κ C σ (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) , where C is the same positive constant as given by (4.5) in Theorem 4.1 and (cid:101) C , (cid:101) C are given by (4.11) in Lemma 4.1 below. so that h ( z ∗ R ) is a good approximation of the high-mode projection P s y ∗ . Moreover, the following error estimate regarding the distance (cid:107) y ∗ − l R (cid:107) L (0 ,T ; H ) , between the liftedtrajectory l R and the optimal trajectory y ∗ , holds (4.8) (cid:107) y ∗ − l R (cid:107) L (0 ,T ; H ) ≤ (cid:2) C α + (cid:101) C T (cid:0) C C α Lip( h ) | V c ) (cid:1)(cid:3) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + 4 κ C σ [1 + 2( C C α Lip( h ) | V c ) ] (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) (cid:17) + 4 (cid:0) C C α Lip( h ) | V c ) (cid:1)(cid:104) (cid:101) C T (cid:107) C (cid:107) (cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) + κ C σ (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:105) , where C and C α are some generic constants given by (4.18) and (4.34) , respectively; and Lip( h ) | V c is the local Lipschitz constant of the PM function h over some bounded set V c ⊂ H c ; see (4.30) and (4.33) . Finally, the last corollary concerns a refinement of the error estimate (4.5) which consists of identi-fying conditions under which the contribution of the high-mode energy remainder (cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) ofthe optimal control, can be removed in the upper bound of (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) . Corollary 4.2.
Assume that the assumptions given in Theorem 4.1 hold. Assume furthermore thatthe linear operator C leaves stable the subspaces H c and H s , i.e. (4.9) C H c ⊂ H c and C H s ⊂ H s . Then, the error estimate (4.5) reduces to: (4.10) (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) ≤ C σ (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) (cid:17) . Similarly, the corresponding results of Corollary 4.1 under the additional condition (4.9) amountsto dropping the terms involving P s u ∗ on the RHS of the estimates (4.7) and (4.8).4.2. Proofs of Theorem 4.1 and Corollaries 4.1 and 4.2.
For the proofs of the above results,we will make use of the following preparatory lemma.
Lemma 4.1.
Given any control u ∈ L (0 , T ; H ) , we denote by y ( t ) the corresponding solution to (2.4) . Let h : H c → H s α be a PM function assumed to be locally Lipschitz, and z ( t ) be the solution tothe corresponding PM-based reduced system (4.2a) driven by P c C P c u and emanating from P c y (0) .Then, there exists (cid:101) C , (cid:101) C > such that (4.11) (cid:107) y c ( t ) − z ( t ) (cid:107) ≤ (cid:101) C (cid:90) t (cid:107) y s ( s ) − h ( y c ( s )) (cid:107) α d s + (cid:101) C (cid:107) C (cid:107) (cid:90) t (cid:107) P s u ( s ) (cid:107) d s, t ∈ [0 , T ] , where y c := P c y , y s := P s y ; and (cid:101) C , (cid:101) C depend in particular on T and the local Lipschitz constant of h ; see (4.23) below.Proof. Let us introduce w ( t ) := y c ( t ) − z ( t ). By projecting (2.4) against the subspace H c , we obtaind y c d t = L c λ y c + P c B ( y c + y s ) + P c C u ( t ) , y c (0) = P c y ∈ H c . This together with (4.1) implies that w satisfies the following problem:(4.12) d w d t = L c λ w + P c (cid:0) B ( y c + y s ) − B ( z + h ( z )) (cid:1) + P c C P s u, w (0) = 0 , recalling that u − P c u = P s u. INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 15
By taking the H -inner product on both sides of (4.12) with w , we obtain:(4.13) 12 d (cid:107) w (cid:107) d t = (cid:104) L c λ w, w (cid:105) + (cid:104) P c (cid:0) B ( y c + y s ) − B ( z + h ( z )) (cid:1) , w (cid:105) + (cid:104) P c C P s u, w (cid:105) . Since B : H α × H α → H is a continuous bilinear mapping, there exists C B > v and v in H α , it holds that(4.14) (cid:107) B ( v ) − B ( v ) (cid:107) = (cid:107) B ( v , v ) − B ( v , v ) (cid:107)≤ (cid:107) B ( v , v ) − B ( v , v ) (cid:107) + (cid:107) B ( v , v ) − B ( v , v ) (cid:107)≤ C B (cid:107) v (cid:107) α (cid:107) v − v (cid:107) α + C B (cid:107) v − v (cid:107) α (cid:107) v (cid:107) α ≤ C B ( (cid:107) v (cid:107) α + (cid:107) v (cid:107) α ) (cid:107) v − v (cid:107) α . Thanks to the above bilinear estimate, we get thus(4.15) (cid:104) P c (cid:0) B ( y c + y s ) − B ( z + h ( z )) (cid:1) , w (cid:105) ≤ C B (cid:0) (cid:107) y c + y s (cid:107) α + (cid:107) z + h ( z ) (cid:107) α (cid:1) (cid:107) y c + y s − z − h ( z ) (cid:107) α (cid:107) w (cid:107) . On the other hand, the assumptions made at the end of Section 2 and in this section regarding thewell-posedness problem associated respectively with Eq. (2.4) and the reduced system (4.2a), ensurethe existence of a bounded set V in H α , such that y ( t ) and z ( t ) + h ( z ( t )) stay in V for all t ∈ [0 , T ].As a consequence, there exists a constant C ( V ) >
0, such that(4.16) C B (cid:0) (cid:107) y c ( t ) + y s ( t ) (cid:107) α + (cid:107) z ( t ) + h ( z ( t )) (cid:107) α (cid:1) ≤ C ( V ) , t ∈ [0 , T ] . Note also that by using the local Lipschitz property of h , we get(4.17) (cid:107) y c ( t ) + y s ( t ) − z ( t ) − h ( z ( t )) (cid:107) α ≤ (cid:107) y c ( t ) − z ( t ) (cid:107) α + (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α + (cid:107) h ( y c ( t )) − h ( z ( t )) (cid:107) α ≤ (1 + Lip( h ) | V c ) (cid:107) y c ( t ) − z ( t ) (cid:107) α + (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α ≤ C (1 + Lip( h ) | V c ) (cid:107) w ( t ) (cid:107) + (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α , t ∈ [0 , T ] , where V c = P c V , and C in the the last inequality denotes the generic positive constant for which(4.18) (cid:107) v (cid:107) α ≤ C (cid:107) v (cid:107) , ∀ v ∈ H c , due to the finite-dimensional nature of H c .By using now the estimates (4.16) and (4.17) in (4.15), we get(4.19) (cid:104) P c (cid:0) B ( y c ( t ) + y s ( t )) − B ( z ( t ) + h ( z ( t ))) (cid:1) , w ( t ) (cid:105)≤ C C ( V )(1 + Lip( h ) | V c ) (cid:107) w ( t ) (cid:107) + C ( V ) (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α (cid:107) w ( t ) (cid:107)≤ C C ( V )(1 + Lip( h ) | V c ) (cid:107) w ( t ) (cid:107) + [ C ( V )] (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α + 12 (cid:107) w ( t ) (cid:107) , where we have applied the standard Young’s inequality ab < a + b to derive the last inequality.Since L λ is assumed to be self-adjoint with dominant eigenvalue β ( λ ), we obtain(4.20) (cid:104) L c λ w ( t ) , w ( t ) (cid:105) = m (cid:88) i =1 β i ( λ ) | w i ( t ) | ≤ β ( λ ) (cid:107) w ( t ) (cid:107) . Note also that(4.21) (cid:104) P c C P s u, w (cid:105) ≤ (cid:107) C (cid:107)(cid:107) P s u ( t ) (cid:107)(cid:107) w ( t ) (cid:107) ≤ (cid:107) C (cid:107) (cid:107) P s u ( t ) (cid:107) + 12 (cid:107) w ( t ) (cid:107) . Using (4.19)–(4.21) in (4.13), we obtain(4.22) 12 d (cid:107) w ( t ) (cid:107) d t ≤ (cid:16) β ( λ ) + C C ( V )(1 + Lip( h ) | V c ) (cid:17) (cid:107) w ( t ) (cid:107) + [ C ( V )] (cid:107) y s ( t ) − h ( y c ( t )) (cid:107) α + 12 (cid:107) C (cid:107) (cid:107) P s u ( t ) (cid:107) . Now, by a standard application of the Gronwall’s inequality, we obtain for all t ∈ [0 , T ],(4.23) (cid:107) w ( t ) (cid:107) = (cid:107) y c ( t ) − z ( t ) (cid:107) ≤ (cid:90) t e β ( λ )+ C C ( V )(1+Lip( h ) | V c )]( t − s ) (cid:18) [ C ( V )] (cid:107) y s ( s ) − h ( y c ( s )) (cid:107) α + (cid:107) C (cid:107) (cid:107) P s u ( s ) (cid:107) (cid:19) d s ≤ e β ( λ )+ C C ( V )(1+Lip( h ) | V c )] T (cid:18) [ C ( V )] (cid:90) t (cid:107) y s ( s ) − h ( y c ( s )) (cid:107) α d s + (cid:107) C (cid:107) (cid:90) t (cid:107) P s u ( s ) (cid:107) d s (cid:19) , taking into account that w (0) = y c (0) − z (0) = 0, by assumption. The estimate (4.11) is thus proved. (cid:3) We present now the proofs of Theorem 4.1 and Corollaries 4.1 and 4.2.
Proof of Theorem 4.1.
Let us denote by y ∗ in C ([0 , T ]; H ) ∩ C ([0 , T ]; H α ) the optimal trajectory tothe optimal control problem ( P ), and by y ∗ R (in the same functional space) the trajectory of Eq. (2.4)corresponding to the control u taken to be the optimal (low-dimensional) controller u ∗ R of the reducedoptimal control problem ( P sub ).Let us also introduce the lifted trajectories(4.24) l R = z ∗ R + h ( z ∗ R ) , and l ∗ = z ∗ + h ( z ∗ ) , where z ∗ R and z ∗ are the solutions to (4.2) driven respectively by P c C u ∗ R ( t ) and P c C P c u ∗ ( t ), t ∈ [0 , T ].Thanks to the second order optimality condition (4.4), the proof boils down to the derivation of asuitable upper bound for ∆ := J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ), which is organized as follows.In Step 1, we reduce the control of ∆ to the control of J ( y ∗ R , u ∗ R ) − J ( l R , u ∗ R )+ J ( l ∗ , u ∗ ) − J ( y ∗ , u ∗ ) byusing the optimality property of the pair ( z ∗ R , u ∗ R ) for the reduced problem ( P sub ). The main interestin doing so relies on the fact that only (cid:107) y ∗ R − l R (cid:107) and (cid:107) y ∗ − l ∗ (cid:107) are then determining in the control of∆; see Step 2. This leads in turn to an upper bound of ∆ expressed in terms of key quantities for thedesign of suboptimal controller in our PM-based theory.In that respect, the upper bound of ∆ derived in (4.36) involves (cid:107) y ∗ R, s − h ( y ∗ R, c ) (cid:107) L (0 ,T ; H ) and (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H ) , the energy (over the interval [0 , T ]) of the high modes unexplained by the PM functionwhen applied respectively to y ∗ R, c and y ∗ c ; and involves (cid:107) y ∗ R, c − z ∗ R (cid:107) L (0 ,T ; H ) and (cid:107) y ∗ c − z ∗ (cid:107) L (0 ,T ; H ) , theerrors associated with the modeling of the y ∗ R, c - and y ∗ c -dynamics by the reduced system (4.2a).Thanks to Lemma 4.1, we can bound the two latter quantities by the former ones together witha term involving the energy contained in the high modes of u ∗ . This is the purpose of Step 3. Thedesired result follows then by rewriting the relevant unexplained energies by using the parameterizationdefects associated with the PM function h and the controllers u ∗ and u ∗ R . Step 1.
Since ( y ∗ , u ∗ ) is an optimal pair for ( P ), we get(4.25) 0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ )= J ( y ∗ R , u ∗ R ) − J ( l R , u ∗ R ) + J ( l R , u ∗ R ) − J ( l ∗ , u ∗ ) + J ( l ∗ , u ∗ ) − J ( y ∗ , u ∗ ) . Since ( z ∗ R , u ∗ R ) is an optimal pair for the reduced problem ( P sub ), we obtain(4.26) J R ( z ∗ R , u ∗ R ) − J R ( z ∗ , P c u ∗ ) ≤ . INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 17
Note also that J ( l R , u ∗ R ) = J R ( z ∗ R , u ∗ R ) , and that according to (C2) J ( l ∗ , u ∗ ) ≥ J R ( z ∗ , P c u ∗ ) , since (cid:107) P c u ∗ (cid:107) ≤ (cid:107) u ∗ (cid:107) .Consequently,(4.27) J ( l R , u ∗ R ) − J ( l ∗ , u ∗ ) ≤ . We obtain then from (4.25) that(4.28) 0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) ≤ J ( y ∗ R , u ∗ R ) − J ( l R , u ∗ R ) + J ( l ∗ , u ∗ ) − J ( y ∗ , u ∗ ) . Step 2.
Let V ⊂ H α be a bounded set such that(4.29) y ∗ R ( t ) , l R ( t ) , y ∗ ( t ) , l ∗ ( t ) ∈ V ∀ t ∈ [0 , T ] . Let also(4.30) V c = P c V. It is clear that P c y ∗ R ( t ), P c y ∗ ( t ), z ∗ R ( t ) and z ∗ ( t ) are contained in V c for all t ∈ [0 , T ].Recalling (C1), we denote by Lip( G ) | V the Lipschitz constant of G : H → R + restricted to thebounded set V . In (4.28), by applying Lipschitz estimates to the G -part of the cost functional J , weobtain(4.31) 0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) ≤ Lip( G ) | V ( (cid:107) y ∗ R − l R (cid:107) L (0 ,T ; H ) + (cid:107) l ∗ − y ∗ (cid:107) L (0 ,T ; H ) ) ≤ √ T Lip( G ) | V ( (cid:107) y ∗ R − l R (cid:107) L (0 ,T ; H ) + (cid:107) l ∗ − y ∗ (cid:107) L (0 ,T ; H ) ) , where the last inequality follows from H¨older’s inequality.Recall that l R ( t ) = z ∗ R ( t ) + h ( z ∗ R ( t )). Let us also rewrite y ∗ R ( t ) as y ∗ R, c ( t ) + y ∗ R, s ( t ) with y ∗ R, c ( t ) = P c y ∗ R ( t ) and y ∗ R, s ( t ) = P s y ∗ R ( t ). We obtain then(4.32) (cid:107) y ∗ R ( t ) − l R ( t ) (cid:107) ≤ (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) + (cid:107) y ∗ R, s ( t ) − h ( z ∗ R ( t )) (cid:107)≤ (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) + (cid:107) y ∗ R, s ( t ) − h ( y ∗ R, c ( t )) (cid:107) + (cid:107) h ( y ∗ R, c ( t )) − h ( z ∗ R ( t )) (cid:107) . Let us denote by Lip( h ) | V c the Lipschitz constant of h : H c → H s α restricted to the bounded set V c .We get(4.33) (cid:107) h ( y ∗ R, c ( t )) − h ( z ∗ R ( t )) (cid:107) α ≤ Lip( h ) | V c (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) α ≤ C Lip( h ) | V c (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) , t ∈ [0 , T ] , where we have used the equivalence between the norms on H c ; see (4.18).Since H α is continuously embedded into H , there exists a generic positive constant C α , such that(4.34) (cid:107) v (cid:107) ≤ C α (cid:107) v (cid:107) α , ∀ v ∈ H α . We obtain then(4.35) (cid:107) h ( y ∗ R, c ( t )) − h ( z ∗ R ( t )) (cid:107) ≤ C C α Lip( h ) | V c (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) . This together with (4.32) leads to (cid:107) y ∗ R ( t ) − l R ( t ) (cid:107) ≤ (1 + C C α Lip( h ) | V c ) (cid:107) y ∗ R, c ( t ) − z ∗ R ( t ) (cid:107) + (cid:107) y ∗ R, s ( t ) − h ( y ∗ R, c ( t )) (cid:107) , t ∈ [0 , T ] . Similarly, (cid:107) l ∗ ( t ) − y ∗ ( t ) (cid:107) ≤ (1 + C C α Lip( h ) | V c ) (cid:107) y ∗ c ( t ) − z ∗ ( t ) (cid:107) + (cid:107) y ∗ s ( t ) − h ( y ∗ c ( t )) (cid:107) , t ∈ [0 , T ] . Reporting the above two estimates into (4.31), we obtain(4.36) 0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) ≤ √ T Lip( G ) | V (cid:16) (cid:107) y ∗ R, s − h ( y ∗ R, c ) (cid:107) L (0 ,T ; H ) + (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H ) + (1 + C C α Lip( h ) | V c ) (cid:0) (cid:107) y ∗ R, c − z ∗ R (cid:107) L (0 ,T ; H ) + (cid:107) y ∗ c − z ∗ (cid:107) L (0 ,T ; H ) (cid:1)(cid:17) . Step 3.
By using Lemma 4.1 (see (4.23) above), we obtain: (cid:107) y ∗ R, c − z ∗ R (cid:107) L (0 ,T ; H ) ≤ √ T C ( V ) e [1+ β ( λ )+ C C ( V )(1+Lip( h ) | V c )] T (cid:107) y ∗ R, s − h ( y ∗ R, c ) (cid:107) L (0 ,T ; H α ) , where we have used P s u ∗ R = 0 since u ∗ R lives in L (0 , T ; H c ); and the same lemma leads to (cid:107) y ∗ c − z ∗ (cid:107) L (0 ,T ; H ) ≤ √ T e [1+ β ( λ )+ C C ( V )(1+Lip( h ) | V c )] T (cid:16) C ( V ) (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) . Now, by reporting these estimates in (4.36) and using again the property of continuous embedding(4.34), we obtain:(4.37)0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) ≤ C ( V, Lip( h ) | V c , T ) (cid:16) (cid:107) y ∗ R, s − h ( y ∗ R, c ) (cid:107) L (0 ,T ; H α ) + (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) , where(4.38) C ( V, Lip( h ) | V c , T ) := 2 C α √ T Lip( G ) | V + 2max { C ( V ) , } T Lip( G ) | V (1 + C C α Lip( h ) | V c ) e [1+ β ( λ )+ C C ( V )(1+Lip( h ) | V c )] T . In terms of parameterization defects defined in (3.5), the above estimate (4.37) can be rewritten as:(4.39) 0 ≤ J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) ≤ C ( V, Lip( h ) | V c , T ) (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) , where Q ( T, y ; u ∗ R ) and Q ( T, y ; u ∗ ) are the parameterization defects of the finite-horizon PM function h when the control in (2.4) is taken to be u ∗ R and u ∗ , respectively.The proof is complete. Proof of Corollary 4.1.
