On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems
OON THE TURNPIKE PROPERTY AND THE RECEDING-HORIZONMETHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROLPROBLEMS ∗ TOBIAS BREITEN † AND
LAURENT PFEIFFER ‡ Abstract.
Optimal control problems with a very large time horizon can be tackled with theReceding Horizon Control (RHC) method, which consists in solving a sequence of optimal controlproblems with small prediction horizon. The main result of this article is the proof of the exponentialconvergence (with respect to the prediction horizon) of the control generated by the RHC methodtowards the exact solution of the problem. The result is established for a class of infinite-dimensionallinear-quadratic optimal control problems with time-independent dynamics and integral cost. Suchproblems satisfy the turnpike property: the optimal trajectory remains most of the time very close tothe solution to the associated static optimization problem. Specific terminal cost functions, derivedfrom the Lagrange multiplier associated with the static optimization problem, are employed in theimplementation of the RHC method.
Key words.
Receding horizon control, model predictive control, value function, optimalitysystems, Riccati equation, turnpike property.
AMS subject classifications.
1. Introduction.1.1. Context.
We consider in this article the following class of linear-quadraticoptimal control problems:( P ) inf y ∈ W (0 , ¯ T ) u ∈ L (0 , ¯ T ; U ) J ¯ T ,Q,q ( u, y ) := (cid:90) ¯ T (cid:96) ( y ( t ) , u ( t )) d t + (cid:104) y ( ¯ T ) , Qy ( ¯ T ) (cid:105) + (cid:104) q, y ( ¯ T ) (cid:105) , subject to: ˙ y ( t ) = Ay ( t ) + Bu ( t ) + f (cid:5) , y (0) = y , where the integral cost (cid:96) is defined by (cid:96) ( y, u ) = (cid:107) Cy (cid:107) Z + (cid:104) g (cid:5) , y (cid:105) V ∗ ,V + α (cid:107) u (cid:107) U + (cid:104) h (cid:5) , u (cid:105) U . Here V ⊂ Y ⊂ V ∗ is a Gelfand triple of real Hilbert spaces [28, page 147], where theembedding of V into Y is dense, V ∗ denotes the topological dual of V and U, Z denotefurther Hilbert spaces. The operator A : D ( A ) ⊂ Y → Y is the infinitesimal generatorof an analytic C -semigroup e At on Y , B ∈ L ( U, V ∗ ), C ∈ L ( Y, Z ), α > Q ∈ L ( Y )is self-adjoint positive semi-definite and D ( A ) denotes the domain of A . The pairs( A, B ) and (
A, C ) are assumed to be stabilizable and detectable, respectively. Theelements y ∈ Y , f (cid:5) ∈ V ∗ , g (cid:5) ∈ V ∗ , h (cid:5) ∈ U , q ∈ Y are given.The following problem, referred to as static optimization problem (or steady-state optimization problem), has a unique solution ( y (cid:5) , u (cid:5) ) with unique associated Lagrangemultiplier p (cid:5) :(1.1) inf ( y,u ) ∈ V × U (cid:96) ( y, u ) , subject to: Ay + Bu + f (cid:5) = 0 . ∗ Submitted to the editors on November 8, 2018. † Institute of Mathematics and Scientific Computing, University of Graz, Austria([email protected]). ‡ Inria and CMAP (UMR 7641), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris,Route de Saclay, 91128 Palaiseau, France (laurent.pfeiff[email protected]).1 a r X i v : . [ m a t h . O C ] J a n T. BREITEN AND L. PFEIFFER
A particularly important feature of ( P ) is the exponential turnpike property. It statesthat there exist two constants M > λ >
0, independent of ¯ T , such that for all t ∈ [0 , ¯ T ], (cid:107) ¯ y ( t ) − y (cid:5) (cid:107) Y ≤ M (cid:0) e − λt + e − λ ( ¯ T − t ) (cid:1) , where ¯ y denotes the optimal trajectory.The trajectory ¯ y is thus made of three arcs, the first and last one being transient short-time arcs and the middle one a long-time arc, where the trajectory remains close to y (cid:5) . We refer the reader to the books [31, 32], where different turnpike propertiesare established for different kinds of systems. We mention in particular the generalcharacterization of the turnpike phenomenon for linear systems in [32, Section 5.34].For linear-quadratic problems, we mention the articles [9, 12] for discrete-time systemsand the articles [21] and [22] containing results for classes of infinite-dimensionalsystems. We also mention the early reference [1] dealing with a tracking problem.Exponential turnpike properties have been established for non-linear systems in [26]and [25].The aim of this article is to analyze the efficiency of the Receding Horizon Control(RHC) method (also called Model Predictive Control method), that we briefly presenthere, a detailed description can be found in Section 5. We consider an implementationof the method with three parameters: a sampling time τ , a prediction horizon T , anda prescribed number of iterations N . The method generates in a recursive way acontrol u RH and its associated trajectory y RH . At the beginning of iteration n of thealgorithm, u RH and y RH have already been computed on (0 , nτ ). Then, an optimalcontrol problem is solved on the interval ( nτ, nτ + T ), with initial condition y RH ( nτ ),with the same integral cost as in ( P ), but with the following terminal cost function:(1.2) φ ( y ) = (cid:104) p (cid:5) , y (cid:105) Y . The restriction of the solution to ( nτ, ( n + 1) τ ) is then concatenated with ( y RH , u RH ).At iteration N , a last optimal control problem is solved on the interval ( N τ, ¯ T ). Thedefinition (1.2) is actually a particular choice of the terminal cost among a generalclass of linear-quadratic functions. For this specific definition, the main result of thearticle is the following estimate:max (cid:0) (cid:107) y RH − ¯ y (cid:107) W (0 , ¯ T ) , (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ; U ) (cid:1) ≤ M e − λ ( T − τ ) (cid:0) e − λT (cid:107) y − y (cid:5) (cid:107) Y + e − λ ( ¯ T − ( Nτ + T )) (cid:107) ˜ q (cid:107) Y (cid:1) , (1.3)with ˜ q = q − p (cid:5) + Qy (cid:5) . The estimate is proven for sampling times and predictionhorizons satisfying 0 < τ ≤ τ ≤ T ≤ ¯ T . The constants τ > M >
0, and λ > y , f (cid:5) , g (cid:5) , h (cid:5) , q , N , τ , T , and ¯ T . Let us mention that the lower bound τ cannot be chosen arbitrarily small. The idea of taking (cid:104) p (cid:5) , y (cid:105) as a terminal costhas been proposed in the recent article [29] in the context of discrete-time problems.The choice of an appropriate terminal cost function is a key issue in the designof an appropriate RHC scheme. When φ is the exact value function, then the RHCmethod generates the exact solution to the problem, as a consequence of the dynamicprogramming principle. The article will give a (positive) answer to the following ques-tion: Does the RHC algorithm generate an efficient control if a good approximationof the value function is used as terminal cost function? The construction of such anapproximation is here possible thanks to the turnpike property. We will see that thederivative of the value function (with respect to the initial condition), evaluated at y (cid:5) , converges to p (cid:5) as ¯ T − t increases. Roughly speaking, the definition (1.2) is a kindof first-order Taylor approximation of the value function, around y (cid:5) .The RHC method is receiving a tremendous amount of attention and it is fre-quently used in control engineering, in particular because it is computationally easier HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS P ) (restricted to ( nτ, nτ + T )) and with the solution to the optimalcontrol problem with short prediction horizon T . This comparison is realized with thehelp of a priori bounds for linear optimality systems in specific weighted spaces. Theanalysis of the optimality systems is an important part of the present article. Thea priori bounds that we have obtained are of general interest. A classical technique(used in particular in [21, 26]), allowing to decouple the optimality systems, plays animportant role.The article is structured as follows. In Section 2, we prove our error bound inweighted spaces for the optimality systems associated with ( P ). Some additionalproperties on linear optimality systems are provided in Section 3. We formulate thenthe class of linear-quadratic problems to be analyzed in Section 4. The turnpikeproperty and some properties of the value function are then established. Section5 deals with the RHC method and contains our main result (Theorem 5.2). Anextension to infinite-horizon problems is realized in Section 6. Finally, we providenumerical results showing the tightness of our error estimate in Section 7. For T ∈ (0 , ∞ ), we make use of the vector space W (0 , T ) = (cid:8) y ∈ L (0 , T ; V ) | ˙ y ∈ L (0 , T ; V ∗ ) (cid:9) . As it is well-known, W (0 , T ) is continuouslyembedded in C ([0 , T ] , Y ). We can therefore equip it with the following norm: (cid:107) y (cid:107) W (0 ,T ) = max (cid:0) (cid:107) y (cid:107) L (0 ,T ; V ) , (cid:107) ˙ y (cid:107) L (0 ,T ; V ∗ ) , (cid:107) y (cid:107) L ∞ (0 ,T ; Y ) (cid:1) . Weighted spaces.