The estimate given by (4.7) can be derived directly from Theorem 4.1 andLemma 4.1 by noting that (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) ≤ (cid:107) y ∗ c − z ∗ (cid:107) L (0 ,T ; H ) + 2 (cid:107) z ∗ − z ∗ R (cid:107) L (0 ,T ; H ) . INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 19
Indeed, the first term on the RHS above can be controlled as follows by Lemma 4.1: (cid:107) y ∗ c − z ∗ (cid:107) L (0 ,T ; H ) ≤ (cid:90) T (cid:16) (cid:101) C (cid:90) t (cid:107) y ∗ s ( s ) − h ( y ∗ c ( s )) (cid:107) α d s + (cid:101) C (cid:107) C (cid:107) (cid:90) t (cid:107) P s u ( s ) (cid:107) d s (cid:17) d t ≤ T (cid:0) (cid:101) C (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H α ) + (cid:101) C (cid:107) C (cid:107) (cid:107) P s u (cid:107) L (0 ,T ; H ) (cid:1) ≤ T (cid:0) (cid:101) C Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:101) C (cid:107) C (cid:107) (cid:107) P s u (cid:107) L (0 ,T ; H ) (cid:1) . For the term (cid:107) z ∗ − z ∗ R (cid:107) L (0 ,T ; H ) , according to the condition (4.6) on the sublinear response andTheorem 4.1, we obtain (cid:107) z ∗ − z ∗ R (cid:107) L (0 ,T ; H ) ≤ κ (cid:107) u ∗ R − P c u ∗ (cid:107) L (0 ,T ; H ) ≤ κ (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) ≤ C κ σ (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) + (cid:107) C (cid:107)(cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) (cid:17) . We obtain then (4.7) by combining the above two estimates.The estimate (4.8) follows from (4.7) by noting that (cid:107) y ∗ − ( z ∗ R + h ( z ∗ R )) (cid:107) L (0 ,T ; H ) ≤ (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) + 2 (cid:107) y ∗ s − h ( z ∗ R ) (cid:107) L (0 ,T ; H ) ≤ (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) + 4 (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H ) + 4 (cid:107) h ( y ∗ c ) − h ( z ∗ R ) (cid:107) L (0 ,T ; H ) ≤ (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) + 4 C α (cid:107) y ∗ s − h ( y ∗ c ) (cid:107) L (0 ,T ; H α ) + 4 (cid:107) h ( y ∗ c ) − h ( z ∗ R ) (cid:107) L (0 ,T ; H ) ;and that (cid:107) h ( y ∗ c ) − h ( z ∗ R ) (cid:107) L (0 ,T ; H ) ≤ C C α Lip( h ) | V c (cid:107) y ∗ c − z ∗ R (cid:107) L (0 ,T ; H ) ;see (4.35) for more details about the derivation of this last inequality (with y ∗ R, c therein replaced by y ∗ c here). Proof of Corollary 4.2.
Note that if C leaves stable the two subspaces H c and H s , then in Lemma 4.1,the equation (4.12) satisfied by the difference w ( t ) := y c ( t ) − z ( t ) is simplified into the following:d w d t = L c λ w + P c (cid:0) B ( y c + y s ) − B ( z + h ( z )) (cid:1) , w (0) = 0 , where the term P c C P s u vanishes here. Consequently, the terms involving P s u in the subsequentestimates are dropped out, leading then to the the estimate given in (4.10).5. We apply in this section and the next, the PM-based reduction approach introduced above for thedesign of suboptimal solutions to an optimal control problem of a Burgers-type equation, in the caseof globally distributed control laws. The more challenging case of locally distributed control laws, isaddressed in Section 7.5.1.
Cost functional of terminal payoff type for a Burgers-type equation, and existenceof optimal solution.
The model considered here takes the following form, which is posed on theinterval (0 , l ) driven by a globally distributed control term C u ( x, t ):(5.1) d y d t = νy xx + λy − γyy x + C u ( x, t ) , ( x, t ) ∈ (0 , l ) × (0 , T ] , where ν, λ and γ are positive parameters, the final time T > C are specified in Section 5.2 below.The equation is supplemented with the Dirichlet boundary condition(5.2) y (0 , t ; u ) = y ( l, t ; u ) = 0 , t ∈ [0 , T ];and appropriate initial condition(5.3) y ( x,
0) = y ( x ) , x ∈ (0 , l ) . The classical Burgers equation (with λ = 0 in (5.1)) has widely served as a theoretical laboratoryto test various methodologies devoted to the design of optimal/suboptimal controllers of nonlineardistributed-parameter systems; see e.g. [7, 29, 72, 75, 101] and references therein. The inclusion of theterm λy here allows for the presence of linearly unstable modes, which lead in turn to the existenceof non-trivial (and nonlinearly) stable steady states for the uncontrolled version of (5.1) providedthat λ is large enough; see [58]. The latter property will be used in the choices of initial data andtargets for the associated optimal control problems analyzed hereafter. From a physical perspective,we mention that (5.1) arises in the modeling of flame front propagation [11]. This model will serve ushere to demonstrate the effectiveness of the PM approach introduced above in the design of suboptimalsolutions to optimal control problems.In that respect, we consider the following cost functional associated with (5.1)–(5.3),(5.4) J ( y, u ) = (cid:90) T (cid:0) (cid:107) y ( · , t ; y , u ) (cid:107) + µ (cid:107) u ( · , t ) (cid:107) (cid:1) d t + µ (cid:107) y ( · , T ; y , u ) − Y (cid:107) , constituted by a running cost along the controlled trajectory and a terminal payoff term defining apenalty on the final state; here µ and µ are some positive constants, Y ∈ L (0 , l ) is some giventarget profile, and (cid:107) · (cid:107) denotes the L (0 , l )-norm.Compared to the cost functional (2.3) associated with the optimal control problem ( P ) given inSection 2, we have added here a terminal payoff term µ (cid:107) y ( · , T ; y , u ) − Y (cid:107) to the running cost (cid:82) T (cid:0) (cid:107) y ( · , t ; y , u ) (cid:107) + µ (cid:107) u ( · , t ) (cid:107) (cid:1) d t . In Section 4, the optimal control problem ( P ) involving onlythe latter type of running cost, has served to identify the determining quantities controlling thedistance to an optimal control of a suboptimal solution to ( P ) built from a PM-reduced system; seeTheorem 4.1 and Corollary 4.2. For a functional cost of type (5.4), error estimates similar to (4.5)and (4.10) can be derived by controlling appropriately the contribution of the terminal payoff term to J ( y ∗ R , u ∗ R ) − J ( y ∗ , u ∗ ) in the estimate (4.31). For instance, the error estimate (4.10) becomes(5.5) (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) ≤ C σ (cid:16)(cid:113) Q ( T, y ; u ∗ R ) (cid:107) y ∗ R, s (cid:107) L (0 ,T ; H α ) + (cid:112) Q ( T, y ; u ∗ ) (cid:107) y ∗ s (cid:107) L (0 ,T ; H α ) (cid:17) + | C T ( y ∗ R,T , Y ) − C T ( y ∗ T , Y ) | σ , where C T ( v, Y ) := µ (cid:107) v − Y (cid:107) , y ∗ R,T = y ∗ R ( T ) and y ∗ T = y ∗ ( T ). We dealt with the simpler situationof a single running cost type functional in Section 4 in order not to overburden the presentation.Furthermore, as we will see in this section and the forthcoming ones, the error estimates derivedin Section 4 are sufficient enough to provide useful (and computable) insights to help analyze theperformances of a PM-based suboptimal controller. The interest of cost functionals such as (5.4) is that they arise naturally when the goal is to drivethe state y ( · ; u ) of (5.1) as close as possible to a target profile Y at the final time T , while keepingthe cost of the control, expressed by µ (cid:82) T (cid:107) u ( t ) (cid:107) d t , as low as possible. Here, the terminal payoff Note that in practice, although the second order optimality condition (4.4) is difficult to check, the error estimatessuch as (4.10) will still demonstrate their relevance for the performance analysis; see Section 5.5.
INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 21 term gives a measurement of the “proximity” to the target Y at the final-time SPDE profile. If onecan make µ = + ∞ , it means the problem is exactly controllable, if not the system is approximatelycontrollable [80].We turn now to the precise description of the optimal control problem considered in this sectionand the next. Adopting the notations of Section 2, the functional spaces are(5.6) H := L (0 , l ) , H := H (0 , l ) ∩ H (0 , l ) , H / := H (0 , l ) , the linear operator L λ : H → H is given by(5.7) L λ y := ν∂ xx y + λy, and the nonlinearity F is expressed by the bilinear term(5.8) B : H / × H / → H ( y, y ) (cid:55)→ B ( y, y ) := − γy∂ x y, with slight abuse of notations, understanding (5.7) and y∂ x y in (5.8) within the appropriate weaksense.The optimal control problem for which we will propose suboptimal solutions takes here the followingform:(5.9) min J ( y, u ) with J defined in (5.4) s.t.( y, u ) ∈ L (0 , T ; H ) × L (0 , T ; H ) solves the problem (5.1)–(5.3) . It can be checked by standard energy estimates that for any given controller u ∈ L (0 , T ; H ),initial datum y ∈ H and any finite T >
0, there exists a unique weak solution y ( · ; y , u ) for theproblem (5.1)–(5.3) such that y ( · ; y , u ) ∈ L (0 , T ; H / ) and y (cid:48) ( · ; y , u ) ∈ L (0 , T ; ( H / ) − ), where( H / ) − = H − (0 , l ) is the dual of H / = H (0 , l ); see e.g. [101] for the standard Burgers equationsubject to affine control.Note also that y ( · ; y , u ) ∈ C ([0 , T ]; H ) thanks to the continuous embedding W := { y | y ∈ L (0 , T ; H / ) and d y d t ∈ L (0 , T ; ( H / ) − ) } ⊂ C ([0 , T ]; H );see e.g. [40, Sect. 5.9 Thm. 3] for more details. This last property implies thus that the cost functional J given by (5.4) is well defined for any pair ( y, u ) ∈ W × L (0 , T ; H ) that satisfies the problem (5.1)–(5.3) in the weak sense (5.10).Within this functional setting, the existence of an optimal pair to (5.9) in W × L (0 , T ; H ), can beachieved by application of the direct method of calculus of variations [38]. The closest application ofsuch a method that serves our purpose can be found in the proof of [101, Prop. 4] for the standardBurgers equation where the author considered cost functional of tracking type; the arguments beingeasily adaptable to cost functional of the form (5.4). We provide below a sketch of such arguments.First note that given a minimizing sequence { ( y n , u n ) } ∈ ( W × L (0 , T ; H )) N , since the cost func-tional J defined by (5.4) is positive (and thus bounded from below) and satisfies J ( y, u ) → ∞ if (cid:107) y (cid:107) L (0 ,T ; H ) → ∞ or (cid:107) u (cid:107) L (0 ,T ; H ) → ∞ , the minimizing sequence lives in a bounded subset of the functional space W × L (0 , T ; H ). Wecan then extract a subsequence, say { ( y n j , u n j ) } , which converges weakly to some element ( y ∗ , u ∗ ) ∈W × L (0 , T ; H ); see e.g. [21, Thm. 3.18]. By using the fact that W is compactly embedded in in the sense recalled in (5.10) below. L (0 , T ; L ∞ (0 , l )) [96], standard energy estimates on the nonlinear term allow to show that actually( y ∗ , u ∗ ) satisfies (5.1)–(5.3) in the following weak sense, i.e. for any ϕ ∈ L (0 , T ; H / ) and any T > (cid:90) T (cid:16)(cid:10) d y ∗ d t , ϕ (cid:11) H − / ; H / − (cid:104) B ( y ∗ , y ∗ ) , ϕ (cid:11) H + ν (cid:10) y ∗ , ϕ (cid:11) H / − (cid:10) λy ∗ + C u ∗ , ϕ (cid:11) H (cid:17) d t = 0 , with y ∗ (0) = y . Invoking now the lower semi-continuity property of the norm in Banach space (see e.g. [21, Prop. 3.5(iii)]) with respect to the convergence in the weak topology, from the functional form of J given in (5.4)we conclude that ( y ∗ , u ∗ ) is an optimal pair for the optimal control problem (5.9). Having ensured theexistence of an optimal pair to (5.9), we turn now to the design of low-dimensional suboptimal pairsbased on the (leading-order) parameterizing manifold introduced in Section 3.2.5.2. Analytic derivation of the h (1) λ -based 2D reduced system for the design of suboptimalcontrollers. We present in this section the analytic derivation of the h (1) λ -based reduced system onwhich we will rely to design suboptimal solutions to problem (5.9). In this respect, we consider theparticular case where the subspace H c of the low-modes is chosen to be the subspace spanned by thefirst two eigenmodes of the linear operator L λ defined in (5.7). Recall that the eigenvalues of L λ aregiven by(5.11) β n ( λ ) := λ − νn π l , n ∈ N , and the corresponding eigenvectors are(5.12) e n ( x ) := (cid:114) l sin (cid:16) nπxl (cid:17) , x ∈ (0 , l ) . Throughout the numerical applications presented hereafter, we will choose λ to be bigger than thecritical value λ c := νπ l such that L λ admits one and only one unstable eigenmode. The subspace H c given by(5.13) H c := span { e , e } , is thus spanned by one unstable and one stable mode.For the regimes considered hereafter, it can be checked that the (NR)-condition is satisfied, leadingin particular to a well-defined h (1) λ . We take as a finite-horizon PM candidate, the manifold function h (1) λ provided by the explicit formula (3.11) that we apply to the PDE (5.1). Recall that according toLemma 3.1, the manifold function h (1) λ provides a natural theoretical PM candidate. Numerical resultsreported in Fig. 2 will support that this choice is in fact relevant for the regimes analyzed hereafterfor the PDE (5.1) leading in particular to manifold functions with parameterization defect less thanunity as required in Definition 3.1.To analyze the performances achieved by the h (1) λ -based reduced system in the design of suboptimalsolutions to (5.9), we place ourselves within the conditions of Corollary 4.2. In particular, we assumethat the continuous linear operator C : H → H leaves stable the subspaces H c and H s :(5.14) C H c ⊂ H c , C H s ⊂ H s . Recall that under such assumptions, the high-mode energy remainder (cid:107) P s u ∗ (cid:107) L (0 ,T ; H ) of the (un-known) optimal controller u ∗ , does not contribute to the estimate of (cid:107) u ∗ R − u ∗ (cid:107) L (0 ,T ; H ) ; leaving theparameterization defect as a key determining parameter in the control of the latter. In particular wewill see in Section 6 that other manifold functions with a smaller parameterization defect than the INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 23 one associated with h (1) λ , lead to a design of better suboptimal solutions to (5.9) than those based on h (1) λ .To be more specific, the operator C when restricted to H c takes the following form(5.15) C e = a e + a e , C e = a e + a e , where the coefficient matrix(5.16) M := (cid:18) a a a a (cid:19) is chosen to be non-trivial to avoid pathological situations.Corresponding to the cost functional (5.4), the cost associated with the h (1) λ -based reduced systemtakes the following form:(5.17) J R ( z, u R ) = (cid:90) T (cid:0) (cid:107) z ( t ) + h (1) λ ( z ( t ; P c y , u R )) (cid:107) + µ (cid:107) u R ( t ) (cid:107) (cid:1) d t + µ (cid:107) z ( T ; P c y , u R ) − P c Y (cid:107) , where Y ∈ H is some prescribed target.Recall that following (4.2), the h (1) λ -based reduced system intended to model the dynamics of thelow modes P c y , takes the following abstract form:(5.18) d z d t = L c λ z + P c B (cid:16) z + h (1) λ ( z ) , z + h (1) λ ( z ) (cid:17) + P c C u R ( t ) , t ∈ (0 , T ] ,z (0) = P c y ∈ H c , where y is the initial datum of the original PDE (5.1), and u R ∈ L (0 , T ; H c ) is a given control of thereduced system.We are thus left with the following reduced optimal control problem associated with (5.9):(5.19) min J R ( z, u R ) s.t. ( z, u R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (5.18) . We turn now to the description of the analytic form of (5.19).
Analytic form of (5.19) . We proceed with the explicit expression of h (1) λ provided by (3.11) thatwe apply to the Burgers-type equation (5.1). In that respect the nonlinear interactions between the H c -modes as projected onto the H s -modes given by B ni i := (cid:104) B ( e i , e i ) , e n (cid:105) , constitute key quantities to determine. In the case of the Burgers-type equation (5.1), they take thefollowing form:(5.20) B ni i = − γ (cid:104) e i ( e i ) x , e n (cid:105) = − αi , n = i + i , − αi sgn( i − i ) , n = | i − i | , , otherwise , where(5.21) α := γπ √ l / . In particular, we have (cid:104) e i ( e i ) x , e n (cid:105) = 0 , for any n ≥ i , i ∈ { , } . By using the above nonlinear interaction relations in (3.11), we obtain thus the following expressionof h (1) λ :(5.22) h (1) λ ( z e + z e ) = α ( λ ) z z e + α ( λ )( z ) e , ( z , z ) ∈ R , where(5.23) α ( λ ) := − γπ √ l / ( β ( λ ) + β ( λ ) − β ( λ )) ,α ( λ ) := − √ γπl / (2 β ( λ ) − β ( λ )) , with the β i ( λ ) given such as given by (5.11). Note that this set of eigenvalues obey the (NR)-conditionfor any λ -value of interest here ( i.e. λ > λ c ). Note also that α ( λ ) < α ( λ ) < λ .Now, by using (5.22), we can rewrite (5.17) into the following explicit form:(5.24) J R ( z, u R ) = (cid:90) T [ G ( z ( t )) + E ( u R ( t ))]d t + C T ( z ( T ) , P c Y ) , where(5.25) G ( z ) = 12 (cid:107) z + h (1) λ ( z ) (cid:107) = 12 [( z ) + ( z ) + ( α ( λ ) z z ) + ( α ( λ ) z ) ] , E ( u R ) = µ (cid:107) u R (cid:107) = µ u R, ) + ( u R, ) ] , and(5.26) C T ( z ( T ) , P c Y ) := µ m (cid:88) i =1 | z i ( T ) − Y i | , with z i := (cid:104) z, e i (cid:105) , u R,i := (cid:104) u R , e i (cid:105) , and Y i := (cid:104) Y, e i (cid:105) , i = 1 , h (1) λ given in (5.22) into (5.18), we obtain finally afterprojection onto H c , the following analytic formulation of (5.18):(5.27) d z d t = β ( λ ) z + α (cid:16) z z + α ( λ ) z z + α ( λ ) α ( λ ) z z (cid:17) + a u R, ( t ) + a u R, ( t ) , d z d t = β ( λ ) z + α (cid:16) − z + 2 α ( λ ) z z + 2 α ( λ ) z (cid:17) + a u R, ( t ) + a u R, ( t ) , where α ( λ ) and α ( λ ) are defined in (5.23), and α = γπ √ l / . Note that for any given initial datum ( z , , z , ) and any T >
0, the h (1) λ -based reduced system(5.27) admits a unique solution in C ([0 , T ]; R ); this is carried out through some simple but specificenergy estimates that are provided in Appendix B for the sake of clarity.5.3. Synthesis of suboptimal controllers by a Pontryagin-maximum-principle approach.
The analytic form (5.27) of the h (1) λ -based reduced system (5.18) allows for the use of standard tech-niques from finite-dimensional optimal control theory to solve the related reduced optimal controlproblem (5.19) [18, 23, 66, 67, 94]. We follow below an indirect approach relying on the Pontryaginmaximum principle (PMP); see e.g. [18, 20, 66, 67, 87, 94]. Usually, the use of the Pontryagin maxi-mum principle allows to identify a set of necessary conditions to be satisfied by an optimal solution.However, as we will see, due to the particular form of the cost functionals considered here and thenature of the reduced control system (5.27), these conditions will turn out to be sufficient to ensurethe existence of a (unique) optimal control for the reduced problem. Relying on a PMP approach INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 25 allows also for theoretical insights that can be gained on the reduced optimal control problem (5.19)from the (costate-based) explicit formula of the (reduced) optimal controller reachable by such anapproach; see (5.32) and Lemmas 5.1 and 5.2 below.In that perspective, let us denote the h (1) λ -based reduced vector field involved in (5.27), by f ( z, u R ) := ( f ( z, u R ) , f ( z, u R )) tr . We introduce now the following Hamiltonian associated with the reduced optimal control problem(5.19):(5.28) H ( z, p, u R ) := G ( z ) + E ( u R ) + p f ( z, u R ) + p f ( z, u R ) , where p := ( p , p ) tr is the costate (or adjoint state) associated with the state z = ( z , z ) tr .It follows from the Pontryagin maximum principle that for a given pair( z ∗ R , u ∗ R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c )to be optimal for the reduced problem (5.19), it must satisfy the following constrained Hamiltoniansystem: d z ∗ R d t = ∇ p H ( z ∗ R , p ∗ R , u ∗ R ) = f ( z ∗ R , u ∗ R ) , d p ∗ R d t = −∇ z H ( z ∗ R , p ∗ R , u ∗ R ) = g ( z ∗ R , p ∗ R ) , (Hamiltonian system for ( z ∗ R , p ∗ R ))(5.29a) ∇ u R H ( z ∗ R , p ∗ R , u ∗ R ) = 0 , (1 st -order optimality condition)(5.29b) p ∗ R ( T ) = ∇ z C T ( z ∗ R ( T ) , P c Y ) , (terminal condition)(5.29c)where ∇ x stands for the gradient operator along the x -direction, p ∗ R = p ∗ R, e + p ∗ R, e is the costateassociated with z ∗ R , and the vector field g = ( g , g ) tr has the following expression(5.30) g ( z, p ) := − z − β ( λ ) p − αp z + 2 αp z − αα ( λ ) p ( z ) − αα ( λ ) p z z − ( α ( λ )) z ( z ) − αα ( λ ) α ( λ ) p ( z ) ,g ( z, p ) := − z − β ( λ ) p − αp z − αα ( λ ) p z z − αα ( λ ) p ( z ) + 6 αα ( λ ) p ( z ) − ( α ( λ )) ( z ) z − αα ( λ ) α ( λ ) p z ( z ) − α ( λ )) ( z ) . Note also that ∇ u R H ( z ∗ R , p ∗ R , u ∗ R ) = (cid:16) µ u ∗ R, + a p ∗ R, + a p ∗ R, , µ u ∗ R, + a p ∗ R, + a p ∗ R, (cid:17) tr . The 1 st -order optimality condition (5.29b) reduces then to(5.31) ( u ∗ R, , u ∗ R, ) = − (cid:16) a p ∗ R, + a p ∗ R, µ , a p ∗ R, + a p ∗ R, µ (cid:17) , which written into a compact form, gives(5.32) u ∗ R = − µ M p ∗ R , where M is the matrix introduced in (5.16). Thanks to the relation (5.31) between u ∗ R and the costate p ∗ R , we get(5.33) a u ∗ R, + a u ∗ R, = − µ (cid:0) ( a ) + ( a ) (cid:1) p ∗ R, − µ ( a a + a a ) p ∗ R, =: f ( p ∗ R, , p ∗ R, ) ,a u ∗ R, + a u ∗ R, = − µ ( a a + a a ) p ∗ R, − µ (cid:0) ( a ) + ( a ) (cid:1) p ∗ R, =: f ( p ∗ R, , p ∗ R, ) . Finally, the terminal condition (5.29c) leads to(5.34) p ∗ R,i ( T ) = µ ( z ∗ R,i ( T ) − Y i ) , i = 1 , . By using the above relations, we can reformulate the set of necessary conditions (5.29) as thefollowing boundary-value problem (BVP) to be satisfied by z ∗ R and p ∗ R :(5.35) d z d t = β ( λ ) z + αz z + αα ( λ ) z ( z ) + αα ( λ ) α ( λ ) z ( z ) + f ( p , p ) , d z d t = β ( λ ) z − α ( z ) + 2 αα ( λ )( z ) z + 2 αα ( λ )( z ) + f ( p , p ) , d p d t = g ( z, p ) , d p d t = g ( z, p ) , subject to the boundary conditions(5.36) z (0) = (cid:104) y , e (cid:105) , z (0) = (cid:104) y , e (cid:105) , p ( T ) = µ ( z ( T ) − Y ) , p ( T ) = µ ( z ( T ) − Y ) , where f and f are given by (5.33), and g ( z, p ) and g ( z, p ) are given by (5.30).Once this BVP is solved, the corresponding controller u ∗ R determined by (5.32) constitutes then anatural candidate to solve the h (1) λ -based reduced optimal control problem (5.19). For the problem athand, since the cost functional (5.17) is quadratic in u R and the dependence on the controller is affinefor the system of equations (5.27), it is known that the controller u ∗ R so obtained is actually the uniqueoptimal controller of the reduced problem (5.19); see e.g. [66, Sect. 5.3] and [98]. This observationalso holds for the other reduced optimal control problems derived in later sections.It is worth mentioning that the solution of the above BVP depends on the coefficient matrix M defined in (5.16) associated with the linear operator C through the expressions of f and f given in(5.33). However, due to the specific form of f and f , different choices of M can lead to the samesolution of the BVP. More precisely, the solutions of (5.35)–(5.36) remain unchanged as long as M stays in the group of 2 × Lemma 5.1.
The solution of (5.35) – (5.36) is the same for any M ∈ O (2) .Proof. The result follows trivially by noting that given any M ∈ O (2), it holds that M tr M = I . Inparticular, the following basic identities hold:( a ) + ( a ) = ( a ) + ( a ) = 1 , a a + a a = 0 . By using the above identities in (5.33), we obtain for any M ∈ O (2) that f ( p ∗ R, , p ∗ R, ) = − µ p ∗ R, f ( p ∗ R, , p ∗ R, ) = − µ p ∗ R, , which is independent of M . The desired result follows. (cid:3) INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 27
In connection to the above lemma, let us make finally the following basic observation, which willbe of some interest in the numerical experiments.
Lemma 5.2.
For any two bounded linear operators C i : H → H (i = 1,2), if they leave invariant thesubspaces H c and H s , and their actions on the low modes differs only by an orthogonal transformation,i.e., C i H c ⊂ H c , C i H s ⊂ H s , P c C = M P c C with M ∈ O (2) , then the optimal pairs ( z ∗ R , u ∗ R ) and ( z ∗ R , u ∗ R ) , corresponding to the reduced optimal control problem (5.19) with C in (5.18) taken to be C and C respectively, satisfy the following relation: z ∗ R = z ∗ R , u ∗ R = M − u ∗ R , J R ( z ∗ R , z ∗ R ) = J R ( z ∗ R , u ∗ R ) . If we assume furthermore that P s C = P s C , then analogous results hold for the original optimal controlproblem (5.9) . Remark 5.1.
The above result is not limited to the two-dimensional nature of H c given by (5.13) ,and can be generalized to a higher dimension m , as long as H c is spanned by the first m eigenmodes,and M lives in O ( m ) . Suboptimal pair ( y ∗ R , u ∗ R ) to (5.9) based on h (1) λ : Numerical aspects. The method used tosolve the reduced optimal control problem (5.19) being clarified in the previous section, we turn now tothe practical aspects concerning the synthesis of an h (1) λ -based suboptimal pair ( y ∗ R , u ∗ R ) to the optimalcontrol problem (5.9) associated with the Burgers-type equation (5.1). This synthesis is organized intwo steps. First, the BVP problem (5.35)–(5.36) is solved to get the h (1) λ -based suboptimal controller u ∗ R according to the costate-based explicit expression (5.32). Second, this suboptimal controller isthen used in (5.1) to get the suboptimal trajectory y ∗ R driven by C u ∗ R . We explain below how thesesteps are numerically carried out.Recall that the uncontrolled Burgers-type equation admits two locally stable steady states y ± (emerging from a pitch-fork bifurcation) when λ is above the critical value λ c = νπ l at which theleading eigenmode e loses its linear stability [58]. In the experiments below we take y + as initial data y , the target Y being specified in Section 5.5.Shooting and collocation methods are commonly used to solve two-point boundary value problems[5, 19, 23, 64, 90]. A convenient collocation code is the Matlab built-in solver bvp4c.m , which is usedto solve the aforementioned BVP (5.35)–(5.36) as well as other BVPs encountered in later sections.The simulation of the Burgers equation (5.1) as driven by the 2D suboptimal controller u ∗ R isthen performed by means of a semi-implicit Euler scheme where at each time step the nonlinearterm yy x = ( y ) x / u ∗ R ( x, t ) are treated explicitly, while the linear term is treatedimplicitly. The Laplacian operator is discretized using a standard second-order central differenceapproximation. The resulting semi-implicit scheme now reads as follows:(5.37) y n +1 j − y nj = (cid:16) ν ∆ d y n +1 j + λy n +1 j − γ ∇ d (cid:0) ( y nj ) (cid:1) + u R,nj (cid:17) δt, j ∈ { , · · · , N x − } , where y nj denotes the discrete approximation of y ( jδx, nδt ); u R,nj , the discrete approximation of u ∗ R ( jδx, nδt ); δx , the mesh size of the spatial discretization; δt , the time step; while ∆ d and ∇ d denote the discrete Laplacian and discrete first-order derivative given respectively by∆ d y nj = y nj − − y nj + y nj +1 ( δx ) ; ∇ d (cid:0) ( y nj ) (cid:1) = ( y nj +1 ) − ( y nj ) δx , j ∈ { , · · · , N x − } . See [65] for more details about bvp4c . We also mention that all the numerical experiments performed in this articlehave been carried out by using the Matlab version (R2011b).
The Dirichlet boundary condition (5.2) becomes y n = y nN x = 0 , where N x + 1 is the number of grid points used for the discretization of the spatial domain [0 , l ].The time-dependent ( N x − Y n , and is intendedto be an approximation of the suboptimal trajectory y ∗ R at time t = nδt . Let us also denote by U n the spatial discretization of u ∗ R ( x, nδt ) for x ∈ [ δx, l − δx ], given by U n := (cid:0) u ∗ R ( δx, nδt ) , · · · , u ∗ R (( N x − δx, nδt ) (cid:1) tr . Then after rearranging the terms, equation (5.37) can be rewritten into the following algebraic system:(5.38) (cid:0) (1 − λδt ) I − νδt A (cid:1) Y n +1 = Y n − γ δt B [ S ( Y n )] + δt U n , where I is the ( N x − × ( N x −
1) identity matrix, A is the tridiagonal matrix associated with thediscrete Laplacian ∆ d , B is the matrix associated with the discrete spatial derivative ∇ d , and S ( Y n )denotes the vector whose entries are the square of the corresponding entries of Y n .Since the eigenvalues of A are given by δx ) (cid:16) cos( jπδxl ) − (cid:17) ( j = 1 , · · · , N x −
1) and the corre-sponding eigenvectors are the discretized version of the first N x − e , · · · , e N x − given in(5.12), the eigenvalues of the matrix M := (1 − λδt ) I − νδt A of the LHS of (5.38) can be obtainedeasily, and the corresponding eigenvectors are still the discretized sine functions. At each time step,the algebraic system (5.38) can thus be solved efficiently using the discrete sine transform . To do so,we first compute the discrete sine transform of the RHS and then divide the elements of the trans-formed vector by the eigenvalues of M to which the inverse discrete sine transform is applied to find Y n +1 ; see e.g. [41, Sect. 3.2] for more details. In the numerical results that follow, the discrete sinetransform has been handled by using the Matlab built-in function dst.m .Finally, it is worthwhile mentioning that we have used a uniform time mesh for the integration ofthe PDE, whereas the u ∗ R is defined on a non-uniform mesh due to the adaptive mesh feature of the bvp4c solver. This discrepancy is resolved by using linear interpolation to obtain the value of u ∗ R atthe uniform mesh used in the PDE scheme.For the sake of comparison, the synthesis of a suboptimal controller based on a two-mode Galerkinapproximation has been carried out following the same steps and the same numerical treatment de-scribed above. The corresponding suboptimal controller u ∗ G associated with the 2D Galerkin-basedreduced optimal problem (A.5) is also obtained via a PMP approach which leads to solving a BVPdescribed in Appendix A.1; see (A.7). The same procedure is applied to higher-dimensional Galerkin-based reduced optimal control problems (A.10) derived in Appendix A.2.5.5. h (1) λ , and control performances: Numeri-cal results. We assess in this section the control performances achieved by the h (1) λ -based suboptimalpair ( y ∗ R , u ∗ R ) of the optimal control problem (5.9) such as synthesized according to the proceduredescribed above. These performances are compared with those achieved by a suboptimal solutioncomputed from the 2D Galerkin-based reduced optimal control problem (A.5). In that respect, thecost (5.4) evaluated at the suboptimal pair ( y ( · ; y , u ∗ R ) , u ∗ R ) will be compared with the cost evaluatedat the suboptimal pair ( y ( · ; y , u ∗ G ) , u ∗ G ), where u ∗ G is the suboptimal controller synthesized from (A.5).We also set the coefficient µ weighting the terminal payoff part of the cost functional (5.4) to besufficiently large so that the comparison of the solution profile at the final time T of (5.37) — drivenby the corresponding synthesized controller — with the prescribed target profile Y , provides a wayto visualize the performance of the synthesized suboptimal controller.The simulations reported below, are performed for δt = 0 .
001 and N x = 251 with l = 1 . π sothat δx ≈ .
02. The system parameters are taken to be ν = 1, γ = 2 .
5, and λ = 3 λ c ≈ .
78. The
INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 29 parameters µ and µ in the cost functional (5.4) are taken to be µ = 1 and µ = 20. For all thesimulations conducted in this article, the relative tolerance for the bvp4c has been set to 10 − andthe BVP mesh size parameter has been set to 1.6E4. The linear operator C : H → H is taken to bethe identity mapping for the sake of simplicity. According to Lemma 5.2, any operator C such that P c C ∈ O (2) and P s C = Id H s can be reduced to this case.The numerical results at the final time T = 3 are reported in Fig. 1. The left panel of this figurepresents for this final time, the solution profile to (5.37) as driven by u ∗ R and u ∗ G , respectively. Forthese simulations, the target profile has been chosen to be given by(5.39) Y = − . (cid:104) y − , e (cid:105) e + 1 . (cid:104) y − , e (cid:105) e . The right panel of Fig. 1 shows the two components of the synthesized suboptimal controllers u ∗ R and u ∗ G .As can be observed, the (approximate) PDE final state y ( T ; u ∗ R ) associated with the controller u ∗ R captures the main qualitative feature of the target, while y ( T ; u ∗ G ) associated with the controller u ∗ G fails in this task. At a more quantitative level, the relative L -errors between the respective drivenPDE final states and the target Y are given by (cid:107) y ( T ; y , u ∗ R ) − Y (cid:107)(cid:107) Y (cid:107) = 22 . , and (cid:107) y ( T ; y , u ∗ G ) − Y (cid:107)(cid:107) Y (cid:107) = 76 . . This discrepancy in the control performance as revealed on the above relative L -errors, goes witha noticeable discrepancy between the respective numerical values of the cost, namely J ( y ( · ; y , u ∗ R ) , u ∗ R ) = 9 . , and J ( y ( · ; y , u ∗ G ) , u ∗ G ) = 30 . . These preliminary results clearly indicate that given a decomposition H c ⊕ H s of H , the slavingrelationships between the H s -modes and the H c -modes such as parameterized by h (1) λ , participate inimproving the control performance of the suboptimal solutions synthesized from a reduced systeminvolving only the (partial) interactions between the H c -modes as modeled by a low-dimensionalGalerkin approximation.To better assess the control performance achieved by the h (1) λ -based suboptimal pair ( y ∗ R , u ∗ R ),we compared with the performance achieved by a (suboptimal) solution to (5.9) based on a high-dimensional Galerkin approximation of (5.1). In that respect, we checked that the cost associated witha suboptimal pair ( y ( · ; y , (cid:101) u ∗ G ) , (cid:101) u ∗ G ), where (cid:101) u ∗ G is a controller synthesized by solving the BVP (A.13)associated with an m -dimensional Galerkin-based reduced optimal problem (A.10), can serve as goodestimate of the cost associated with the (genuine) optimal solution to the problem (5.9) provided that m is sufficiently large. We indeed observed that increasing the dimension beyond m = 16 does notresult in significant change of the cost value (up to six significant digits) and we thus retained theresults obtained for m = 16 as reference for providing a good approximation of the optimal solutionto (5.9). For m = 16, the corresponding values of the cost (5.4), and the relative L -error for the finaltime solution profile are given by J ( y ( · ; y , (cid:101) u ∗ G ) , (cid:101) u ∗ G ) = 8 . , and (cid:107) y ( T ; y , (cid:101) u ∗ G ) − Y (cid:107)(cid:107) Y (cid:107) = 13 . . These values when compared with those obtained for the two-dimensional h (1) λ -based reduced prob-lem (5.19) indicates that the two-dimensional controller u ∗ R already provides a fairly good controlperformance but at a much cheaper expense.On the other hand, the quantitative discrepancy observed on the cost values and relative L -errorsbetween the results based on (5.19) and those for the original optimal control problem (as indicatedby the results based on the high-dimensional Galerkin reduced problem) can be attributed to two x Target pro fi le and controlled PDE solutions y ( T ; u ∗ R ) y ( T ; u ∗ G ) Yy t The corresponding controllers u ∗ R, u ∗ R, u ∗ G, u ∗ G, Figure 1.