Let µ ∈ R be given, let T ∈ (0 , ∞ ). We denote by L µ (0 , T ; U )the space of measurable functions u : (0 , T ) → U such that (cid:107) u (cid:107) L µ (0 ,T ; U ) := (cid:107) e µ · u ( · ) (cid:107) L (0 ,T ; U ) = (cid:16) (cid:90) T (cid:107) e µt u ( t ) (cid:107) U d t (cid:17) / < ∞ . Observing that the mapping u ∈ L µ (0 , T ; U ) (cid:55)→ e µ · u ∈ L (0 , T ; U ) is an isometry,we deduce that L µ (0 , T ; U ) is a Banach space. Since e µ · is bounded from above andfrom below by a positive constant, we have that for all measurable u : (0 , T ) → U , u ∈ L (0 , T ; U ) if and only if u ∈ L µ (0 , T ; U ). The spaces L (0 , T ; U ) and L µ (0 , T ; U )are therefore the same vector space, equipped with two different norms. We define ina similar way the space L µ (0 , T ; X ), for a given Hilbert space X . Similarly, we definethe space L ∞ µ (0 , T ; Y ) of measurable mappings from y : (0 , T ) → Y such that (cid:107) y (cid:107) L ∞ µ (0 ,T ; Y ) := (cid:107) e µ · y ( · ) (cid:107) L ∞ (0 ,T ; Y ) < ∞ . We finally define the Banach space W µ (0 , T ) as the space of measurable mappings y : (0 , T ) → V such that e µ · y ∈ W (0 , T ). One can check that for all measurablemappings y : (0 , T ) → V , y ∈ W (0 , T ) if and only if y ∈ W µ (0 , T ). T. BREITEN AND L. PFEIFFER
For T ∈ (0 , ∞ ) and µ ∈ R , we introduce the space(1.4) Λ T,µ = W µ (0 , T ) × L µ (0 , T ; U ) × W µ (0 , T ) , equipped with the norm (cid:107) ( y, u, p ) (cid:107) Λ T,µ = max (cid:0) (cid:107) y (cid:107) W µ (0 ,T ) , (cid:107) u (cid:107) L µ (0 ,T ; U ) , (cid:107) p (cid:107) W µ (0 ,T ) (cid:1) .For T ∈ (0 , ∞ ), we define the space(1.5) Υ T,µ = Y × L µ (0 , T ; V ∗ ) × L µ (0 , T ; V ∗ ) × L µ (0 , T ; U ) × Y that we equip with the norm (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ = max (cid:0) (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) L µ (0 ,T ; V ∗ × V ∗ × U ) , e µT (cid:107) q (cid:107) Y (cid:1) . Let us emphasize the fact that the component q appears with a weight e µT in theabove norm. The spaces Λ T, and Λ T,µ (resp. Υ T, and Υ T,µ ) are the same vectorspace, equipped with two different norms. In the following lemma, the equivalencebetween these two norms is quantified.
Lemma
For all µ and µ with µ ≤ µ , there exists a constant M > suchthat for all T , for all ( y, u, p ) ∈ Λ T, , (cid:107) ( y, u, p ) (cid:107) Λ T,µ ≤ M (cid:107) ( y, u, p ) (cid:107) Λ T,µ , (cid:107) ( y, u, p ) (cid:107) Λ T,µ ≤ M e ( µ − µ ) T (cid:107) ( y, u, p ) (cid:107) Λ T,µ , and such that, similarly, for all ( y , f, g, h, q ) ∈ Υ T, , (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ , (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ ≤ M e ( µ − µ ) T (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . Proof of Lemma 1.1.
Let y ∈ W (0 , T ) and u ∈ L (0 , T ; U ). For proving thelemma, it suffices to prove the existence of M >
0, independent of T , y , and u , suchthat(1.6) (cid:107) u (cid:107) L µ (0 ,T ; U ) ≤ M (cid:107) u (cid:107) L µ (0 ,T ; U ) , (cid:107) u (cid:107) L µ (0 ,T ; U ) ≤ M e ( µ − µ ) T (cid:107) u (cid:107) L µ (0 ,T ; U ) , and such that(1.7) (cid:107) y (cid:107) W µ (0 ,T ) ≤ M (cid:107) y (cid:107) W µ (0 ,T ) , (cid:107) y (cid:107) W µ (0 ,T ) ≤ M e ( µ − µ ) T (cid:107) y (cid:107) W µ (0 ,T ) . The inequalities (1.6) can be easily verified (with M = 1). One can also easily verifythat (cid:107) y (cid:107) L µ (0 ,T ; V ) ≤ M (cid:107) y (cid:107) L µ (0 ,T ; V ) , (cid:107) y (cid:107) L µ (0 ,T ; V ) ≤ M e ( µ − µ ) T (cid:107) y (cid:107) L µ (0 ,T ; V ) (cid:107) y (cid:107) L ∞ µ (0 ,T ; Y ) ≤ M (cid:107) y (cid:107) L ∞ µ (0 ,T ; Y ) , (cid:107) y (cid:107) L ∞ µ (0 ,T ; Y ) ≤ M e ( µ − µ ) T (cid:107) y (cid:107) L ∞ µ (0 ,T ; Y ) . Let z ( t ) = e µ t y ( t ) and z ( t ) = e µ t y ( t ). For proving (1.7), it remains to compare (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) and (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) . We have z ( t ) = e ( µ − µ ) t z ( t ) and thus ˙ z ( t ) =( µ − µ ) z ( t ) + e ( µ − µ ) t ˙ z ( t ). We deduce that (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) ≤ M (cid:107) z (cid:107) L (0 ,T ; V ) + (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) ≤ M (cid:107) z (cid:107) W (0 ,T ) = M (cid:107) y (cid:107) W µ (0 ,T ) . HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS z ( t ) = ( µ − µ ) z ( t ) + e ( µ − µ ) t ˙ z ( t ). We deduce that (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) ≤ M (cid:107) z (cid:107) L (0 ,T ; V ) + e ( µ − µ ) T (cid:107) ˙ z (cid:107) L (0 ,T ; V ∗ ) ≤ M e ( µ − µ ) T (cid:107) z (cid:107) W (0 ,T ) = M e ( µ − µ ) T (cid:107) y (cid:107) W µ (0 ,T ) . The inequalities (1.7) follow. This concludes the proof.
Throughout the article we assume that the following fourassumptions hold true.(A1) The operator − A can be associated with a V - Y coercive bilinear form a : V × V → R which is such that there exist λ > δ ∈ R satisfying a ( v, v ) ≥ λ (cid:107) v (cid:107) V − δ (cid:107) v (cid:107) Y , for all v ∈ V .(A2) [Stabilizability] There exists an operator F ∈ L ( Y, U ) such that the semigroup e ( A + BF ) t is exponentially stable on Y .(A3) [Detectability] There exists an operator K ∈ L ( Z, Y ) such that the semigroup e ( A − KC ) t is exponentially stable on Y .Assumptions (A2) and (A3) are well-known and analysed for infinite-dimensionalsystems, see e.g. [8]. Consider the algebraic Riccati equation: for all y and y ∈ D ( A ),(1.8) (cid:104) A ∗ Π y , y (cid:105) Y + (cid:104) Π Ay , y (cid:105) Y + (cid:104) Cy , Cy (cid:105) Z − α (cid:104) B ∗ Π y , B ∗ Π y (cid:105) U = 0 . Due to the (exponential) stabilizability and detectability assumptions, it is well-known(see [8, Theorem 6.2.7] and [18, Theorem 2.2.1]) that (1.8) has a unique nonnegativeself-adjoint solution Π ∈ L ( Y, V ) ∩ L ( V ∗ , Y ). Additionally, the semigroup generatedby the operator A π := A − α BB ∗ Π is exponentially stable on Y . We fix now, for therest of the article, a real number λ such that0 < λ < ¯ λ := − sup µ ∈ σ ( A π ) Re( µ ) . (1.9)With (A1) holding the operator A associated with the form a generates an analyticsemigroup that we denote by e At , see e.g. [24, Sections 3.6 and 5.4]. Let us set A = A − λ I . Then − A has a bounded inverse in Y , see [24, page 75], and in particularit is maximal accretive, see [24]. We have D ( A ) = D ( A ) and the fractional powers of − A are well-defined. In particular, D (( − A ) ) = [ D ( − A ) , Y ] := ( D ( − A ) , Y ) , the real interpolation space with indices 2 and , see [5, Proposition 6.1, Part II,Chapter 1]. Assumption (A4) below will only be used in the proof Lemma 4.1, wherethe existence and uniqueness of a solution ( y (cid:5) , u (cid:5) ) to the static problem is established.It is not necessary for the analysis of optimality systems done in Sections 2 and 3.(A4) It holds that [ D ( − A ) , Y ] = [ D ( − A ∗ ) , Y ] = V .
2. Linear optimality systems.
The section is dedicated to the analysis of thefollowing optimality system:(2.1) y (0) = y in Y ˙ y − ( Ay + Bu ) = f in L µ (0 , T ; V ∗ ) − ˙ p − A ∗ p − C ∗ Cy = g in L µ (0 , T ; V ∗ ) αu + B ∗ p = − h in L µ (0 , T ; U ) p ( T ) − Qy ( T ) = q in Y , where µ ∈ {− λ, , λ } , T > Q ∈ L ( Y ) is self-adjoint and positive semi-definite,and ( y , f, g, h, q ) ∈ Υ T,µ . Given two times t < t , we introduce the operator T. BREITEN AND L. PFEIFFER H : W ( t , t ) × L ( t , t ; U ) × W ( t , t ) → L ( t , t ; V ∗ × V ∗ × U ), defined by H ( y, u, p ) = (cid:0) ˙ y − ( Ay + Bu ) , − ˙ p − A ∗ p − C ∗ Cy, αu + B ∗ p (cid:1) . The dependence of H with respect to t and t is not indicated and the underlyingvalues of t and t are always clear from the context. The operator H enables usto write the three intermediate equations of (2.1) in the compact form H ( y, u, p ) =( f, g, − h ).The main result of the section is the following theorem, which is proved in sub-section 2.2. Theorem
Let
Q ⊂ L ( Y ) be a bounded set of self-adjoint and positive semi-definite operators. For all T > , for all Q ∈ Q , for all ( y , f, g, h, q ) ∈ Υ T, , thereexists a unique solution ( y, u, p ) to system (2.1) . Moreover, for all µ ∈ {− λ, , λ } ,there exists a constant M independent of T , Q , and ( y , f, g, h, q ) such that (2.2) (cid:107) ( y, u, p ) (cid:107) Λ T,µ ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . Remark µ = 0, is rather classical in the litera-ture and can be established by analyzing the associated optimal control problem (seeLemma 2.8). The main novelty of our result is the estimate (2.2) in weighted spaces,with a constant M which is independent of T . Let us mention that a similar resulthas been obtained in [15, Theorem 3.1], for negative weights. The proof is based ona Neumann-series argument. Let us mention that the range of admissible weights inthat reference is different from ours (compare in particular with [15, Corollary 3.16]). We prove in this subsection Theorem2.1 in the case where Q = Π (Lemma 2.5). We begin with a useful result on forwardand backward linear systems with a right-hand side in L µ (0 , T ; V ∗ ) (Lemma 2.4). Lemma
For all µ ≤ λ , A π + µI generates an exponentially stable semigroup.For all µ ≥ − λ , A ∗ π − µI generates an exponentially stable semigroup.Proof. Let ˜ λ ∈ ( λ, ¯ λ ). Since the semigroup e A π t is analytic, the spectrum de-termined growth condition is satisfied, see e.g. [27]. Hence, (cid:107) e A π t (cid:107) L ( Y ) ≤ M e − ˜ λt ,where M does not depend on t . Therefore, (cid:107) e ( A π + µI ) t (cid:107) L ( Y ) ≤ M e ( − ˜ λ + µ ) t , whichproves the exponential stability of A π + µI since − ˜ λ + µ < − λ + µ ≤
0. Moreover,( e ( A π + µI ) t ) ∗ = e ( A π + µI ) ∗ t (see [20, page 41]), thus the operator A ∗ π − µI generates aexponentially stable semigroup as well, for µ ≥ − λ . Lemma