Left panel : PDE solution profiles at the final time T = 3 driven respectively by thesuboptimal controllers u ∗ R and u ∗ G with initial profile y taken to be y + (the locally stable positive steadystate of the uncontrolled PDE); the target Y (in solid black) is taken to be − . (cid:104) y − , e (cid:105) e +1 . (cid:104) y − , e (cid:105) e . Right panel : The controller u ∗ R = u ∗ R, e + u ∗ R, e synthesized by the finite-horizon PM-based reducedoptimal control problem (5.19); and the controller u ∗ G = u ∗ G, e + u ∗ G, e synthesized by the Galerkin-based reduced optimal control problem (A.5). Here, the system parameters are taken to be l = 1 . π , ν = 1, γ = 2 . λ = 3 λ c . The time step in the PDE solver is δt = 0 .
001 and spatial mesh size δx ≈ .
02. The parameters µ and µ in the cost functional (5.4) are taken to be µ = 1 and µ = 20.The corresponding costs are J ( y ( · ; y , u ∗ R ) , u ∗ R ) = 9 .
75 and J ( y ( · ; y , u ∗ G ) , u ∗ G ) = 30 . main factors according to the theoretical results of Section 4; see Corollary 4.2 and in particular theerror estimate (4.10). The first factor is related to the parameterization defect associated with thefinite-horizon PM used here, namely h (1) λ ; and the second concerns the energy kept in the high modesof the solution either driven by the suboptimal controller u ∗ R or the optimal controller u ∗ itself.For the remaining part of this section, we report on detailed numerical results which further empha-size the practical relevance of the aforementioned theoretic results provided by Corollary 4.2. Thesenumerical results shown in Figs. 2 and 3 are carried out by varying the final time T in the range [0 . , T is varied, associated with the suboptimal pairs( y ∗ R , u ∗ R ) on one hand (blue curve), and associated with the suboptimal pairs ( (cid:101) y ∗ G , (cid:101) u ∗ G ), on the otherhand (black curve). As one can observe up to T = 3, the suboptimal controllers u ∗ R synthesizedfrom the h (1) λ -based reduced problem (5.19) gives access to suboptimal solutions whose cost values areclose to those achieved by the optimal ones . Such good performances starts however to noticeablydeteriorate as T increases from T = 3.The reasons of this deterioration are actually rich of teaching, as we explain now. If the errorestimate (4.10) is meaningful, analyzing its main constitutive elements should help understand whatcauses this deterioration. In that respect, we computed (i) the corresponding parameterization de-fects associated with h (1) λ and a given suboptimal controller u ∗ R , and (ii) the energy contained inthe high modes of the PDE solution either driven by the suboptimal controller u ∗ R (leading to the As approximated from the 16-dimensional Galerkin-based reduced optimal problem (A.10). Note that, given a suboptimal controller, the computation of the parameterization defects here and in latter sections,has been performed by integrating the discrete form (5.37) of (5.1), and by using the formula (3.5), where the H -normhas been used in place of the (cid:107) · (cid:107) α -norm; see Definition 3.1 and Section 5.1 for the functional spaces defined in (5.6). INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 31 suboptimal trajectory y ∗ R ) or the (sub)optimal controller (cid:101) u ∗ G (leading to the (sub)optimal trajectory (cid:101) y ∗ G ).As a first result, the panels (b)–(f) of Fig. 2 show that h (1) λ provides a finite-horizon PM for thewhole range of T analyzed here. The parameterization defects of h (1) λ is furthermore robust withrespect to variations of T , reaching a (nearly) constant value of about 0.57 for T ≥
1. At the sametime, a substantial growth of the energy contained in the high modes of the suboptimal trajectories y ∗ R ( i.e. (cid:107) P s y ∗ R ( t ) (cid:107) H (0 ,l ) ), is observed from T = 3 to T = 5 while (cid:107) P s (cid:101) y ∗ G ( t ) (cid:107) H (0 ,l ) does not changesignificantly; see Fig. 3. A closer look at the numbers reveals that Q ( T, y ; u ∗ R ) = 0 . , (cid:107) P s y ∗ R (cid:107) L (0 ,T ; H (0 ,l )) = 2 . ,Q ( T, y ; (cid:101) u ∗ G ) = 0 . , (cid:107) P s (cid:101) y ∗ G (cid:107) L (0 ,T ; H (0 ,l )) = 2 . , (cid:41) for T = 3 ,Q ( T, y ; u ∗ R ) = 0 . , (cid:107) P s y ∗ R (cid:107) L (0 ,T ; H (0 ,l )) = 3 . ,Q ( T, y ; (cid:101) u ∗ G ) = 0 . , (cid:107) P s (cid:101) y ∗ G (cid:107) L (0 ,T ; H (0 ,l )) = 2 . , (cid:41) for T = 5 , which clearly shows that the RHS of the error estimate (4.10) experiences a growth of about 15% when T increases from T = 3 to T = 5. This growth of the RHS of (4.10) comes with a growth related tothe low-mode part of the LHS of (4.10), i.e. (cid:107) P c ( u ∗ R − (cid:101) u ∗ G ) (cid:107) L (0 ,T ; L (0 ,l )) , of about 10%. This deviationfrom (cid:101) u ∗ G , observed on its low-mode part, is consistent with the substantial growth observed on thecost value J ( y ∗ R , u ∗ R ) as shown in Fig. 2 (a).To summarize, the error estimate (4.10) given in Corollary 4.2 provides useful (and computable)insights that can be used to guide the design of PM-based suboptimal controllers with good controlperformance. In particular, it addresses the importance of constructing PMs with small parameteri-zation defects on one hand, while keeping small the energy contained in the high-modes, on the other.While the latter factor can be conceivably alleviated by increasing the dimension of the reduced phasespace H c , finite-horizon PMs with smaller parameterization defects than proposed by h (1) λ can be thusexpected to be even more useful for the design of low-dimensional suboptimal controllers with goodperformances. The next section addresses the construction of such finite-horizon PMs. Remark 5.2.
We mention that the numerical results reported in Fig. 1 have been compared with thoseobtained by solving the reduced optimal control problem (5.19) with the
BOCOP toolbox [17] . For theparameters used, the relative error under the L -norm between the controllers numerically obtained bythis toolbox and by our calculations has been observed to be within a margin of . . For the sake ofreproducibility of the results for (5.19) , we provide the following numerical values of the componentsof Y used in (5.39) : (cid:104) Y, e (cid:105) = 0 . and (cid:104) Y, e (cid:105) = − . . As illustrated in the previous section in the context of a Burgers-type equation, the finite-horizonPM h (1) λ based on the simple one-layer backward forward system (3.6), can be used efficiently to obtainlow-dimensional suboptimal controllers with relatively good performances for certain cases. Figures 2and 3 indicate that these performances can be altered when the parameterization defects associatedwith h (1) λ is not specially small, while the energy contained in the high modes of the solution — eitherdriven by the suboptimal controller u ∗ R or the optimal controller u ∗ itself — get large, in agreementwith the theoretical predictions of Corollary 4.2. The error estimate (4.10) suggests that other finite-horizon PMs with smaller parameterization defects than h (1) λ should help in the synthesis of suboptimal In contrast to the indirect method adopted above,
BOCOP uses a direct method combining discretization and interior-point methods to solve the reduced optimal control problem (5.19) as implemented in the solver
IPOPT [102]; see thewebpage http://bocop.org for more information. T c o s t Cost values (a) h (1) λ -based16D Galerkin-based T = 1 t (b) T = 2 t (c) T = 3 t (d) Parameterization defects of h (1) λ as the fi nal time T changes T = 4 t (e) T = 5 t (f) Figure 2. (a) : The values of the corresponding cost functional J defined by (5.4) associated withthe suboptimal pair ( y ∗ R , u ∗ R ) as well as the suboptimal pair ( (cid:101) y ∗ G , (cid:101) u ∗ G ) as the final time T various in[0 . , u ∗ R denotes the suboptimal controller synthesized by the h (1) λ -based reduced problem and (cid:101) u ∗ G the 16-dimensional Galerkin based one; (b)-(f ) : The parameterization defect associated with thefinite-horizon PM h (1) λ over the time interval [0 , T ] for various values of T . The parameters are thesame as given in Fig. 1. Energy in the high modes with T = 3 t h (1) λ -based16D Galerkin-based t Energy in the high modes with T = 5 Figure 3.
Energy contained in the high modes of the suboptimal trajectories y ∗ R and (cid:101) y ∗ G for T = 3(left panel) and T = 5 (right panel). The plotted curves are (cid:107) P s y ∗ R ( t ) (cid:107) H (0 ,l ) (blue) and (cid:107) P s (cid:101) y ∗ G ( t ) (cid:107) H (0 ,l ) (black). The parameters are the same as given in Fig. 1. controllers with better performances. The main purpose of this section is to build effectively such PMsthat in particular add higher-order terms to h (1) λ (Theorem 6.1 below) which will turn out to play acrucial role to improve the performances of the h (1) λ -based suboptimal controllers encountered so far;see Remark 6.1 below. INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 33
Higher-order finite-horizon PMs based on two-layer backward-forward system: An-alytic derivation.
We follow [26, Sect. 11] and consider the following two-layer backward-forwardsystem associated with the uncontrolled version of (5.1):d y (1) c d s = L c λ y (1) c , s ∈ [ − τ, , y (1) c ( s ) | s =0 = ξ, (6.1a) d y (2) c d s = L c λ y (2) c + P c B ( y (1) c , y (1) c ) , s ∈ [ − τ, , y (2) c ( s ) | s =0 = ξ, (6.1b) d y (2) s d s = L s λ y (2) s + P s B ( y (2) c , y (2) c ) , s ∈ [ − τ, , y (2) s ( s ) | s = − τ = 0 , (6.1c)where L c λ := P c L λ , L s λ := P s L λ , and ξ ∈ H c .Similar to the one-layer backward-forward system (3.6), the above system is integrated using a two-step backward-forward integration procedure where Eqns. (6.1a)-(6.1b) are integrated first backward,and Eq. (6.1c) is then integrated forward. We will emphasize the dependence on ξ of the high-modecomponent y (2) s of this system as y (2) s [ ξ ].Theorem 6.1 below identifies non-resonance conditions (NR2) under which the pullback limit of y (2) s [ ξ ] exists as τ → ∞ . In particular, it provides an analytical expression of this pullback limit. Asit will be supported by the numerical results of Section 6.2, this pullback limit will turn out to giveaccess to finite-horizon PMs for a broad class of targets. Theorem 6.1.
Consider the two-layer backward-forward system (6.1) associated with the uncontrolledBurgers-type equation (5.1) , i.e. with C = 0 . Let H c be the subspace spanned by the first two eigenmodes e and e of the corresponding linear operator L λ defined in (5.7) . Assume that the eigenvalues of L λ satisfy the following non-resonance conditions: (NR2) β ( λ ) + β ( λ ) − β ( λ ) > , β ( λ ) + 2 β ( λ ) − β ( λ ) > , β ( λ ) − β ( λ ) > , β ( λ ) + β ( λ ) − β ( λ ) > , β ( λ ) + β ( λ ) − β ( λ ) > , β ( λ ) − β ( λ ) > , β ( λ ) − β ( λ ) > . Then the pullback limit of the solution y (2) s [ ξ ] to (6.1) exists and is given by: (6.2) h (2) λ ( ξ ) := lim τ → + ∞ y (2) s [ ξ ]( − τ,
0) = (cid:90) −∞ e − τ (cid:48) L s λ P s B (cid:0) y (2) c ( τ (cid:48) ) , y (2) c ( τ (cid:48) ) (cid:1) d τ (cid:48) , ∀ ξ ∈ H c . Under the above conditions, h (2) λ has furthermore the following analytic expression: (6.3) h (2) λ ( ξ e + ξ e ) = h (2) , λ ( ξ , ξ ) e + h (2) , λ ( ξ , ξ ) e , ( ξ , ξ ) ∈ R , where h (2) , λ ( ξ , ξ ) := (cid:104) h (2) λ ( ξ e + ξ e ) , e (cid:105) = A ξ ξ + B ( ξ ) + C ξ ( ξ ) + D ( ξ ) ξ , (6.4a) h (2) , λ ( ξ , ξ ) := (cid:104) h (2) λ ( ξ e + ξ e ) , e (cid:105) = E ( ξ ) + F ( ξ ) ξ + G ( ξ ) , (6.4b) with (6.5) A = − αβ ( λ ) + β ( λ ) − β ( λ ) , B = − α (3 β ( λ ) − β ( λ ))( β ( λ ) + β ( λ ) − β ( λ )) , C = 3 α ( β ( λ ) + 2 β ( λ ) − β ( λ ))( β ( λ ) + β ( λ ) − β ( λ )) , D = 3 α (3 β ( λ ) − β ( λ ))( β ( λ ) + β ( λ ) − β ( λ ))( β ( λ ) + 2 β ( λ ) − β ( λ ))+ 3 α (3 β ( λ ) − β ( λ ))(3 β ( λ ) + β ( λ ) − β ( λ ))( β ( λ ) + 2 β ( λ ) − β ( λ )) , E = − αβ ( λ ) − β ( λ ) , F = − α (2 β ( λ ) + β ( λ ) − β ( λ ))(2 β ( λ ) − β ( λ )) , G = − α (4 β ( λ ) − β ( λ ))(2 β ( λ ) + β ( λ ) − β ( λ ))(2 β ( λ ) − β ( λ )) , and (6.6) α = γπ √ l / . Remark 6.1.
Note that the analytic expression of h (2) λ given in (6.3) can be written as the sum of h (1) λ given by (5.22) associated with the one-layer backward-forward system (3.6) , and some other higher-order terms. It is worth noting that the extra five terms contained in the expression of h (2) λ resultfrom the nonlinear self-interactions between the low modes as brought by P c B (cid:0) y (1) c , y (1) c (cid:1) in (6.1b) .Numerical results of Section 6.2 below, support the fact that these extra terms can be interpreted ascorrective terms to h (1) λ . Indeed, as we will illustrate for the optimal control problem (5.9) , these termscan help design suboptimal low-dimensional controller of better performances than those built from h (1) λ -based reduced system; the h (2) λ -based reduced system bringing extra higher-order terms correspondingto “low-high” and “high-high” interactions absent from the h (1) λ -based reduced system. This last pointcan be observed by comparing (5.27) with (6.17) below, where both reduced systems are derived fromthe abstract formulation (4.2) by setting the PM function h to be h (1) λ or h (2) λ , respectively.Proof. A simple integration of (6.1) shows that for any τ > ξ ∈ H c the solution to the backward-forward system (6.1) is given by: y (1) c ( s ) = e sL c λ ξ, (6.7a) y (2) c ( s ) = e sL c λ ξ − (cid:90) s e ( s − τ (cid:48) ) L c λ P c B (cid:0) y (1) c ( τ (cid:48) ) , y (1) c ( τ (cid:48) ) (cid:1) d τ (cid:48) , (6.7b) y (2) s [ ξ ]( − τ, s ) = (cid:90) s − τ e ( s − τ (cid:48) ) L s λ P s B (cid:0) y (2) c ( τ (cid:48) ) , y (2) c ( τ (cid:48) ) (cid:1) d τ (cid:48) , (6.7c)for all s ∈ [ − τ, y (2) s [ ξ ]( − τ,
0) takes the form given in (6.2) provided that theconcerned integral exists. We show below that the (NR2)-condition is necessary and sufficient for such Using the symbols introduced here, h (1) λ ( ξ , ξ ) = A ξ ξ e + E ( ξ ) e from (5.22). INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 35 an integral to exist. In that respect, the fact that H c is spanned by the first two eigenmodes facilitatesome of the manipulations as described below.First, note that the projections of y (1) c onto e and e , give respectively,(6.8) y (1)1 ( s ) := (cid:104) y (1) c ( s ) , e (cid:105) = e β ( λ ) s ξ , y (1)2 ( s ) := (cid:104) y (1) c ( s ) , e (cid:105) = e β ( λ ) s ξ , where ξ i := (cid:104) ξ, e i (cid:105) , i = 1 , y (2) c against e and e , we need to recall that the nonlinear interactionlaws (5.20), give here(6.9) B = 0 , B = 2 α, B = − α, B = − α, B = B = 0 , which leads to (cid:104) B ( y (1) c , y (1) c ) , e (cid:105) = (cid:10) B (cid:0) y (1)1 e + y (1)2 e , y (1)1 e + y (1)2 e (cid:1) , e (cid:11) = y (1)1 y (1)2 B − y (1)1 y (1)2 B = αy (1)1 y (1)2 , (cid:104) B ( y (1) c , y (1) c ) , e (cid:105) = (cid:0) y (1)1 (cid:1) B = − α (cid:0) y (1)1 (cid:1) . The projection of y (2) c against e and e are then given by(6.10) y (2)1 ( s ) := (cid:104) y (2) c ( s ) , e (cid:105) = e β ( λ ) s ξ − α (cid:90) s e β ( λ )( s − τ (cid:48) ) y (1)1 ( τ (cid:48) ) y (1)2 ( τ (cid:48) )d τ (cid:48) ,y (2)2 ( s ) := (cid:104) y (2) c ( s ) , e (cid:105) = e β ( λ ) s ξ + α (cid:90) s e β ( λ )( s − τ (cid:48) ) ( y (1)1 ( τ (cid:48) )) d τ (cid:48) . Relying again on to the nonlinear interaction laws (5.20), we have(6.11) B = 0 , B = − α, B = − α, B = 0 ,B = B = B = 0 , B = − α,B nij = 0 , ∀ i, j ∈ { , } , n ≥ , which leads to(6.12) y (2)3 [ ξ ]( − τ, s ) := (cid:104) y (2) s [ ξ ]( − τ, s ) , e (cid:105) = − α (cid:90) s − τ e β ( λ )( s − τ (cid:48) ) y (2)1 ( τ (cid:48) ) y (2)2 ( τ (cid:48) )d τ (cid:48) ,y (2)4 [ ξ ]( − τ, s ) := (cid:104) y (2) s [ ξ ]( − τ, s ) , e (cid:105) = − α (cid:90) s − τ e β ( λ )( s − τ (cid:48) ) ( y (2)2 ( τ (cid:48) )) d τ (cid:48) . By using the expressions of y (2)1 and y (2)2 given in (6.10) (and using also (6.8)), it can be checkedthat the limit h (2) , λ := lim τ → + ∞ y (2)3 [ ξ ]( − τ,
0) exists if and only if the first four inequalities in the(NR2)-condition hold, while h (2) , λ is given by (6.4a) under these conditions. Similarly, the limit h (2) , λ := lim τ → + ∞ y (2)4 [ ξ ]( − τ,
0) exists if and only if the last three inequalities in the (NR2)-conditionhold, and h (2) , λ is given by (6.4b) under these conditions. The theorem is proved. (cid:3) Controller synthesis based on h (2) λ , and control performances: Analytic derivation andnumerical results. Analytic derivation of the h (2) λ -based reduced optimal control problem. Following (4.2), the h (2) λ -based reduced system intended to model the dynamics of the low modes P c y of (5.1), takes the following abstract form:(6.13) d z d t = L c λ z + P c B (cid:16) z + h (2) λ ( z ) , z + h (2) λ ( z ) (cid:17) + C u R , t ∈ (0 , T ] ,z (0) = P c y ∈ H c , where y is the initial datum for the original PDE (5.1).Analogous to (5.17), the cost functional associated with the reduced system (6.13) is given by(6.14) (cid:98) J R ( z, u R ) = (cid:90) T (cid:0) (cid:107) z ( t ) + h (2) λ ( z ( t )) (cid:107) + µ (cid:107) u R ( t ) (cid:107) (cid:1) d t + C T ( z ( T ) , P c Y ) , where C T ( z ( T ) , P c Y ) := µ (cid:80) mi =1 | z i ( T ) − Y i | is the terminal payoff term as defined in (5.26), with Y being some prescribed target for (5.1).By using the analytic expression of h (2) λ given in (6.3)-(6.5), the cost functional (6.14) can be writteninto the following explicit form:(6.15) (cid:98) J R ( z, u R ) = (cid:90) T (cid:104) G ( z ( t )) + µ E ( u R ( t )) (cid:105) d t + C T ( z ( T ) , P c Y ) , where(6.16) G ( z ) = 12 (cid:107) z + h (2) λ ( z ) (cid:107) = 12 (cid:104) ( z ) + ( z ) + ( h (2) , λ ( ξ , ξ )) + ( h (2) , λ ( ξ , ξ )) (cid:105) , E ( u R ) = µ (cid:107) u R (cid:107) = µ u R, ) + ( u R, ) ] , with z i := (cid:104) z, e i (cid:105) and u R,i := (cid:104) u R , e i (cid:105) , i = 1 , h (2) λ ( ξ e + ξ e ) = h (2) , λ ( ξ , ξ ) e + h (2) , λ ( ξ , ξ ) e in (6.13) and projecting this equation against e and e respectively, we obtain, after simplificationby using the nonlinear interaction laws (5.20), the following analytic formulation of the h (2) λ -basedreduced system (6.13):(6.17)d z d t = β ( λ ) z + α (cid:16) z z + z h (2) , λ ( z , z ) + h (2) , λ ( z , z ) h (2) , λ ( z , z ) (cid:17) + a u R, ( t ) + a u R, ( t ) , d z d t = β ( λ ) z − αz + 2 α (cid:16) z h (2) , λ ( z , z ) + z h (2) , λ ( z , z ) (cid:17) + a u R, ( t ) + a u R, ( t ) , with h (2) , λ ( z , z ) and h (2) , λ ( z , z ) given by (6.4)-(6.5). The resulting reduced optimal control problem based on h (2) λ is thus:(6.18) min (cid:98) J R ( z, u R ) s.t. ( z, u R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (6.17) . By following similar arguments as provided in Section 5.2 and applying the Pontryagin maximumPrinciple, we can conclude that for a given pair( (cid:98) z ∗ R , (cid:98) u ∗ R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) Using this analytic formulation, we mention that the Cauchy problem for (6.17) can be dealt with by carryingout similar (but more tedious) energy estimates as presented in Appendix B for the two-dimensional h (1) λ -based reducedsystem (5.27). INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 37 to be optimal for the h (2) λ -reduced optimal problem (6.18), it is necessary and sufficient to satisfythe following set of conditions:(6.19) ( (cid:98) u ∗ R, , (cid:98) u ∗ R, ) = − (cid:16) a (cid:98) p ∗ R, + a (cid:98) p ∗ R, µ , a (cid:98) p ∗ R, + a (cid:98) p ∗ R, µ (cid:17) , where ( (cid:98) p ∗ R, , (cid:98) p ∗ R, ) is the costate associated with ( (cid:98) z ∗ R, , (cid:98) z ∗ R, ), both determined by solving the followingBVP:(6.20) d z d t = β ( λ ) z + α (cid:16) z z + z h (2) , λ ( z , z ) + h (2) , λ ( z , z ) h (2) , λ ( z , z ) (cid:17) − p , d z d t = β ( λ ) z − α ( z ) + 2 α (cid:16) z h (2) , λ ( z , z ) + z h (2) , λ ( z , z ) (cid:17) − p , d p d t = g ( z, p ) , d p d t = g ( z, p ) , subject to the boundary condition(6.21) z (0) = (cid:104) y , e (cid:105) , z (0) = (cid:104) y , e (cid:105) , p ( T ) = µ ( z ( T ) − Y ) , p ( T ) = µ ( z ( T ) − Y ) , where g ( z, p ) := − z − h (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z − h (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z − p (cid:18) β ( λ ) + αz + αz ∂h (2) , λ ( z , z ) ∂z + α ∂h (2) , λ ( z , z ) ∂z h (2) , λ ( z , z )+ αh (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z (cid:19) − αp (cid:18) − z + h (2) , λ ( z , z ) + z ∂h (2) , λ ( z , z ) ∂z + z ∂h (2) , λ ( z , z ) ∂z (cid:19) ,g ( z, p ) := − z − h (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z − h (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z − αp (cid:18) z + h (2) , λ ( z , z ) + z ∂h (2) , λ ( z , z ) ∂z + ∂h (2) , λ ( z , z ) ∂z h (2) , λ ( z , z )+ h (2) , λ ( z , z ) ∂h (2) , λ ( z , z ) ∂z (cid:19) − p (cid:18) β ( λ ) + 2 αz ∂h (2) , λ ( z , z ) ∂z + 2 αh (2) , λ ( z , z ) + 2 αz ∂h (2) , λ ( z , z ) ∂z (cid:19) . The vector field ( g , g ) given above has been determined by evaluating −∇ z (cid:98) H ( z, p, u ), with thefollowing Hamiltonian (cid:98) H , formed by application of the PMP to (6.18) (cid:98) H ( z, p, u ) := G ( z ) + E ( u ) + p (cid:98) f ( z, u ) + p (cid:98) f ( z, u ) , where ( (cid:98) f , (cid:98) f ) denotes the vector field constituting the RHS of the z -equations in (6.20). The sufficient part is again due to the fact that the cost functional (6.14) is quadratic in u R and the dependenceon the controller is affine for the system of equations (6.17); see e.g. [66, Sect. 5.3] and [98]. Numerical results.
The above BVP is solved again using bvp4c , and the resulting two-dimensionalsuboptimal controller (cid:98) u ∗ R is obtained according to (6.19). As before, the corresponding suboptimaltrajectory (cid:98) y ∗ R of the PDE (5.1) is computed by driving (5.1) with (cid:98) u ∗ R , following the numerical proceduredescribed in Section 5.4. x Target pro fi le and controlled PDE solutions (a) y ( T ; u ∗ G ) y ( T ; u ∗ R ) y ( T ; (cid:31) u ∗ R ) Y t Corresponding controllers (c) u ∗ R, u ∗ R, (cid:31) u ∗ R, (cid:31) u ∗ R, u ∗ G, u ∗ G, t Parameterization defects of fi nite-horizon PMs (d) for h (1) λ for h (2) λ x (b) y Figure 4. (a) : Final state at T = 3 of the PDE solution profiles driven respectively by the subopti-mal controllers u ∗ G , u ∗ R and (cid:98) u ∗ R with initial profile taken to be y + as shown in (b) ; also shown in (a) isthe target state Y given by (6.22). (c) : The suboptimal controllers u ∗ G = u ∗ G, e + u ∗ G, e synthesizedfrom the Galerkin-based reduced optimal control problem (A.5); u ∗ R = u ∗ R, e + u ∗ R, e synthesized fromthe h (1) λ -based reduced optimal control problem (5.19); and (cid:98) u ∗ R = (cid:98) u ∗ R, e + (cid:98) u ∗ R, e synthesized fromthe h (2) λ -based reduced optimal control problem (6.18). (d) : Finite-horizon parameterization defectsof h (1) λ and h (2) λ associated with the PDE (5.1) driven respectively by u ∗ R and (cid:98) u ∗ R over the time interval[0 , The corresponding control performance is shown in Fig. 4, where the performance of the suboptimalcontrollers u ∗ R and u ∗ G associated with respectively the two-dimensional h (1) λ -based reduced optimalcontrol problem (5.19) and the two-dimensional Galerkin-based one (A.5) are also reported for com-parison. In panel (a) of Fig. 4, we present the PDE final time solution profile y ( T, (cid:98) u ∗ R ), y ( T, u ∗ R ), and y ( T, u ∗ G ) driven respectively by (cid:98) u ∗ R , u ∗ R and u ∗ G , for T = 3. For these simulations, the target profile Y has been chosen to be again spanned by the first two eigenfunctions, but given this time by(6.22) Y = − . (cid:104) y + , e (cid:105) e − . (cid:104) y + , e (cid:105) e ;the initial profile is taken to be the positive steady state y + for the uncontrolled PDE as used inSection 5.5, see panel (b) . The two components of the synthesized suboptimal controllers are shownin panel (c) , and the parameterization defects associated with respectively h (1) λ and h (2) λ are shown inpanel (d) . The corresponding cost values and final-time relative L -errors are given in Table 2 below.The results of Fig. 4 (a) and Table 2 illustrate that for a given reduced phase space — here thetwo-dimensional vector space H c — the slaving relationship of the high-modes (not in H c ) by the lowmodes (in H c ) as parameterized by h (2) λ can turn out to be superior than the one proposed by h (1) λ for INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 39
Table 2.
Cost values and final-time relative L -errors associated with the suboptimalcontrollers u ∗ G u ∗ R (cid:98) u ∗ R (cid:101) u ∗ G (with m = 16) J ( y ( · ; y + , u ) , u ) 108.65 12.48 5.07 5.02Relative L -error: (cid:107) y ( T ; y + , u ) − Y (cid:107) / (cid:107) Y (cid:107) .
60% 107 .
23% 15 .
07% 11 . J is the one defined in (5.4) associated with the optimal control problem(5.9). This cost is assessed at the suboptimal pairs ( y ( · ; y + , u ) , u ) with u taken to beeither u ∗ G , u ∗ R , (cid:98) u ∗ R , or (cid:101) u ∗ G . The target Y is given by (6.22). The suboptimal controller u ∗ G is synthesized from the 2D Galerkin-based reduced optimal control problem (A.5); u ∗ R from the h (1) λ -based (5.19); (cid:98) u ∗ R from the h (2) λ -based (6.18); and (cid:101) u ∗ G from the m -dimensional Galerkin-based one (A.10) with m = 16. The latter serves as a benchmarkhere. The model parameters are those used for Fig. 1.the synthesis of suboptimal solutions to (5.9), and can turn out to be clearly advantageous comparedto suboptimal solutions for which no slaving relationship whatsoever is involved such as for those builtfrom the 2D Galerkin-based reduced optimal control problem (A.5). Again, Corollary 4.2 and theerror estimate (4.10) provide theoretical insights that help understand why improving the quality ofsuch a slaving relationship participates to improve the performance of a suboptimal controller. Forinstance, the improvement in getting closer to the prescribed target Y (Fig. 4 (d) ) — accompaniedwith a noticeable reduction of the cost values (Table 2) — occurs when the PDE (5.1) is driven bythe h (2) λ -based suboptimal controller (cid:98) u ∗ R instead of the h (1) λ -based one u ∗ R , and goes with a parameter-ization defect (overall) smaller for h (2) λ than for h (1) λ (Fig. 4 (d) ). Interestingly, this reduction of theparameterization defect comes with the higher-order terms contained in h (2) λ (see Theorem 6.1) thatcan be thus reasonably interpreted as correction terms to the parameterization proposed by h (1) λ ; seealso Remark 6.1.However, such a statement has to be nuanced and an h (2) λ -based reduced system does not alwayslead to the significant advantages in the design of suboptimal solutions such as illustrated in Fig. 4.The caveat relies on the fact that the parameterization defect associated with h (2) λ also depends on thetarget profile. For instance, with the sign-changing target (5.39) used in the experiments of Section5.5, the suboptimal solutions designed from (6.18) achieve comparable performances to those designedfrom (5.19).These remarks motivate further analysis to arbitrate whether the success achieved for the targetprescribed in (6.22) are pathological or robust, to some extent. For that purpose, we considereddeformations of the target (6.22) taken to be of the form(6.23) Y σ ,σ = − σ (cid:104) y + , e (cid:105) e − σ (cid:104) y + , e (cid:105) e , with σ ∈ [0 . , .
7] and σ ∈ [0 . , . h (2) λ -based (resp. h (1) λ -based) reduced optimal problem to provide the corresponding h (2) λ -based (resp. h (1) λ -based) suboptimalsolutions. As a benchmark , these solutions are compared with those obtained from the m -dimensionalGalerkin-based reduced optimal problem (A.10) with m = 16. The results are reported in Fig. 5 andin Fig. 6 below. Figure 5 shows for each ( σ , σ ) the corresponding relative L -errors at the final-time solution profiles compared with the target Y σ ,σ ; and Figure 6 shows the cost values associated Here, 4 significant digits of the cost J are ensured with m = 16 by comparing with cost values associated withhigher-dimensional suboptimal controller synthesized from (A.10). with the suboptimal controllers u ∗ R and (cid:98) u ∗ R , on one hand, and (cid:101) u ∗ G obtained from the m -dimensionalGalerkin-based reduced problem, on the other. σ σ Relative L -error of the PDE controlled state driven by u ∗ R compared to the target σ σ Relative L -error of the PDE controlled state driven by b u ∗ R compared to the target σ σ Relative L -error of the PDE controlled state driven by e u ∗ G compared to the target Figure 5. ( σ , σ )-dependence of the relative L -error of the PDE final state y ( T, y + ; u ) comparedto the target Y σ ,σ given by (6.23). Here the controller u is taken to be either u ∗ R ( upper panel ), or (cid:98) u ∗ R ( middle panel ), or (cid:101) u ∗ G ( lower panel ); the parameters σ and σ are taken to be σ ∈ [0 . , . σ ∈ [0 , . T = 3. The markers “+” in the plots correspond to the resultsshown in Fig. 4, for ( σ , σ ) = (0 . , . Figures 5 and 6 show that the good performance achieved by the h (2) λ -based suboptimal controllershown in Fig. 4 (a) , is not isolated and can be even further improved within a broad region of the( σ , σ )-parameter space when Y σ ,σ is changed accordingly. Compared to the bad performancesobserved on Fig. 5 (top panel) for the h (1) λ -based suboptimal controllers, these h (2) λ -based resultsprovide strong evidence that the higher-order terms brought by h (2) λ with respect to h (1) λ , act ascorrective terms in the high-mode parametrization proposed by h (1) λ .These numerical results together with the theoretic results of Corollary 4.2 suggest that in order todesign reduced problems whose solutions would provide even better control performance than thosereported here, one can try to construct finite-horizon PMs with smaller parameterization defects thanthose achieved by h λ . In that respect, the discussions and results of [26, Sect. 8.3-8.5], presented inthe context of asymptotic PMs, can be valuable. In connection to the discussion concerning Figs. 2and 3 in Section 5.5, the searching for better slaving relationships between the H s -modes and the H c -modes can be combined with the usage of higher dimensional reduced phase spaces H c so thatthe energy kept in the high modes gets reduced. The next section shows that a moderate increase INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 41 of dim( H c ) can actually already help improve the performances based on h (1) λ , in the case of locallydistributed control laws. J ( b u ∗ R ) J ( e u ∗ G ) J ( u ∗ R ) σ Costs associated with the controllers u ∗ R , b u ∗ R and e u ∗ G σ c o s t Figure 6. ( σ , σ )-dependence of the cost values J ( y, u ) given by (5.4) when u = u ∗ R , u = (cid:98) u ∗ R and u = (cid:101) u ∗ G , respectively. The parameters σ and σ vary in [0 . , .