For all µ ≤ λ , for all T ∈ (0 , ∞ ) , for all y ∈ Y , for all f ∈ L µ (0 , T ; V ∗ ) , the following system: (2.3) ˙ y = A π y + f, y (0) = y has a unique solution in W µ (0 , T ) . Moreover, there exists a constant M > indepen-dent of T , y , and f such that (cid:107) y (cid:107) W µ (0 ,T ) ≤ M (cid:0) (cid:107) y (cid:107) Y + (cid:107) f (cid:107) L µ (0 ,T ; V ∗ ) (cid:1) .For all µ ≥ − λ , for all T ∈ (0 , ∞ ) , for all q ∈ Y , for all Φ ∈ L µ (0 , T ; V ∗ ) , thefollowing system: − ˙ r = A ∗ π r + Φ , r ( T ) = q has a unique solution in W µ (0 , T ) .Moreover, there exists a constant M > independent of T , q , and Φ such that (cid:107) r (cid:107) W µ (0 ,T ) ≤ M (cid:0) (cid:107) Φ (cid:107) L µ (0 ,T ; V ∗ ) + e µT (cid:107) q (cid:107) Y (cid:1) .Proof. Let us prove the first statement. Let y ∈ W (0 , T ). Defining y µ := e µ · y ∈ W µ (0 , T ) and f µ := e µ · f ∈ L µ (0 , T ; V ∗ ), we observe that y solves (2.3) if and only if HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS y µ is the solution to the following system:(2.4) ˙ y µ = ( A π + µI ) y µ + f µ , y µ (0) = y . Since µ ≤ λ , the operator A π + µI generates an exponentially stable semigroup, byLemma 2.3. Standard regularity results for analytic semigroups ensure the existenceand uniqueness of a solution to (2.4), as well as the existence of a constant M > T , y , and f such that (cid:107) y µ (cid:107) W (0 ,T ) ≤ M (cid:0) (cid:107) y (cid:107) + (cid:107) f µ (cid:107) L (0 , ∞ ) (cid:1) , whichis the estimate that was to be proved.The second statement can be proved similarly with a time-reversal argument.We are now ready to analyze (2.1) in the case where Q = Π. The key idea is todecouple the system with the help of the variable r = p − Π y . This variable is indeedthe solution to a backward differential equation which is independent of y , u , and p .Let us mention that this remarkable property only holds in the case Q = Π. Lemma
For all µ ∈ [ − λ, λ ] , for all T > , for all ( y , f, g, h, q ) ∈ Υ T,µ , thereexists a unique ( y, u, p ) ∈ Λ T,µ solution to (2.1) with Q = Π . Moreover, there existsa constant M > , independent of T and ( y , f, g, h, q ) such that (2.5) (cid:107) ( y, u, p ) (cid:107) Λ T,µ ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . Remark M is a positive constant whosevalue may change from an inequality to the next one. When an estimate involving aconstant M independent of some variables (for example T ) has to be proved, then allconstants M used in the corresponding proof are also independent of these variables. Proof of Lemma 2.5.
Let Φ ∈ L µ (0 , T ; V ∗ ) be defined by Φ = Π f − α Π Bh + g .Let us denote by r ∈ W µ (0 , T ) the unique solution to the system − ˙ r = A ∗ π r + Φ, t ∈ [0 , T ), r ( T ) = q . By Lemma 2.4, there exists a constant M , independent of T and( y , f, g, h, q ) such that(2.6) (cid:107) r (cid:107) W µ (0 ,T ) ≤ M (cid:0) (cid:107) Φ (cid:107) L µ (0 ,T ; V ∗ ) + e µT (cid:107) q (cid:107) Y (cid:1) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . By Lemma 2.4, the following system has a unique solution y ∈ W µ (0 , T ):(2.7) ˙ y = A π y − α Bh + f − α BB ∗ r, y (0) = y . Since (cid:13)(cid:13) − α Bh + f − α BB ∗ r (cid:13)(cid:13) L µ (0 ,T ; V ∗ ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ , we have that(2.8) (cid:107) y (cid:107) W µ (0 ,T ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . Let us set p = Π y + r . Since Π ∈ L ( Y, V ) ∩ L ( V ∗ , Y ), we have that Π y ∈ L (0 , T ; V ) ∩ H (0 , ∞ ; Y ). Therefore, using (2.6) and (2.8), we obtain that p ∈ W µ (0 , T ) with (cid:107) p (cid:107) W µ (0 ,T ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . We finally define u = − α ( h + B ∗ p ). We deducefrom the estimate on p that (cid:107) u (cid:107) L µ (0 ,T ; U ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,µ . The bound (2.5)is proved.Let us check that ( y, u, p ) is a solution to the linear system (2.1). It follows fromthe definition of u that αu + B ∗ p = − h . Using p = Π y + r and (2.7), we obtain that Ay + Bu + f = Ay − α Bh − α BB ∗ p + f = Ay − α Bh − α BB ∗ Π y − α BB ∗ r + f = ˙ y. T. BREITEN AND L. PFEIFFER
It remains to verify that the adjoint equation is satisfied. We obtain with the defini-tions of p , y , Φ, and A π that p ( T ) − Π y ( T ) = q and that˙ p = Π ˙ y + ˙ r = Π A π y + (cid:0) − α Π Bh + Π f (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) =Φ − g + (cid:0) − α Π BB ∗ (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) = A ∗ π − A ∗ r + ˙ r. Using ˙ r + A ∗ π r + Φ = 0 and (1.8), we obtain that˙ p = Π A π y + Φ − g + A ∗ π r − A ∗ r + ˙ r = Π A π y − A ∗ r − g = (cid:0) Π A − α Π BB ∗ Π (cid:1) y − A ∗ r − g = − (cid:0) A ∗ Π + C ∗ C (cid:1) y − A ∗ r − g = − A ∗ p − C ∗ Cy − g. Therefore, the adjoint equation is satisfied and ( y, u, p ) is a solution to (2.2).It remains to show uniqueness. To this end, it suffices to consider the case where( y , f, g, h, q ) = (0 , , , , y, u, p ) be a solution to (2.1). Let r = p − Π y . Onecan easily see that − ˙ r = A ∗ π r , r ( T ) = 0, thus r = 0. Then, one has to check that y satisfies (2.7), with y = 0, f = 0, h = 0, and r = 0. Therefore, y = 0. Finally, weobtain that p = r + Π y = 0 and that u = − α ( h + B ∗ p ) = 0. Uniqueness is proved. We give a proof of Theorem 2.1 in this subsection. Weconsider successively the cases µ = 0, µ = − λ , and µ = λ . Theorem 2.1, in the case where µ = 0, can beestablished by analyzing the optimal control problem associated with (2.1). This isthe result of Lemma 2.8 below. The proof is classical and uses very similar argumentsto the ones used in [6, Proposition 3.1].We begin with a classical lemma, following from the detectability assumption. Lemma
There exists a constant
M > such that for all T > , for all y ∈ Y , for all u ∈ L (0 , T ; U ) , for all f ∈ L (0 , T ; V ∗ ) , the solution y ∈ W (0 , T ) tothe system ˙ y = Ay + Bu + f, y (0) = y satisfies the following estimate: (cid:107) y (cid:107) W (0 ,T ) ≤ M (cid:0) (cid:107) y (cid:107) Y + (cid:107) u (cid:107) L (0 ,T ; U ) + (cid:107) f (cid:107) L (0 ,T ; V ∗ ) + (cid:107) Cy (cid:107) L (0 ,T ; Z ) (cid:1) . Proof.