7] and in [0 , . T = 3. Synthesis of m -Dimensional Locally Distributed Suboptimal Controllers In this last section, we consider the more challenging case of optimal locally distributed controlproblems associated with the Burgers-type equation (5.1). This situation corresponds to the casewhere the linear operator C is associated with the characteristic function χ Ω of a subdomain Ω ⊂ [0 , l ],such that for any u ∈ H = L (0 , l ), the action of C on u is defined by:(7.1) C u ( x ) = χ Ω ( x ) u ( x ) , ∀ x ∈ [0 , l ] . As used in the fully distributed case in the previous sections, we will consider for some prescribed(time-independent) target Y , cost functionals of terminal payoff type such as:(7.2) J TP ( y, u ) = (cid:90) T (cid:0) (cid:107) y ( t ; y , u ) (cid:107) + µ (cid:107) u ( t ) (cid:107) (cid:1) d t + µ (cid:107) y ( T ; y , u ) − Y (cid:107) , but also cost functionals of tracking type :(7.3) J track ( y, u ) = (cid:90) T (cid:0) (cid:107) y ( t ; y , u ) − Y (cid:107) + µ (cid:107) u ( t ) (cid:107) (cid:1) d t, where in both cases, µ and µ are some positive parameters.The optimal control problem takes thus one of the following forms:(7.4) min J TP ( y, u ) with J TP defined in (7.2) s.t.( y, u ) ∈ L (0 , T ; H ) × L (0 , T ; H ) solves the problem (5.1)–(5.3) . or(7.5) min J track ( y, u ) with J track defined in (7.3) s.t.( y, u ) ∈ L (0 , T ; H ) × L (0 , T ; H ) solves the problem (5.1)–(5.3) . The goal of this last section is to show that the PM-approach introduced above provides an ef-ficient way to design suboptimal solutions for such optimal control problems associated with locallydistributed control laws. For simplicity, we will focus on the performance achieved by the h (1) λ -based re-duced system for the design of such suboptimal solutions, that is the following m -dimensional reducedsystem(7.6) d z d t = L c λ z + P c B (cid:16) z + h (1) λ ( z ) , z + h (1) λ ( z ) (cid:17) + P c χ Ω u R ( t ) , t ∈ (0 , T ] , will be at the core of our synthesis of suboptimal controllers.It is worthwhile to note that in general, the choice of the reduced dimension, m , depends typicallyon the system parameters such as the viscosity ν , the domain size l and the control parameter λ ;and m is chosen so that the resolved modes explain a sufficient large portion of the energy containedin the PDE solution. For the particular case of locally distributed control laws, the size and thelocation of the subdomain Ω plays also a determining role in sizing “a good” m . For instance, thesmaller the subdomain Ω will be, the larger the dimension m will need to be in order to obtain areduced system useful for the design of good suboptimal controllers. Intuitively, this is related to thefact that further eigenmodes are needed in order to obtain a reasonably good approximation of thecharacteristic function χ Ω when the size of the support Ω is further reduced. This intuition will benumerically confirmed in Section 7.3 below, where a reduction of 40 percent of the domain compared tothe globally distributed case analyzed in Section 5.5, led to a choice of m = 4 for a design of suboptimalcontrollers with comparable performances than those achieved in Section 5.5, from two-dimensionalreduced systems.We now describe the h (1) λ -based reduced optimal control that will serve us to design the correspond-ing suboptimal controllers. First, note that the cost functional associated with (7.6) takes one of thefollowing forms(7.7) J TP R ( z, u R ) = (cid:90) T (cid:16) (cid:107) z + h (1) λ ( z ) (cid:107) + µ (cid:107) u R (cid:107) (cid:17) d t + µ (cid:107) z ( T ; z , u R ) − P c Y (cid:107) , or(7.8) J track R ( z, u R ) = (cid:90) T (cid:16) (cid:107) z + h (1) λ ( z ) − Y (cid:107) + µ (cid:107) u R (cid:107) (cid:17) d t, depending on whether (7.2) or (7.3) is considered.The reduced optimal control problem for (7.4) reads then as follows:(7.9) min J TP R ( z, u R ) s.t. ( z, u R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (7.6) . Accordingly, the reduced optimal control problem for (7.5) reads:(7.10) min J track R ( z, u R ) s.t. ( z, u R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (7.6) . Analytic derivation of m -dimensional h (1) λ -based reduced systems for the design ofsuboptimal controllers. In this subsection, we derive explicit forms of the reduced suboptimalcontrol problems (7.9) and (7.10). Details are presented for (7.9), while the analogous derivation for(7.10) is left to the interested reader. For this purpose, let us first examine the existence of the finite-horizon PM candidate h (1) λ . We know from Section 3.2 that the pullback limit h (1) λ associated with INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 43 the backward-forward system (3.6) exists when the (NR)-condition holds. For the Burgers equationconsidered here, due to the nonlinear interaction relations (5.20), the (NR)-condition reads as follows:(7.11) ∀ n > m, ∀ i ∈ { , · · · , m } , (cid:16) n − i ∈ { , · · · , m } (cid:17) = ⇒ (cid:16) β i ( λ ) + β n − i ( λ ) − β n ( λ ) > (cid:17) . By using the analytic expression of the eigenvalues as given in (5.11), we get(7.12) β i ( λ ) + β n − i ( λ ) − β n ( λ ) = λ + νπ ( n − i − ( n − i ) ) l , which is positive for all values of λ of interest here ( λ > λ c := νπ l ). Consequently, the pullback limit h (1) λ always exists for such given λ , and its analytic form provided in (3.11) reads as follows for theproblem considered here:(7.13) h (1) λ ( ξ ) = (cid:88) n>m h (1) ,nλ ( ξ ) e n , where(7.14) h (1) ,nλ ( ξ ) = (cid:88) i + i = n ≤ i ,i ≤ m ξ i ξ i β i ( λ ) + β i ( λ ) − β n ( λ ) (cid:68) B ( e i , e i ) , e n (cid:69) . From (7.14), it is clear that h (1) ,nλ = 0 for all n > m . Note also that it follows from the nonlinearinteraction laws (5.20) that (cid:68) B ( e i , e n − i ) , e n (cid:69) + (cid:68) B ( e n − i , e i ) , e n (cid:69) = − nα, where α = γπ √ l / . By using this identity, we can rewrite h (1) ,nλ for n = m + 1 , · · · , m as follows:(7.15) h (1) ,nλ ( ξ ) = − nα ( n − / (cid:88) i = n − m ξ i ξ n − i β i ( λ ) + β n − i ( λ ) − β n ( λ ) , if n is odd , − nα (cid:18) ( n − / (cid:88) i = n − m ξ i ξ n − i β i ( λ ) + β n − i ( λ ) − β n ( λ ) + ( ξ n/ ) β n/ ( λ ) − β n ( λ ) (cid:19) , if n is even . where the convention that the sum is zero when the lower bound of the summation index is greaterthan its upper bound, has been adopted.Let us denote by M the matrix whose components are given by(7.16) M ( i, j ) := (cid:104) χ Ω e i , e j (cid:105) , ≤ i, j ≤ m. Let us also introduce(7.17) v R ( t ) := M tr u R ( t ) . By rewriting the reduced system (7.6) as(7.18) d z i d t = β i ( λ ) z i + (cid:68) B (cid:16) z + h (1) λ ( z ) , z + h (1) λ ( z ) (cid:17) , e i (cid:69) + v R,i ( t ) , t ∈ (0 , T ] , i = 1 , · · · , m, and by using the expansions z = m (cid:88) i =1 z i e i , h (1) λ ( z ) = m (cid:88) n = m +1 h (1) ,nλ ( z ) e n , along with the nonlinear interaction relations (5.20), the above system of equations becomes:(7.19) d z i d t = β i ( λ ) z i + ( a ) (cid:122) (cid:125)(cid:124) (cid:123) iα (cid:16) − (cid:98) i/ (cid:99) (cid:88) j =1 ω i,j z j z i − j + m (cid:88) j = i +1 z j z j − i (cid:17) + ( b ) (cid:122) (cid:125)(cid:124) (cid:123) iα m (cid:88) j = m − i +1 z j h (1) ,j + iλ ( z )+ iα m − i (cid:88) n = m +1 h (1) ,nλ ( z ) h (1) ,n + iλ ( z ) (cid:124) (cid:123)(cid:122) (cid:125) ( c ) + v R,i ( t ) , t ∈ (0 , T ] , i = 1 , · · · , m, where (cid:98) x (cid:99) denotes the largest integer less than x ; h (1) ,nλ is provided by (7.15); and the coefficients ω i,j are given by ω i,j := (cid:40) , if i is odd, or if i is even and j (cid:54) = i/ / , if i is even and j = i/ . In the above system, the terms gathered in ( a ) correspond to the self-interactions between the lowmodes: (cid:104) B ( z, z ) , e i (cid:105) , the terms gathered in ( b ) correspond to the cross-interactions between the lowand (unresolved) high modes such as parameterized by h (1) λ : (cid:104) B ( z, h (1) λ ( z )) , e i (cid:105) + (cid:104) B ( h (1) λ ( z ) , z ) , e i (cid:105) ,and the terms gathered in ( c ) correspond to the self-interactions between the high modes (still suchas parameterized by h (1) λ ) as projected onto H c : (cid:104) B ( h (1) λ ( z ) , h (1) λ ( z )) , e i (cid:105) .Note that in the case m = 2 the system (7.19) takes the same functional form as the h (1) λ -basedreduced system (5.27) derived in Section 5.2 for the globally distributed control case, only the matricesgiven in (5.16) and (7.16) differ. We refer again to Appendix B for an analysis of the Cauchy problemassociated with (7.19), leaving to the interested reader the generalization to the m -dimensional case.7.2. Synthesis of m -dimensional locally distributed suboptimal controllers. We apply oncemore the Pontryagin maximum principle to derive boundary value problems to be satisfied by an h (1) λ -based suboptimal controller. We focus again on the case with terminal payoff given by (7.9), andindicate necessary changes for the case of tracking type (7.10) at the end of this subsection.Let us denote the RHS of (7.19) by f ( z, v R ). The Hamiltonian associated with the cost functional(7.7) reads then as follows:(7.20) H ( z, p, u R ) := 12 (cid:107) z + h (1) λ ( z ) (cid:107) + µ (cid:107) u R (cid:107) + p tr f ( z, v R )where p := ( p , · · · , p m ) tr is the costate, and v R = M tr u R ; see (7.17).Recall also that the terminal payoff, denoted by C T ( z ( T ) , P c Y ), reads in this case:(7.21) C T ( z ( T ) , P c Y ) := µ m (cid:88) i =1 | z i ( T ) − Y i | . It follows from the Pontryagin maximum principle that for a given pair( z ∗ R , v ∗ R ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 45 to be optimal for the reduced problem (7.9), it must satisfy the following conditions for all i = 1 , · · · , m (see e.g. [66, Chap. 5]): d z ∗ R d t = ∇ p H ( z ∗ R , p ∗ R , v ∗ R ) = f ( z ∗ R , v ∗ R ) , (7.22a) d p ∗ R d t = −∇ z H ( z ∗ R , p ∗ R , v ∗ R ) = g ( z ∗ R , p ∗ R ) , (7.22b) ∇ u R H ( z ∗ R , p ∗ R , v ∗ R ) = 0 , (7.22c) p ∗ R ( T ) = ∇ z C T ( z ∗ R ( T ) , P c Y ) , (7.22d)where v ∗ R = M tr u ∗ R ; p ∗ R = (cid:80) mi =1 p ∗ R,i e i denotes the costate associated with z ∗ R ; and the vector field( g , · · · , g m ) tr is defined by(7.23) g i ( z, p ) := − ∂H ( z, p, v R ) ∂z i = − z i − m (cid:88) n = m +1 h (1) ,nλ ( z ) ∂h (1) ,nλ ( z ) ∂z i − m (cid:88) j =1 p j ∂f j ( z, v R ) ∂z i , i = 1 , · · · , m. Here the partial derivatives ∂h (1) ,nλ ( z ) ∂z i can be obtained by using the expression of h (1) ,nλ given in (7.15)which leads to(7.24) ∂h (1) ,nλ ( z ) ∂z i = − jαz n − i β i ( λ ) + β n − i ( λ ) − β n ( λ ) , if n ∈ { m + 1 , · · · , m } and i ∈ { n − m, · · · , m } ,0 , otherwise . The formula for ∂f j ( z,v R ) ∂z i can be obtained by taking the corresponding partial derivative of the RHSof (7.19) form which we obtain after simplifications(7.25) ∂f j ( z, v R ) ∂z i = β j ( λ ) δ ij + jα ( I aj,i + I bj,i + I cj,i ) , where δ ij denotes the Kronecker delta, and I aj,i , I bj,i and I cj,i are given by(7.26) I aj,i = ∂∂z i (cid:16) − (cid:98) j/ (cid:99) (cid:88) k =1 ω j,k z k z j − k + m (cid:88) k = j +1 z k z k − j (cid:17) = z i − j , if i > j,z i + j , if i = j and i + j ≤ m,z i + j − z j − i , if i < j and i + j ≤ m, − z j − i , if i < j and i + j > m, , otherwise;(7.27) I bj,i = ∂∂z i (cid:16) m (cid:88) k = m − j +1 z k h (1) ,k + jλ ( z ) (cid:17) = h (1) ,i + jλ + m (cid:88) k = m − j +1 z k ∂h (1) ,k + jλ ( z ) ∂z i , if i + j > m, m (cid:88) k = m − j +1 z k ∂h (1) ,k + jλ ( z ) ∂z i , if i + j ≤ m ;and(7.28) I cj,i = ∂∂z i (cid:18) m − j (cid:88) n = m +1 h (1) ,nλ ( z ) h (1) ,n + jλ ( z ) (cid:19) = m − j (cid:88) n = m +1 (cid:18) ∂h (1) ,nλ ( z ) ∂z i h (1) ,n + jλ ( z ) + h (1) ,nλ ( z ) ∂h (1) ,n + jλ ( z ) ∂z i (cid:19) . We derive next a relation between u ∗ R and p ∗ R , which when used in (7.22) leads to a BVP for ( z ∗ R , p ∗ R )to be solved in order to find u ∗ R . To this end, note that from the expression of the Hamiltonian H given in (7.20), we obtain the following expression of ∇ u R H ( z ∗ R , p ∗ R , u ∗ R ), which written component-wise, gives: ∂H∂u R,i ( z ∗ R , p ∗ R , u ∗ R ) = µ u ∗ R,i + m (cid:88) j =1 p ∗ R,j ∂f j ∂u R,i ( z ∗ R , M tr u ∗ R ) = µ u ∗ R,i + m (cid:88) j =1 p ∗ R,j M ( i, j ) , i ∈ { , · · · , m } . The first-order optimality condition (7.22c) leads to(7.29) u ∗ R = − µ M p ∗ R , where M is given by (7.16).It follows then that the controller v ∗ R in (7.22a) takes the form:(7.30) v ∗ R = M tr u ∗ R = − µ M tr M p ∗ R . To summarize, corresponding to the h (1) λ -based reduced optimal control problem (7.9), we havederived the following BVP to be satisfied by the optimal trajectory z ∗ R and its costate p ∗ R :d z ∗ R,i d t = f i (cid:16) z ∗ R , v ∗ R (cid:17) , t ∈ (0 , T ] , (7.31a) d p ∗ R,i d t = g i ( z ∗ R , p ∗ R ) , t ∈ (0 , T ] , (7.31b) z ∗ R,i (0) = y ,i , p ∗ R,i ( T ) = µ ( z ∗ R i ( T ) − Y i ) , i = 1 , · · · , m, (7.31c)where v ∗ R is given by (7.30), y ,i is the projection of the initial data y for the underlying PDE (5.1)against e i , and the boundary condition for p ∗ R is derived from the terminal condition (7.22d) by usingthe expression of the terminal payoff C T given in (7.21). Once (7.31) is solved, the m -dimensionalcontroller u ∗ R given by (7.29) constitutes our h (1) λ -based suboptimal controller for the optimal controlproblem (7.4). Note that u ∗ R synthesized this way turns out to be the unique optimal controller forthe reduced problem (7.9) for the same reasons pointed out in Section 5.3.The corresponding BVP associated with the reduced optimal control problem (7.10) can be derivedin the same fashion; and we indicate below the necessary changes. In this case, the Hamiltonianassociated with the cost functional (7.8) reads:(7.32) (cid:101) H ( z, p, u R ) := 12 (cid:107) z + h (1) λ ( z ) − Y (cid:107) + µ (cid:107) u R (cid:107) + p tr f ( z, v R ) . The resulting BVP reads: d z ∗ R,i d t = f i (cid:16) z ∗ R , v ∗ R (cid:17) , t ∈ (0 , T ] , (7.33a) d p ∗ R,i d t = (cid:101) g i ( z ∗ R , p ∗ R ) , t ∈ (0 , T ] , (7.33b) z ∗ R,i (0) = y ,i , p ∗ R,i ( T ) = 0 , i = 1 , · · · , m, (7.33c)where f ( z, v R ) denotes the RHS of (7.19), v ∗ R is still given by (7.30), but in contrast to g i given by(5.30), the components (cid:101) g i of the vector field involved in the RHS of the p -equations of (7.33), are now INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 47 given by(7.34) (cid:101) g i ( z, p ) := − ∂ (cid:101) H∂z i = − ( z i − Y i ) − m (cid:88) n = m +1 ( h (1) ,nλ ( z ) − Y n ) ∂h (1) ,nλ ( z ) ∂z i − m (cid:88) j =1 p j ∂f j ( z, v R ) ∂z i , i = 1 , · · · , m. Once the above BVP (7.33) is solved, we take u ∗ R given by (7.29) with p ∗ R obtained from (7.33) asthe h (1) λ -based suboptimal controller for the optimal control problem (7.5).7.3. Control performances: Numerical results.
To assess the ability of the h (1) λ -based reducedoptimal control problems (7.9) and (7.10) in synthesizing suboptimal controllers of good performancefor respectively the optimal control problems (7.4) and (7.5), we consider the case where the charac-teristic function χ Ω is supported on the subdomain Ω = [0 . l, . l ], and the target is taken to be thetarget Y used in (5.39) for the experiments of Section 5.5. As pointed out prior to Section 7.1, toachieve performances comparable to those achieved in Section 5.5, it turned out that four-dimensional h (1) λ -based reduced systems were required for the design of suboptimal controllers, instead of the two-dimensional reduced systems of Section 5.5. As explained above, this increase of the dimension of theresolved subspace H c results from the spatial localization of the controller dealt with here. Figure 7.