Let z ∈ W (0 , T ) be the solution to˙ z = Az + Bu + f + KC ( y − z ) , z (0) = y , where K is given by Assumption (A3). The above system can be re-written as follows:˙ z = ( A − KC ) z + Bu + f + KCy, z (0) = y . Since ( A − KC ) is exponentially stable, there exists a constant M , independent of T , y , u , f , and y such that (cid:107) z (cid:107) W (0 ,T ) ≤ M (cid:0) (cid:107) y (cid:107) Y + (cid:107) Bu + f + KCy (cid:107) L (0 ,T ; V ∗ ) (cid:1) ≤ M (cid:0) (cid:107) y (cid:107) Y + (cid:107) u (cid:107) L (0 ,T ; U ) + (cid:107) f (cid:107) L (0 ,T ; V ∗ ) + (cid:107) Cy (cid:107) L (0 ,T ; Z ) (cid:1) . (2.9)Observing that e := z − y is the solution to ˙ e = ( A − KC ) e , e (0) = 0, we obtain that e = 0 and that z = y . Thus y satisfies (2.9), as was to be proved. HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS Lemma
For all
T > , for all Q ∈ Q , for all ( y , f, g, h, q ) ∈ Υ T, , thefollowing optimal control problem ( LQ ) inf y ∈ W (0 ,T ) u ∈ L (0 ,T ; U ) (cid:104) (cid:90) T (cid:107) Cy ( t ) (cid:107) Z + (cid:104) g ( t ) , y ( t ) (cid:105) + α (cid:107) u ( t ) (cid:107) U + (cid:104) h ( t ) , u ( t ) (cid:105) U d t + (cid:104) y ( T ) , Qy ( T ) (cid:105) Y + (cid:104) q, y ( T ) (cid:105) Y (cid:105) , subject to: ˙ y = Ay + Bu + f, y (0) = y , has a unique solution ( y, u ) . There exists a unique associated adjoint variable p , whichis such that ( y, u, p ) is the unique solution to (2.1) . Moreover, there exists a constant M , independent of T , Q , and ( y , f, g, h, q ) such that (2.10) (cid:107) ( y, u, p ) (cid:107) Λ T, ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, . Proof.
We follow the same lines as in [6, Lemma 3.2]. Let us first bound the valueof the problem. Let y ∈ W (0 , T ) be the solution to˙ y = ( A + BF ) y + f, y (0) = y , where F is given by Assumption (A2). Since ( A + BF ) is exponentially stable, thereexists a constant M such that (cid:107) y (cid:107) W (0 ,T ) ≤ M max (cid:0) (cid:107) y (cid:107) Y , (cid:107) f (cid:107) L (0 ,T ; V ∗ ) (cid:1) . Let us set u = F y . We have (cid:107) u (cid:107) L (0 ,T ; U ) ≤ M max (cid:0) (cid:107) y (cid:107) Y , (cid:107) f (cid:107) L (0 ,T ; V ∗ ) (cid:1) . Then, onecan easily check the existence of a constant M such that J ( u, y ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) T, . Now, we prove the existence of a solution to the problem. Let ( y n , u n ) n ∈ N ∈ W (0 , T ) × L (0 , T ; U ) be a minimizing sequence such that for all n ∈ N , J ( y n , u n ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) T, . We now look for a lower bound for J , so that we can further obtain a bound on( y n , u n ). We have J ( y n , u n ) ≥ (cid:107) Cy n (cid:107) L (0 ,T ; Z ) − (cid:107) g (cid:107) L (0 ,T ; V ∗ ) (cid:107) y n (cid:107) W (0 ,T ) + α (cid:107) u n (cid:107) L (0 ,T ; U ) − (cid:107) h (cid:107) L (0 ,T ; U ) (cid:107) u n (cid:107) L (0 ,T ; U ) − (cid:107) q (cid:107) Y (cid:107) y n ( T ) (cid:107) Y ≥ (cid:107) Cy n (cid:107) L (0 ,T ; Z ) + α (cid:16) (cid:107) u n (cid:107) L (0 ,T ; U ) − (cid:107) h (cid:107) L (0 ,T ; U ) α (cid:17) − (cid:107) h (cid:107) L (0 ,T ; U ) α − ε (cid:0) (cid:107) g (cid:107) L (0 ,T ; V ∗ ) + (cid:107) q (cid:107) Y (cid:1) − ε (cid:107) y n (cid:107) W (0 ,T ) . Therefore, there exists a constant M such that (cid:107) Cy n (cid:107) ≤ M (cid:16) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, + 1 √ ε (cid:0) (cid:107) g (cid:107) L (0 ,T ; V ∗ ) + (cid:107) q (cid:107) Y (cid:1) + √ ε (cid:107) y n (cid:107) W (0 ,T ) (cid:17) , (2.11) (cid:107) u n (cid:107) ≤ M (cid:16) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, + 1 √ ε (cid:0) (cid:107) g (cid:107) L (0 ,T ; V ∗ ) + (cid:107) q (cid:107) Y (cid:1) + √ ε (cid:107) y n (cid:107) W (0 ,T ) (cid:17) . (2.12)0 T. BREITEN AND L. PFEIFFER
Applying Lemma 2.7 and estimate (2.11), we obtain that (cid:107) y n (cid:107) W (0 ,T ) ≤ M (cid:0) (cid:107) y (cid:107) Y + (cid:107) f (cid:107) L (0 ,T ; V ∗ ) + (cid:107) u n (cid:107) L (0 ,T ; U ) + (cid:107) Cy n (cid:107) L (0 ,T ; Z ) (cid:1) ≤ M (cid:16) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, + √ ε (cid:107) y n (cid:107) W (0 ,T ) + 1 √ ε (cid:0) (cid:107) g (cid:107) + (cid:107) q (cid:107) Y (cid:1)(cid:17) . Let us fix ε = M ) , where M is the constant obtained in the last inequality. Itfollows that there exists (another) constant M > (cid:107) y n (cid:107) W (0 ,T ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, . Combined with (2.12), we obtain that (cid:107) u n (cid:107) L (0 ,T ; U ) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, . The sequence ( y n , u n ) n ∈ N is therefore bounded in W (0 , T ) × L (0 , T ; U ) and has aweak limit point ( y, u ) satisfying(2.14) max (cid:0) (cid:107) y (cid:107) W (0 ,T ) , (cid:107) u (cid:107) L (0 ,T ; U ) (cid:1) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, . One can prove the optimality of ( y, u ) with the same techniques as those used for theproof of [7, Proposition 2].Consider now the solution p to the adjoint system(2.15) − ˙ p − A ∗ p − C ∗ Cy = g, p ( T ) − Qy ( T ) = q. The optimality conditions for the problem yield αu + B ∗ p + h = 0, see e.g. [16]. Itfollows that ( y, u, p ) is a solution to (2.1).Let us prove the uniqueness. If ( y, u, p ) is a solution to (2.1), then one can provethat ( y, u ) is a solution to problem ( LQ ) with associated costate p . Therefore, itsuffices to prove the uniqueness of the solution to (2.1). To this end, it suffices toconsider the case where ( y , f, g, h, q ) = (0 , , , , y, u, p ) be a solution to(2.1). Then ( y, u ) is a solution to ( LQ ) and one can check that (2.14) holds. Thus,( y, u ) = (0 ,
0) and then, p = 0, which proves the uniqueness.It remains to prove the a priori bound. Observe that ( y, u, p ) is the solution to(2.16) y (0) = y in Y ˙ y − ( Ay + Bu ) = f in L (0 , T ; V ∗ ) − ˙ p − A ∗ p − C ∗ Cy = g in L (0 , T ; V ∗ ) αu + B ∗ p = − h in L (0 , T ; U ) p ( T ) − Π y ( T ) = ˜ q in Y , where ˜ q = ( Q − Π) y ( T ) + q . By (2.14), we have (cid:107) ˜ q (cid:107) Y ≤ M (cid:0) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, .Thus by Lemma 2.5, (cid:107) ( y, u, p ) (cid:107) Λ T, ≤ M (cid:107) ( y , f, g, h, ˜ q ) (cid:107) Υ T, ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, ,which concludes the proof. Proof of Theorem 2.1: the case µ = − λ . Let ( y , f, g, h, q ) ∈ Υ T, − λ . The fol-lowing inequality can be easily checked: (cid:107) ( f, g, h ) (cid:107) L (0 ,T ) ≤ e λT (cid:107) ( f, g, h ) (cid:107) L − λ (0 ,T ) .Therefore, by Lemma 2.8, the system (2.1) has a unique solution ( y, u, p ), satisfying (cid:107) ( y, u, p ) (cid:107) Λ T, ≤ M max (cid:0) (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) L (0 ,T ) , (cid:107) q (cid:107) Y (cid:1) ≤ M max (cid:0) (cid:107) y (cid:107) Y , e λT (cid:107) ( f, g, h ) (cid:107) L − λ (0 ,T ) , (cid:107) q (cid:107) Y (cid:1) . HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS (cid:107) y ( T ) (cid:107) Y ≤ M max (cid:0) (cid:107) y (cid:107) Y , e λT (cid:107) ( f, g, h ) (cid:107) L − λ (0 ,T ) , (cid:107) q (cid:107) Y (cid:1) and then that e − λT (cid:107) y ( T ) (cid:107) Y ≤ M max (cid:0) e − λT (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) L − λ (0 ,T ) , e − λT (cid:107) q (cid:107) Y (cid:1) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, − λ , (2.17)since e − λT ≤
1. The key idea now is to observe that y (0) = y , H ( y, u, p ) = ( f, g, − h ),and p ( T ) − Π y ( T ) = ˜ q , where ˜ q = ( Q − Π) y ( T ) + q . Thus, by Lemma 2.5, (cid:107) ( y, u, p ) (cid:107) Λ T, − λ ≤ M (cid:107) ( y , f, g, h, ˜ q ) (cid:107) Υ T, − λ ≤ M (cid:0) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, − λ + e − λT (cid:107) ( Q − Π) (cid:107) L ( Y ) (cid:107) y ( T ) (cid:107) Y (cid:1) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, − λ + M e − λT (cid:107) y ( T ) (cid:107) Y , (2.18)since Q is bounded. Estimate (2.2) follows, combining (2.17) and (2.18). The approach that we propose for dealingwith the case µ = λ requires some more advanced tools, that we introduce now. Fora given θ ∈ (0 , T ), we make use of the following mixed weighted space: (cid:107) u (cid:107) L λ, − λ (0 ,T ; U ) = (cid:107) e ρ ( · ) u ( · ) (cid:107) L (0 ,T ; U ) , where ρ ( t ) = λt, for t ∈ [0 , T − θ ], ρ ( t ) = 2 λ ( T − θ ) − λt, for t ∈ [ T − θ, T ] . Observe that ρ is continuous and piecewise affine, with ˙ ρ ( t ) = λ for t ∈ [0 , T − θ ) and˙ ρ ( t ) = − λ for t ∈ ( T − θ, T ]. In a nutshell: We use a positive weight on (0 , T − θ ) and anegative weight on ( T − θ, T ). We define similarly the space L − λ,λ (0 , T ; V ∗ × V ∗ × U )— that we often denote by L − λ,λ (0 , T ) — and the space W λ, − λ (0 , T ). The spacesΛ λ, − λ and Υ λ, − λ are defined in a similar way as before, with the corresponding norms (cid:107) ( y, u, p ) (cid:107) Λ λ, − λ = max (cid:0) (cid:107) y (cid:107) W λ, − λ (0 ,T ) , (cid:107) u (cid:107) L λ, − λ (0 ,T ; U ) , (cid:107) p (cid:107) W λ, − λ (0 ,T ) (cid:1) , (cid:107) ( y , f, g, h, q ) (cid:107) Υ λ, − λ = max (cid:0) (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) L λ, − λ (0 ,T ) , e ρ ( T ) (cid:107) q (cid:107) Y (cid:1) . The following lemma is a generalization of Lemma 2.5 for mixed weighted spaces.
Lemma
For all
T > , for all ( y , f, g, h, q ) ∈ Υ λ, − λ , the unique solution ( y, u, p ) to (2.1) with Q = Π satisfies the following bound: (2.19) (cid:107) ( y, u, p ) (cid:107) Λ λ, − λ ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ λ, − λ , where M is independent of T , θ , and ( y , f, g, h, q ) .Proof. We only give the main lines of the proof. One can obtain estimate (2.19)with the same decoupling as the one introduced in Lemma 2.5. The decoupled vari-ables y and r can then be estimated in W λ, − λ (0 , T ), after an adaptation of Lemma2.4 for right-hand sides in L λ, − λ (0 , T ; V ∗ ). Proof of Theorem 2.1: the case µ = λ . Let us first fix some constants. We denoteby M the constant involved in estimate (2.10). We denote by M the constantinvolved in Lemma 2.9. Note that M ≥ M ≥
1. Finally, M denotes anupper bound on (cid:107) Q − Π (cid:107) L ( Y ) . Let us set M = 2 M M ≥ θ > M M e − λθ ≤
1. The first four steps of this proof deal with the case2
T. BREITEN AND L. PFEIFFER where T ≥ θ . We will consider the case T < θ in Step 5. Take now T ≥ θ and( y , f, g, h, q ) ∈ Υ T,λ . Since Υ
T,λ is embedded in Υ T, , the existence of a solution to(2.1) in Λ T, is guaranteed. Let us denote it by (¯ y, ¯ u, ¯ p ). Step 1: construction of the mappings χ and χ .The main idea of the proof consists in obtaining an estimate of ¯ y ( T ) with a fixed-point argument. To this end, we introduce two affine mappings, χ and χ , definedas follows: χ : y T ∈ Y (cid:55)→ y ( T − θ ) ∈ Y , where y is the solution to(2.20) y (0) = y in Y H ( y, u, p ) = ( f, g, − h ) in L λ, − λ (0 , T ; V ∗ × V ∗ × U ) p ( T ) − Π y ( T ) = ( Q − Π) y T + q in Y .