Left panel : The PDE solution field driven by the suboptimal controller u ∗ R synthesizedby solving the h (1) λ -based reduced problem (7.9). Right panel : The suboptimal controller u ∗ R subjectto the action of χ Ω . The support of the characteristic function χ Ω is taken to be Ω = [0 . l, . l ]. H c istaken to be spanned by the first four leading eigenmodes ( m = 4); the target Y is given by (5.39); andthe initial datum is 0 . y + . The parameters are l = 1 . π , λ = 7 λ c , ν = 0 . γ = 2 .
5, and the final timeis T = 3. The parameters µ and µ in the cost functional (7.2) are taken to be µ = 1 and µ = 20. Figures 7 and 8 show the performances achieved by the resulting four-dimensional h (1) λ -based sub-optimal controllers, corresponding to the cost functional of terminal-payoff type (7.2). The left panelof Fig. 7 shows the PDE solution field driven by the corresponding suboptimal controller field shownon the right panel of the same figure. The left panel of Fig. 8 shows the final-time solution profile,while the right panel shows the corresponding parameterization defect associated with h (1) λ . The cor-responding cost value and relative L -error of the final time solution profile compared with the target x Target pro fi le and controlled PDE solution y ( T ; u ∗ R ) Y t Parameterization defects of h (1) λ Figure 8.
Final time solution profile of the PDE driven by χ Ω u ∗ R compared with the target Y isgiven by (5.39) ( left panel ); and the parameterization defects associated with the finite-horizon PM, h (1) λ , given by (7.14) for m = 4 ( right panel ). Parameters are the same as in Fig. 7. are given by J TP ( y ( · ; y , u ∗ R ) , u ∗ R ) = 1 . , (cid:107) y ( T ; y , u ∗ R ) − Y (cid:107)(cid:107) Y (cid:107) = 9 . . As a comparison, by using an m -dimensional Galerkin-based reduced system with m = 16 to designsuboptimal solutions to (7.4), the corresponding cost value and relative L -error are given by J TP ( y ( · ; y , (cid:101) u ∗ G ) , (cid:101) u ∗ G ) = 1 . , (cid:107) y ( T ; y , (cid:101) u ∗ G ) − Y (cid:107)(cid:107) Y (cid:107) = 6 . . The above numerical results indicate thus that the 4-dimensional h (1) λ -based reduced problem (7.9)can be used to design a very good suboptimal controller (for the prescribed target Y given by (5.39))for the optimal control problem (7.4) with performance comparable to the (more standard) higher-dimensional Galerkin-based reduced systems. This success goes with the relatively small parameteri-zation defect as well as with the relatively small energy kept in the high-modes (not shown); see rightpanel of Fig. 8. Note that for these experiments, the system parameters are chosen to be l = 1 . π , λ = 7 λ c , ν = 0 . γ = 2 .
5, while the final time is taken to be T = 3. The parameters µ and µ inthe cost functional (7.2) are taken to be µ = 1 and µ = 20. The initial datum is a scaled version ofthe corresponding positive steady state y + of the uncontrolled PDE, namely y = 0 . y + .The performances of the 4-dimensional h (1) λ -based suboptimal controller for (7.10) associated withthe cost functional of tracking type (7.3) are illustrated in Figs. 9 and 10. The experimental conditionsare here chosen to be: l = 1 . π , λ = 3 λ c , ν = 0 . γ = 2 .
5, while the final time is still taken to be T = 3. The parameter µ in the cost functional (7.3) is taken to be µ = 0 .
02 and the initial datumis y = 0 . y + .For these experiments, the corresponding cost value and relative L -error are given by J track ( y ( · ; y , u ∗ R ) , u ∗ R ) = 0 . , (cid:107) y ( T ; y , u ∗ R ) − Y (cid:107)(cid:107) Y (cid:107) = 12 . . For a high-dimensional Galerkin-based reduced problem with m = 16, the corresponding cost valueand relative L -error are given by J track ( y ( · ; y , (cid:101) u ∗ G ) , (cid:101) u ∗ G ) = 0 . , (cid:107) y ( T ; y , (cid:101) u ∗ G ) − Y (cid:107)(cid:107) Y (cid:107) = 10 . . Here again, a fairly good performance of the suboptimal controller as synthesized by solvingthe 4-dimensional h (1) λ -based reduced problem (7.10), is achieved. Due to the deterioration of the for the optimal control (7.5). INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 49 parameterization defect of h (1) λ that can be observed by comparing the right panel of Fig. 10 withthe right panel of Fig. 8, the error estimate (4.10) suggests that such a success has to come with anoticeable reduction of the energy contained in the high modes of the PDE solution driven by thesuboptimal controller synthesized for (7.10) compared to the PDE solution driven by the suboptimalcontroller synthesized for (7.9). Such theoretical prediction based on Corollary 4.2 can actually beempirically confirmed by looking at the numerical values of these high-mode energies (not shown).Finally, it is worth mentioning that similar to the globally distributed case, the performances ofthe h (1) λ -based reduced systems and the associated parameterization defects of h (1) λ depend on thetarget and the length of the time horizon; cf. Figs. 2, 5 and 6. The dependence on the PDE initialdatum turned out also to be an important factor. In particular, it has been observed that for bothproblems (7.4) and (7.5) the parameterization defects deteriorate when the scaling factors δ used in theconstruction of the initial datum y = δy + increases. Based on the results of Section 6 for the globallydistributed case, it can be reasonably expected that PM functions such as h (2) λ that bring higher-orderterms compared to h (1) λ (cf. Theorem 6.1) can allow to reach better performance for a broader range ofinitial data and target profiles; the parameterization defects being reasonably expected to get smaller. Figure 9.
Left panel : The PDE solution field driven by the suboptimal controller u ∗ R syn-thesized by solving the h (1) λ -based reduced problem (7.10). Right panel : The suboptimal con-troller subject to the action of χ Ω . The support of the characteristic function χ Ω is taken to beΩ = [0 . l, . l ].The resolved modes are taken to be the first four leading eigenmodes ( m = 4), the tar-get is Y = − . (cid:104) y − , e (cid:105) e + 1 . (cid:104) y − , e (cid:105) e and the initial datum is 0 . y + . The parameters are l = 1 . π , λ = 3 λ c , ν = 0 . γ = 2 .
5, and the final time is T = 3. The parameter µ in the cost functional (7.3)is taken to be µ = 0 . Acknowledgments
We are grateful to Monique Chyba and to Bernard Bonnard for their interest in our works onparameterizing manifolds, which led the authors to propose this article. MDC is also grateful to DenisRousseau and Michael Ghil for the unique environment they provided to complete this work, at theCERES-ERTI, ´Ecole Normale Sup´erieure, Paris. This work has been partly supported by the NationalScience Foundation grant DMS-1049253 and Office of Naval Research grant N00014-12-1-0911. x Target pro fi le and controlled PDE solution y ( T ; u ∗ R ) Y t Parameterization defects of h (1) λ Figure 10.
Final time solution profile of the PDE driven by χ Ω u ∗ R compared with the target Y isgiven by (5.39) ( left panel ); and the parameterization defects associated with the finite-horizon PM, h (1) λ , given by (7.14) for m = 4 ( right panel ). Parameters are the same as in Fig. 9. Appendix A. Suboptimal Controller Synthesis Based on Galerkin Projections andPontryagin Maximum Principle
To assess the performance of the PM-based reduced systems considered in Sections 5 and 6 in syn-thesizing suboptimal controllers in the context of a Burgers-type equation, we derive in this appendixsuboptimal control problems associated with the globally distributed optimal control problem (5.9)based on Galerkin approximations. Section A.1 concerns a two-mode Galerkin approximation; andSection A.2 deals with the more general m -dimensional case. The former serves as a basis of compar-ison to analyze the performance achieved by the PM-based approach, while the latter can in principleprovide a good indication of the true optimal controller of the underlying optimal control problems bytaking the dimension sufficiently large. Results for the general m -dimensional case will also be usedin Section 7 to derive Galerkin-based reduced systems for the locally distributed problems (7.4) and(7.5).A.1. Suboptimal controller based on a 2D Galerkin reduced optimal problem.
We firstpresent the reduced optimal control problem based on a two-mode Galerkin approximation of theunderlying PDE (5.1), which can be derived by simply setting h (1) λ in (5.18)–(5.17) to zero. Thecorresponding operational forms for the cost functional and reduced system for the low modes canbe obtained from (5.24)–(5.27) by setting α ( λ ) and α ( λ ) to be zero. The resulting cost functionalreads:(A.1) J G ( v, u G ) = (cid:90) T (cid:2) G G ( v ( t )) + E ( u G ( t )) (cid:3) d t + C T ( v ( T ) , P c Y ) , where v = v e + v e ∈ L (0 , T ; H c ) is the state variable, u G = u G, e + u G, e ∈ L (0 , T ; H c ) is thecontrol, C T is the terminal payoff term defined by (5.26), and(A.2) G G ( v ) := 12 (cid:107) v (cid:107) = 12 [( v ) + ( v ) ] , E ( u G ) := µ (cid:107) u G (cid:107) = µ u G, ) + ( u G, ) ] . The equations for v and v are given by:(A.3) d v d t = β ( λ ) v + αv v + a u G, ( t ) + a u G, ( t ) , d v d t = β ( λ ) v − α ( v ) + a u G, ( t ) + a u G, ( t ) , which is subjected to the initial conditions:(A.4) v (0) = (cid:104) y , e (cid:105) , v (0) = (cid:104) y , e (cid:105) , INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 51 where α = γπ √ l / .The corresponding Galerkin-based reduced optimal control problem for (5.9) reads:(A.5) min J G ( v, u G ) s.t. ( v, u G ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (A.3)–(A.4) . It follows again from the Pontryagin maximum principle that for a given pair( v ∗ G , u ∗ G ) ∈ L (0 , T ; H c ) × L (0 , T ; H c )to be optimal for the problem (A.5), it must satisfy the following conditions:d v ∗ G, d t = β ( λ ) v ∗ G, + αv ∗ G, v ∗ G, + a u ∗ G, ( t ) + a u ∗ G, ( t ) , (A.6a) d v ∗ G, d t = β ( λ ) v ∗ G, − α ( v ∗ G, ) + + a u ∗ G, ( t ) + a u ∗ G, ( t ) , (A.6b) d p ∗ G, d t = − v ∗ G, − β ( λ ) p ∗ G, − αp ∗ G, v ∗ G, + 2 αp ∗ G, v ∗ G, , (A.6c) d p ∗ G, d t = − v ∗ G, − β ( λ ) p ∗ G, − αp ∗ G, v ∗ G, , (A.6d) ( u ∗ G, , u ∗ G, ) tr = − (cid:16) a p ∗ G, ( t ) + a p ∗ G, ( t ) µ , a p ∗ G, ( t ) + a p ∗ G, ( t ) µ (cid:17) tr = − µ M tr p ∗ G , (A.6e)where v ∗ G, = (cid:104) v ∗ G , e i (cid:105) , u ∗ G,i = (cid:104) u ∗ G , e i (cid:105) , i = 1 ,
2, and p ∗ G = p ∗ G, e + p ∗ G, e denotes the costate associatedwith v ∗ G .Thanks to (A.6e), we can express the controller u ∗ G,i in (A.6a)–(A.6b) in terms of the costate p ∗ G,i ,leading thus to the following BVP for v ∗ G and p ∗ G :(A.7) d v d t = β ( λ ) v + αv v + f ( p , p ) , d v d t = β ( λ ) v − α ( v ) + f ( p , p ) , d p d t = − v − β ( λ ) p − αp v + 2 αp v , d p d t = − v − β ( λ ) p − αp v , subject to the boundary condition(A.8) v (0) = (cid:104) y , e (cid:105) , v (0) = (cid:104) y , e (cid:105) , p ( T ) = µ ( v ( T ) − Y ) , p ( T ) = µ ( v ( T ) − Y ) , where f and f are defined by (5.33), and the boundary condition for the costate is derived in thesame way as in (5.34) thanks to the Pontryagin maximum principle. Once this BVP is solved, thecorresponding controller u ∗ G is determined by (A.6e) which provides the unique optimal controllerfor the Galerkin-based reduced optimal control problem (A.5), due again to the fact that the costfunctional (A.1) is quadratic in u G and the dependence on the controller is affine for the system ofequations (A.3); see e.g. [66, Sect. 5.3] and [98]. Note also that analogous results to those presentedin Lemma 5.2 hold for the reduced optimal control problem (A.5) as well.A.2. Suboptimal controller based on an m -dimensional Galerkin reduced optimal prob-lem. We derive now a more general reduced optimal control problem based on higher-dimensionalGalerkin approximation, where the subspace H c is taken to be spanned by the first m eigenmodes:(A.9) H c := span { e , · · · , e m } . The main interest is that by choosing m sufficiently large, such a reduced problem can serve in principleto provide a good estimate of the true optimal controllers of the globally distributed optimal controlproblem (5.9), which can be taken then as a benchmark for the numerical experiments reported inSections 5 and 6. Analogous reduced problems associated with the locally distributed cases (7.4) and(7.5) considered in Section 7 can be derived in the same way (and actually the corresponding resultsare the same as those presented in Section 7.2 by setting h (1) λ therein to be zero).The Galerkin-based reduced optimal control problem (A.5) when generalized to the case with m controlled modes reads:(A.10) min (cid:101) J G ( v, (cid:101) u G ) s.t. ( v, (cid:101) u G ) ∈ L (0 , T ; H c ) × L (0 , T ; H c ) solves (A.11)–(A.12) below , where H c is the m -dimensional reduced phase space defined in (A.9), and (cid:101) J G ( v, (cid:101) u G ) = (cid:90) T (cid:2) m (cid:88) i =1 ( v i ) + µ m (cid:88) i =1 ( (cid:101) u G,i ) (cid:3) d t + µ m (cid:88) i =1 | v i ( T ) − Y i | . The system of equations that v ( · ; (cid:101) u G ) satisfies is given by:(A.11) d v i d t = β i ( λ ) v i + (cid:68) B (cid:16) m (cid:88) i =1 v i e i , m (cid:88) i =1 v i e i (cid:17) , e i (cid:69) + [ M tr (cid:101) u G ( t )] i , i = 1 , · · · , m, which is subjected to the initial conditions:(A.12) v i (0) = (cid:104) y , e i (cid:105) , i = 1 , · · · , m, where the matrix M m × m is the representation of the linear operator P c C under the basis e , · · · , e m , i.e. the elements of M are given by a ij = (cid:104) C e i , e j (cid:105) (see (5.16) for the case m = 2) and [ M tr (cid:101) u G ( t )] i denotes the i th -component of the vector M tr (cid:101) u G ( t ).As before, by using the Pontryagin maximum principle, we can derive the following BVP to besatisfied by any optimal pair ( v ∗ G , (cid:101) u ∗ G ) of (A.10):d v i d t = β i ( λ ) v i + iα (cid:16) − (cid:98) i/ (cid:99) (cid:88) j =1 ω i,j v j v i − j + m (cid:88) j = i +1 v j v j − i (cid:17) − µ [ M tr M p ] i , i = 1 , · · · , m, (A.13a) d p i d t = − v i − m (cid:88) j =1 p j ∂f j ( v, p ) ∂v i , i = 1 , · · · , m, (A.13b) v i (0) = y ,i , p i ( T ) = µ ( v i ( T ) − Y i ) , i = 1 , · · · , m, (A.13c)where the optimal controller (cid:101) u ∗ G is related to the corresponding costate p ∗ G by(A.14) (cid:101) u ∗ G = − µ M p ∗ G , see (A.6e) for the case m = 2. Here, f i , i = 1 , · · · , m , denotes the RHS of (A.13a) and we have usedthe nonlinear interactions (5.20) to derive the quadratic parts of f i . The formula for ∂f j ( v,p ) ∂v i is givenby:(A.15) ∂f j ( v, p ) ∂v i = β j ( λ ) δ ij + jαI j,i , INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 53 where δ ij denotes the Kronecker delta, and(A.16) I j,i = ∂∂v i (cid:16) − (cid:98) j/ (cid:99) (cid:88) k =1 ω j,k v k v j − k + m (cid:88) k = j +1 v k v k − j (cid:17) = v i − j , if i > j,v i + j , if i = j and i + j ≤ m,v i + j − v j − i , if i < j and i + j ≤ m, − v j − i , if i < j and i + j > m, , otherwise;with (cid:98) x (cid:99) being the largest integer less than x and the coefficients ω i,j given by ω i,j := (cid:40) , if i is odd, or if i is even and j (cid:54) = i/ / , if i is even and j = i/ . Appendix B. Global Well-posedness for the Two-dimensional h (1) λ -based ReducedSystem (5.27)In this appendix, we show that for any given initial datum and any fixed T >
0, the h (1) λ -basedreduced system (5.27) admits a unique mild solution in the space C ([0 , T ]; R ). The result followsfrom classical ODE theory [2] once we can establish a priori bounds for the solution ( z ( t ) , z ( t )).Similar (but more tedious) estimates can be used to deal with the Cauchy problem associated withthe h (2) λ -based reduced system (6.17) derived in Section 6 and the more general m -dimensional h (1) λ -based reduced system (7.19) encountered in Section 7.Let us first recall that the two-dimensional h (1) λ -based reduced system is given by:d z d t = β ( λ ) z + α [ z z + α ( λ ) z z + α ( λ ) α ( λ ) z z ] + a u R, ( t ) + a u R, ( t ) , (B.1a) d z d t = β ( λ ) z + α [ − z + 2 α ( λ ) z z + 2 α ( λ ) z ] + a u R, ( t ) + a u R, ( t ) , (B.1b)where u R ( · ) := u R, ( · ) e + u R, ( · ) e ∈ L (0 , T ; H c ) with T > α ( λ )and α ( λ ) are defined in (5.23), α = γπ √ l / , and a ij , 1 ≤ i, j ≤
2, are elements of the coefficientsmatrix M associated with the operator C ; see (5.15)–(5.16).We check below by energy estimates that no finite time blow-up can occur for solutions to thesystem (B.1) emanating from any initial datum ( z , , z , ) ∈ R . For this purpose, let us define R := max (cid:40) | z , | , α | αα ( λ ) | , (cid:115) | β ( λ ) || αα ( λ ) | (cid:41) and C := (cid:90) T | a u R, ( t ) + a u R, ( t ) | d t. We claim that(B.2) | z ( t ) | ≤ e C/R R ∀ t ∈ [0 , T ] . It is clear that we only need to deal with those values of t such that | z ( t ) | > R . Assume that thereexists such time instances, otherwise we are done. Let us fix an arbitrary interval [ t ∗ , t ∗ ] ⊂ [0 , T ] suchthat(B.3) | z ( t ) | ≥ R ∀ t ∈ [ t ∗ , t ∗ ] . For any
T >
0, a given continuous function z : [0 , T ] → R is called a mild solution to the reduced system (5.27)if it satisfies the corresponding integral form of the system: z ( t ) = z (0) + (cid:82) t F ( s, z ( s )) d s , for all t ∈ [0 , T ], where z := ( z , z ) tr and F denotes the RHS of (5.27). Since R ≥ | z , | and z depends continuously on t , we can reduce t ∗ such that z ( t ∗ ) = R while thecondition (B.3) remains true.Now by multiplying z ( t ) on both sides of (B.1b), we obtain(B.4) 12 d[( z ) ]d t = c ( t )( z ) , ∀ t ∈ [ t ∗ , t ∗ ] , where c ( t ) := (cid:16) β ( λ ) − α ( z ) z + 2 αα ( λ )( z ) + 2 αα ( λ )( z ) + a u R, ( t ) + a u R, ( t ) z (cid:17) . It follows then that(B.5) [ z ( t ∗ )] = e (cid:82) t ∗ t ∗ c ( t )d t [ z ( t ∗ )] . Since | z ( t ) | ≥ R for all t ∈ [ t ∗ , t ∗ ] by the choices of t ∗ and t ∗ , we get (cid:90) t ∗ t ∗ c ( t ) d t ≤ β ( λ )( t ∗ − t ∗ )+ (cid:90) t ∗ t ∗ [ αR +2 αα ( λ )]( z ) d t +2 αα ( λ ) R ( t ∗ − t ∗ )+ (cid:82) t ∗ t ∗ | a u R, ( t ) + a u R, ( t ) | d tR , where we have used | − αz | ≤ αR and 2 αα ( λ )( z ) ≤ αα ( λ ) R , which follow from the definition of R and the fact that α > α ( λ ) < R and the facts that α > α ( λ ) < α ( λ ) <
0, we get αR + 2 αα ( λ ) ≤ β ( λ )( t ∗ − t ∗ ) + 2 αα ( λ ) R ( t ∗ − t ∗ ) ≤ . We obtain then (cid:90) t ∗ t ∗ c ( t )d t ≤ (cid:82) t ∗ t ∗ | a u R, ( t ) + a u R, ( t ) | d tR ≤ CR .