The mapping χ is defined as follows: χ : y T − θ ∈ Y (cid:55)→ y ( T ) ∈ Y , where y ∈ W ( T − θ, T ) is the solution to y ( T − θ ) = y T − θ in Y H ( y, u, p ) = ( f, g, − h ) in L ( T − θ, T ; V ∗ × V ∗ × U ) p ( T ) − Qy ( T ) = q in Y .
The existence and uniqueness of a solution to the above system follows from Lemma2.8, after a shifting of the time variable. Observe that ¯ y ( T − θ ) = χ (¯ y ( T )) and that¯ y ( T ) = χ (¯ y ( T − θ )). It follows that ¯ y ( T ) is a fixed point of χ ◦ χ . Step 2: on the Lipschitz-continuity of χ and χ .Let y T and ˜ y T ∈ Y . We have χ (˜ y T ) − χ ( y T ) = y ( T − θ ), where y is the solution to y (0) = 0 in Y H ( y, u, p ) = (0 , ,
0) in L λ, − λ (0 , T ; V ∗ × V ∗ × U ) p ( T ) − Π y ( T ) = ( Q − Π)(˜ y T − y T ) in Y .
By Lemma 2.9, (cid:107) ( y, u, p ) (cid:107) Λ λ, − λ ≤ M e ρ ( T ) (cid:107) Q − Π (cid:107) L ( Y ) (cid:107) ˜ y T − y T (cid:107) Y ≤ M M e ρ ( T ) (cid:107) ˜ y T − y T (cid:107) Y . Thus, e ρ ( T − θ ) (cid:107) y ( T − θ ) (cid:107) Y ≤ (cid:107) y (cid:107) W λ, − λ (0 ,T ) ≤ M M e ρ ( T ) (cid:107) ˜ y T − y T (cid:107) Y . Observing that e ρ ( T ) − ρ ( T − θ ) = e − λθ , we finally obtain that (cid:107) χ (˜ y T ) − χ ( y T ) (cid:107) Y = (cid:107) y ( T − θ ) (cid:107) Y ≤ M M e − λθ (cid:107) ˜ y T − y T (cid:107) Y , which proves that χ is Lipschitz-continuous. Now, let us take y T − θ and ˜ y T − θ in Y .We have χ (˜ y T − θ ) − χ (˜ y T − θ ) = y ( T ), where y ∈ W ( T − θ, T ) is the solution to y ( T − θ ) = ˜ y T − θ − y T − θ in Y H ( y, u, p ) = (0 , ,
0) in L ( T − θ, T ; V ∗ × V ∗ × U ) p ( T ) − Qy ( T ) = 0 in Y .
We obtain with Lemma 2.8 that (cid:107) y (cid:107) W ( T − θ,T ) ≤ M (cid:107) ˜ y T − θ − y T − θ (cid:107) Y and thus (cid:107) χ (˜ y T − θ ) − χ ( y T − θ ) (cid:107) Y = (cid:107) y ( T ) (cid:107) ≤ M (cid:107) ˜ y T − θ − y T − θ (cid:107) Y , proving that χ is Lipschitz-continuous. As a consequence, the mapping χ ◦ χ isLipschitz-continuous, with modulus M M M e − λθ ≤ M M e − λθ ≤ . HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS Step 3: on the invariance of B Y (cid:0) R (cid:1) , with R = M e − λ ( T − θ ) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ .Let y T ∈ B Y ( R ). Consider the solution y to system (2.20). By Lemma 2.9, we have(2.21) (cid:107) y (cid:107) W λ, − λ (0 ,T ) ≤ M max (cid:0) (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) L λ, − λ (0 ,T ) , e ρ ( T ) (cid:107) ( Q − Π) y T + q (cid:107) Y (cid:1) . Let us estimate the last term in the above expression. We have e ρ ( T ) (cid:107) ( Q − Π) y T + q (cid:107) Y ≤ e λT − λθ (cid:0) M (cid:107) y T (cid:107) Y + (cid:107) q (cid:107) Y (cid:1) ≤ e − λθ M M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ + e λT (cid:107) q (cid:107) Y ≤ (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ + e λT (cid:107) q (cid:107) Y . (2.22)Observe that (cid:107) ( f, g, h ) (cid:107) L λ, − λ (0 ,T ) ≤ (cid:107) ( f, g, h ) (cid:107) L λ (0 ,T ) . Combining (2.21), (2.22), andthis last observation, we obtain that (cid:107) y (cid:107) W λ, − λ (0 ,T ) ≤ M max (cid:0) (cid:107) y (cid:107) Y , (cid:107) ( f, g, h ) (cid:107) λ , (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ + e λT (cid:107) q (cid:107) Y (cid:1) ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ . It follows then that (cid:107) χ ( y T ) (cid:107) Y = e − ρ ( T − θ ) (cid:107) e ρ ( T − θ ) y ( T − θ ) (cid:107) Y ≤ e − λ ( T − θ ) (cid:107) y (cid:107) W λ, − λ (0 ,T ) ≤ M e − λ ( T − θ ) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ . (2.23)Applying now Lemma 2.8, we obtain that (cid:107) χ ◦ χ ( y T ) (cid:107) Y ≤ M max (cid:0) (cid:107) χ ( y T ) (cid:107) Y , (cid:107) ( f, g, h ) | ( T − θ,T ) (cid:107) , (cid:107) q (cid:107) Y (cid:1) . (2.24)Observing that e λ ( T − θ ) (cid:107) ( f, g, h ) | ( T − θ,T ) (cid:107) L ( T − θ,T ) ≤ (cid:107) ( f, g, h ) (cid:107) L λ (0 ,T ) , we deducefrom (2.23) and (2.24) that (cid:107) χ ◦ χ ( y T ) (cid:107) Y ≤ M e − λ ( T − θ ) max (cid:0) M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ , (cid:107) ( f, g, h ) (cid:107) λ , e λT (cid:107) q (cid:107) Y (cid:1) ≤ M e − λ ( T − θ ) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ . We have proved that (cid:107) χ ◦ χ ( y T ) (cid:107) Y ≤ R . Step 4: proof of (2.2) (when T ≥ θ ).We have proved in the second step of the proof that χ ◦ χ is a contraction. Therefore,¯ y ( T ) is the unique fixed-point of χ ◦ χ in Y . We have established in the third part ofthe proof that B Y ( R ) is invariant by χ ◦ χ . Therefore, by the fixed-point theorem,the mapping χ ◦ χ has a unique fixed point in B Y ( R ) which is then necessarily ¯ y ( T ).Observe now that (¯ y, ¯ u, ¯ p ) is the solution to (2.20), with y T = ¯ y ( T ). Denoting by M the constant involved in estimate (2.5), we obtain that (cid:107) (¯ y, ¯ u, ¯ p ) (cid:107) Λ T,λ ≤ M (cid:107) ( y , f, g, h, ( Q − Π)¯ y ( T ) + q ) (cid:107) Υ T,λ ≤ M (cid:0) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ + M e λT (cid:107) ¯ y ( T ) (cid:107) Y (cid:1) ≤ M (1 + M M e λθ ) (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ . This concludes the proof, in the case T ≥ θ .4 T. BREITEN AND L. PFEIFFER
Step 5: proof of (2.2) (when
T < θ ).By Lemma 1.1 and Lemma 2.8, we have (cid:107) ( y, u, p ) (cid:107) Λ T,λ ≤ M e λT (cid:107) ( y, u, p ) (cid:107) Λ T, ≤ M e λθ (cid:107) ( y , f, g, h, q ) (cid:107) Υ T, ≤ M (cid:107) ( y , f, g, h, q ) (cid:107) Υ T,λ , which proves (2.2) and concludes the proof of the theorem.