By reporting the above estimate in (B.5) and using | z ( t ∗ ) | = R , we obtain | z ( t ∗ ) | ≤ e C/R | z ( t ∗ ) | = e C/R R, and (B.2) is thus proven.Note also that by multiplying z ( t ) on both sides of (B.1a), we obtain for any t ∈ [0 , T ] at which z ( t ) (cid:54) = 0 that(B.6) 12 d[( z ) ]d t = ( z ) (cid:16) β ( λ ) + αz + αα ( λ )( z ) + αα ( λ ) α ( λ )( z ) + a u R, ( t ) + a u R, ( t ) z (cid:17) . It follows then from the boundedness of z and (B.6) that z can grow at most exponentially. Conse-quently, no finite time blow-up can occur for the h (1) λ -based reduced system (B.1). References [1] Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynamics ,303–325 (1990)[2] Amann, H.: Ordinary Differential Equations: An Introduction to Nonlinear Analysis, de Gruyter Studies in Math-ematics , vol. 13. Walter de Gruyter & Co. (1990)[3] Armaou, A., Christofides, P.D.: Feedback control of the Kuramoto-Sivashinsky equation. Physica D (1-2),49–61 (2000)[4] Armaou, A., Christofides, P.D.: Dynamic optimization of dissipative PDE systems using nonlinear order reduction.Chemical Engineering Science (24), 5083–5114 (2002)[5] Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for OrdinaryDifferential Equations, Classics in Applied Mathematics , vol. 13. SIAM, Philadelphia, PA (1995)[6] Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolicequations. Mathematical and Computer Modelling , 1–19 (2001) INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 55 [7] Baker, J., Armaou, A., Christofides, P.D.: Nonlinear control of incompressible fluid flow: Application to Burgers’equation and 2D channel flow. Journal of Mathematical Analysis and Applications , 230–255 (2000)[8] Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations.Springer (2008)[9] Beeler, S.C., Tran, H.T., Banks, H.T.: Feedback control methodologies for nonlinear systems. Journal of Opti-mization Theory and Applications (1), 1–33 (2000)[10] Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite DimensionalSystems. Springer (2007)[11] Berestycki, H., Kamin, S., Sivashinsky, G.: Metastability in a flame front evolution equation. Interfaces and FreeBoundaries (4), 361–392 (2001)[12] Bergmann, M., Cordier, L.: Optimal control of the cylinder wake in the laminar regime by trust-region methodsand pod reduced-order models. Journal of Computational Physics (16), 7813–7840 (2008)[13] Betts, J.T.: Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics (2), 193–207 (1998)[14] Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Advances inDesign and Control , vol. 19, second edn. SIAM, Philadelphia, PA (2010)[15] Bewley, T.R., Moin, P., Temam, R.: DNS-based predictive control of turbulence: an optimal benchmark forfeedback algorithms. Journal of Fluid Mechanics , 179–225 (2001)[16] Bewley, T.R., Temam, R., Ziane, M.: A general framework for robust control in fluid mechanics. Physica D (3),360–392 (2000)[17] Bonnans, F.J., Martinon, P., Gr´elard, V.: Bocop - A collection of examples. Tech. rep., INRIA (2012). URL http://hal.inria.fr/hal-00726992 . RR-8053[18] Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory,
Math´ematiques & Applications(Berlin) , vol. 40. Springer (2003)[19] Bonnard, B., Faubourg, L., Tr´elat, E.: M´ecanique C´eleste et Contrˆole des V´ehicules Spatiaux,
Math´ematiques &Applications (Berlin) , vol. 51. Springer-Verlag (2006)[20] Boscain, U., Piccoli, B.: Optimal Syntheses for Control Systems on 2-D Manifolds,
Math´ematiques & Applications(Berlin) , vol. 43. Springer (2004)[21] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, NewYork (2011)[22] Brunovsk´y, P.: Controlling the dynamics of scalar reaction diffusion equations by finite-dimensional controllers. In:Modelling and Inverse Problems of Control for Distributed Parameter Systems (Laxenburg, 1989),
Lecture Notesin Control and Inform. Sci. , vol. 154, pp. 22–27. Springer, Berlin (1991)[23] Bryson Jr., A.E., Ho, Y.C.: Applied Optimal Control. Hemisphere Publishing Corp. Washington, D. C. (1975)[24] Cannarsa, P., Tessitore, M.E.: Infinite-dimensional Hamilton-Jacobi equations and Dirichlet boundary controlproblems of parabolic type. SIAM Journal on Control and Optimization (6), 1831–1847 (1996)[25] Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractors for Infinite-Dimensional Non-autonomous DynamicalSystems, Applied Mathematical Sciences , vol. 182. Springer, New York (2013)[26] Chekroun, M.D., Liu, H., Wang, S.: On stochastic parameterizing manifolds: Pullback characterization and non-Markovian reduced equations. Preprint http://arxiv.org/pdf/1310.3896v1.pdf (2013)[27] Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: Random attractors and time-dependentinvariant measures. Physica D (21), 1685–1700 (2011)[28] Chen, C.C., Chang, H.C.: Accelerated disturbance damping of an unknown distributed system by nonlinearfeedback. AIChE Journal (9), 1461–1476 (1992)[29] Choi, H., Temam, R., Moin, P., Kim, J.: Feedback control for unsteady flow and its application to the stochasticBurgers equation. J. Fluid Mech. , 509–543 (1993)[30] Christofides, P.D., Armaou, A., Lou, Y., Varshney, A.: Control and Optimization of Multiscale Process Systems.Springer (2008)[31] Christofides, P.D., Daoutidis, P.: Nonlinear control of diffusion-convection-reaction processes. Computers & Chem-ical Engineering , S1071–S1076 (1996)[32] Christofides, P.D., Daoutidis, P.: Finite-dimensional control of parabolic PDE systems using approximate inertialmanifolds. J. Math. Anal. Appl. (2), 398–420 (1997)[33] Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integral Manifolds and Inertial Manifolds for DissipativePartial Differential Equations, Applied Mathematical Sciences , vol. 70. Springer-Verlag, New York (1989)[34] Crandall, M.G., Ishii, H., Lions, P.L.: Users guide to viscosity solutions of second order partial differential equations.Bulletin of the American Mathematical Society (1), 1–67 (1992) [35] Da Prato, G., Debussche, A.: Dynamic programming for the stochastic Burgers equation. Annali di MatematicaPura ed Applicata (1), 143–174 (2000)[36] Da Prato, G., Debussche, A.: Dynamic programming for the stochastic Navier-Stokes equations. MathematicalModelling and Numerical Analysis , 459–475 (2000)[37] Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces, vol. 293. CambridgeUniversity Press (2002)[38] Dacorogna, B.: Direct Methods in the Calculus of Variations, vol. 78. Springer (2007)[39] Ded`e, L.: Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal controlproblems. SIAM Journal on Scientific Computing , 997–1019 (2010)[40] Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics , vol. 19. American MathematicalSociety, Providence, RI (2010)[41] Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. Mat. Res. Soc. Symp.Proceedings , 39–46 (1998)[42] Fattorini, H.O.: Boundary control systems. SIAM J. Control (3), 349–385 (1968)[43] Fattorini, H.O.: Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and itsApplications , vol. 62. Cambridge University Press (1999)[44] Flandoli, F.: Riccati equation arising in a boundary control problem with distributed parameters. SIAM J. Controland Optimization (1), 76–86 (1984)[45] Foias, C., Manley, O., Temam, R.: Modelling of the interaction of small and large eddies in two-dimensionalturbulent flows. RAIRO Mod´el. Math. Anal. Num´er. (1), 93–118 (1988)[46] Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations (2), 309–353 (1988)[47] Franke, T., Hoppe, R.H.W., Linsenmann, C., Wixforth, A.: Projection based model reduction for optimal designof the time-dependent Stokes system. In: Constrained Optimization and Optimal Control for Partial DifferentialEquations, pp. 75–98. Springer (2012)[48] Fursikov, A.V.: Optimal Control of Distributed Systems: Theory and Applications, Translations of MathematicalMonographs , vol. 187. American Mathematical Society (2000)[49] Grepl, M.A., K¨archer, M.: Reduced basis a posteriori error bounds for parametrized linear-quadratic ellipticoptimal control problems. C. R. Acad. Sci. Paris, Ser. I (15), 873–877 (2011)[50] Gunzburger, M.: Adjoint equation-based methods for control problems in incompressible, viscous flows. Flow,Turbulence and Combustion (3-4), 249–272 (2000)[51] Gunzburger, M.D.: Sensitivities, adjoints and flow optimization. International Journal for Numerical Methods inFluids (1), 53–78 (1999)[52] Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics , vol. 840. Springer-Verlag, Berlin (1981)[53] Hinze, M., Kunisch, K.: On suboptimal control strategies for the Navier-Stokes equations. In: ESAIM: Proceedings,vol. 4, pp. 181–198 (1998)[54] Hinze, M., Kunisch, K.: Three control methods for time-dependent fluid flow. Flow, Turbulence and Combustion , 273–298 (2000)[55] Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Mathematical Modelling:Theory and Applications , vol. 23. Springer (2009)[56] Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems:error estimates and suboptimal control. In: Dimension Reduction of Large-Scale Systems,
Lect. Notes Comput.Sci. Eng. , vol. 45, pp. 261–306. Springer, Berlin (2005)[57] Holmes, P., Lumley, J.L., Berkooz, G., Rowley, C.W.: Turbulence, Coherent Structures, Dynamical Systems andSymmetry, second edn. Cambridge University Press, Cambridge (2012)[58] Hsia, C.H., Wang, X.: On a Burgers’ type equation. Discrete Contin. Dyn. Syst., Ser. B (5), 1121–1139 (2006)[59] Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications, vol. 15. SIAM(2008)[60] Ito, K., Kunisch, K.: Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynam-ical systems. SIAM J. Numer. Anal. (6), 2867–2891 (2008)[61] Ito, K., Ravindran, S.: Optimal control of thermally convected fluid flows. SIAM Journal on Scientific Computing (6), 1847–1869 (1998)[62] Ito, K., Ravindran, S.S.: Reduced basis method for optimal control of unsteady viscous flows. International Journalof Computational Fluid Dynamics (2), 97–113 (2001) INITE-HORIZON PARAMETERIZING MANIFOLDS AND SUBOPTIMAL CONTROL OF NONLINEAR PDES 57 [63] Ito, K., Schroeter, J.D.: Reduced order feedback synthesis for viscous incompressible flows. Mathematical andComputer Modelling , 173–192 (2001)[64] Keller, H.B.: Numerical Solution of Two Point Boundary Value Problems, Regional Conference Series in AppliedMathematics , vol. 24. SIAM (1976)[65] Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and the Maltab PSE. ACM Transactionson Mathematical Software (3), 299–316 (2001)[66] Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications (2012)[67] Knowles, G.: An Introduction to Applied Optimal Control, Mathematics in Science and Engineering , vol. 159.Academic Press Inc., New York (1981)[68] Kokotovi´c, P., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design,
Classicsin Applied Mathematics , vol. 25. SIAM (1999)[69] Kokotovic, P., O’Malley Jr., R., Sannuti, P.: Singular perturbations and order reduction in control theoryanoverview. Automatica (2), 123–132 (1976)[70] Kokotovic, P.V.: Applications of singular perturbation techniques to control problems. SIAM review (4), 501–550(1984)[71] Kokotovic, P.V., Sannuti, P.: Singular perturbation method for reducing the model order in optimal control design.Automatic Control, IEEE Transactions on (4), 377–384 (1968)[72] Krstic, M., Magnis, L., Vazquez, R.: Nonlinear control of the viscous burgers equation: Trajectory generation,tracking, and observer design. Journal of Dynamic Systems, Measurement, and Control (2), 021,012 (2009)[73] Kunisch, K., Volkwein, S.: Control of the Burgers’ equation by a reduced-order approach using proper orthogonaldecomposition. J. Optim. Theory and Appl. , 345–371 (1999)[74] Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluiddynamics. SIAM J. Numer. Anal. , 492–515 (2002)[75] Kunisch, K., Volkwein, S., Xie, L.: HJB-POD-based feedback design for the optimal control of evolution problems.SIAM Journal on Applied Dynamical Systems (4), 701–722 (2004)[76] Lebiedz, D., Rehberg, M.: A numerical slow manifold approach to model reduction for optimal control of multipletime scale ODE. arXiv preprint arXiv:1302.1759 (2013)[77] Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971)[78] Lions, J.L.: Some Aspects of the Optimal Control of Distributed Parameter Systems. SIAM (1972)[79] Lions, J.L.: Perturbations Singuli`eres dans les Probl`emes aux Limites et en Contrˆole Optimal, Lecture Notes inMathematics , vol. 323. Springer (1973)[80] Lions, J.L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. (1), 1–68(1988)[81] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh¨auser (1995)[82] Ly, H.V., Tran, H.T.: Modeling and control of physical processes using proper orthogonal decomposition. Mathe-matical and computer modelling , 223–236 (2001)[83] Ma, T., Wang, S.: Phase Transition Dynamics. Springer (2014)[84] Medjo, T.T., Tebou, L.T.: Adjoint-based iterative method for robust control problems in fluid mechanics. SIAMJ. Numer. Anal. (1), 302–325 (2004)[85] Medjo, T.T., Temam, R., Ziane, M.: Optimal and robust control of fluid flows: some theoretical and computationalaspects. Applied Mechanics Reviews (1), 010,802 (2008)[86] Motte, I., Campion, G.: A slow manifold approach for the control of mobile robots not satisfying the kinematicconstraints. Robotics and Automation, IEEE Transactions on (6), 875–880 (2000)[87] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of OptimalProcesses. Translated by D. E. Brown. A Pergamon Press Book. The Macmillan Co., New York (1964)[88] Ravindran, S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition.International journal for numerical methods in fluids (5), 425–448 (2000)[89] Ravindran, S.S.: Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decompo-sition. SIAM Journal on Scientific Computing (6), 1924–1942 (2002)[90] Roberts, S.M., Shipman, J.S.: Two-point boundary value problems: shooting methods. American Elsevier Pub-lishing Co., Inc., New York (1972)[91] Rosa, R.: Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation. J. Dynam. Differential Equations (1), 61–86 (2003)[92] Rosa, R., Temam, R.: Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertialmanifold theory. In: Foundations of Computational Mathematics (Rio de Janeiro, 1997), pp. 382–391. Springer,Berlin (1997) [93] Sano, H., Kunimatsu, N.: An application of inertial manifold theory to boundary stabilization of semilinear diffusionsystems. J. Math. Anal. Appl. (1), 18–42 (1995)[94] Sch¨attler, H., Ledzewicz, U.: Geometric Optimal Control: Theory, Methods and Examples, InterdisciplinaryApplied Mathematics , vol. 38. Springer, New York (2012)[95] Shvartsman, S.Y., Kevrekidis, I.G.: Nonlinear model reduction for control of distributed systems: A computer-assisted study. AIChE Journal (7), 1579–1595 (1998)[96] Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. American Mathematical Soc. (1984)[97] Temam, R.: Inertial manifolds. The Mathematical Intelligencer (4), 68–74 (1990)[98] Tr´elat, E.: Optimal control and applications to aerospace: some results and challenges. J. Optim. Theory Appl. (3), 713–758 (2012)[99] Tr¨oltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, GraduateStudies in Mathematics , vol. 112. American Mathematical Society (2010)[100] Tr¨oltzsch, F., Volkwein, S.: POD a posteriori error estimates for linear-quadratic optimal control problems. Com-put. Optim. Appl. , 83–115 (2009)[101] Volkwein, S.: Distributed control problems for the Burgers equation. Computational Optimization and Applications (2), 115–140 (2001)[102] W¨achter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scalenonlinear programming. Mathematical Programming (1), 25–57 (2006)(MC) Department of Mathematics, University of Hawai‘i at Manoa, Honolulu, HI 96822, USA, andDepartment of Atmospheric & Oceanic Sciences, University of California, Los Angeles, CA 90095-1565,USA
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