3. Additional results on optimality systems.
In this subsection, we analyzefurther the optimality system associated with the linear-quadratic problem ( LQ ) when( f, g, h ) = (0 , , y, u ) ∈ W (0 , T ) × L (0 , T ), wedenote(3.1) J T,Q,q ( u, y ) = (cid:90) T (cid:107) Cy ( t ) (cid:107) Z + α (cid:107) u ( t ) (cid:107) U d t + (cid:104) y ( T ) , Qy ( T ) (cid:105) Y + (cid:104) q, y ( T ) (cid:105) Y and consider the problem( P ) V T,Q,q ( y ) = inf y ∈ W (0 ,T ) u ∈ L (0 ,T ; U ) J T,Q,q ( u, y )subject to: ˙ y = Ay + Bu, y (0) = y . Problem ( P ) is a particular case of problem ( P ) with ( f (cid:5) , g (cid:5) , h (cid:5) ) = (0 , , y , T, Q, q ):( OS ) y (0) = y , H ( y, u, p ) = (0 , , , p ( T ) − Qy ( T ) = q. Since the solution ( y, u, p ) is a linear mapping of ( y , q ), there exist two linear operatorsΠ( T, Q ) and G ( T, Q ) such that(3.2) p (0) = Π( T, Q ) y + G ( T, Q ) q. Let us mention that Π(
T, Q ) can be described as the solution to a differential Riccatiequation (see [5, Part IV]).
Lemma
There exists a constant
M > such that for all T > and for all Q ∈ Q , (cid:107) Π( T, Q ) (cid:107) L ( Y ) ≤ M , (cid:107) G ( T, Q ) (cid:107) L ( Y ) ≤ M e − λT , and (cid:107) Π( T, Q ) − Π (cid:107) L ( Y ) ≤ M (cid:107) Q − Π (cid:107) L ( Y ) e − λT . As a consequence of the last estimate, we obtain that Π(
T, Q ) −→ T →∞ Π and thatΠ( T, Π) = Π. Let us mention that the third inequality has been obtained in [21,Corollary 2.7] for finite-dimensional systems and that our result improves the onegiven in the same reference (see [21, Lemma 3.9]), where a rate equal to λ (instead of2 λ ) is established for parabolic systems. Proof of Lemma 3.1.
Applying Theorem 2.1 with µ = − λ , we obtain that (cid:107) e − λ · p ( · ) (cid:107) L ∞ (0 ,T ; Y ) ≤ (cid:107) ( y, u, p ) (cid:107) Λ T, − λ ≤ M max (cid:0) (cid:107) y (cid:107) Y , e − λT (cid:107) q (cid:107) Y (cid:1) and thus (cid:107) p (0) (cid:107) Y ≤ M max (cid:0) (cid:107) y (cid:107) Y , e − λT (cid:107) q (cid:107) Y (cid:1) . It follows that (cid:107) Π( T, Q ) (cid:107) L ( Y ) ≤ M and that (cid:107) G ( T, Q ) (cid:107) L ( Y ) ≤ M e − λT , as was to be proved. HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS q = 0. Applying Theorem 2.1 (with µ = λ ), we obtain that (cid:107) e λ · y ( · ) (cid:107) L ∞ (0 ,T ; Y ) ≤ M (cid:107) y (cid:107) Y . Thus (cid:107) y ( T ) (cid:107) Y ≤ M e − λT (cid:107) y (cid:107) Y .Let us set r ( t ) = p ( t ) − Π y ( t ). We have r ( T ) = ( Q − Π) y ( T ), therefore (cid:107) r ( T ) (cid:107) Y ≤ M (cid:107) Q − Π (cid:107) L ( Y ) e − λT (cid:107) y (cid:107) Y . Using the algebraic Riccati equation (1.8) and the fact that Π ∈ L ( V ∗ , Y ) ∩ L ( Y, V ),one can check that r ∈ W (0 , T ) and that − ˙ r = A ∗ π r . Since A ∗ π + λI generates abounded semigroup, we finally deduce that (cid:107) (Π( T, Q ) − Π) y (cid:107) Y = (cid:107) r (0) (cid:107) Y ≤ M e − λT (cid:107) r ( T ) (cid:107) Y ≤ M e − λT (cid:107) Q − Π (cid:107) L ( Y ) (cid:107) y (cid:107) Y , which concludes the proof. Lemma
Let (¯ y, ¯ u ) be the solution to ( P ) with associated costate ¯ p . Let ( y, u ) ∈ W (0 , T ) × L (0 , T ; U ) be such that ˙ y = Ay + Bu . Then, there exists aconstant M , independent of T , Q , q , y , y , and u such that ≤ J T,Q,q ( u, y ) − V T,Q,q ( y ) − (cid:104) ¯ p (0) , y (0) − y (cid:105) Y ≤ M max (cid:0) (cid:107) y − ¯ y (cid:107) W (0 ,T ) , (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:1) . (3.3) Proof.
We have J T,Q,q ( u, y ) − V T,Q,q ( y ) = J T,Q,q ( u, y ) − J T,Q,q (¯ u, ¯ y )= (cid:90) T (cid:16) (cid:107) C ( y − ¯ y ) (cid:107) Z + α (cid:107) u − ¯ u (cid:107) U + (cid:104) C ∗ C ¯ y, y − ¯ y (cid:105) + α (cid:104) ¯ u, u − ¯ u (cid:105) (cid:17) d t + (cid:104) y ( T ) − ¯ y ( T ) , Q ( y ( T ) − ¯ y ( T )) (cid:105) Y + (cid:104) Q ¯ y ( T ) + q, y ( T ) − ¯ y ( T ) (cid:105) Y . (3.4)The three quadratic terms can be bounded from above as follows:0 ≤ (cid:90) T (cid:107) C ( y − ¯ y ) (cid:107) Z + α (cid:107) u − ¯ u (cid:107) U d t + (cid:104) y ( T ) − ¯ y ( T ) , Q ( y ( T ) − ¯ y ( T )) (cid:105) Y ≤ M max (cid:0) (cid:107) y − ¯ y (cid:107) W (0 ,T ) , (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:1) . (3.5)Let us focus on the remaining terms in the right-hand of (3.4). Using the relations C ∗ C ¯ y = − ˙¯ p − A ∗ ¯ p and α ¯ u = − B ∗ ¯ p and integrating by parts, we obtain that (cid:90) T (cid:104) C ∗ C ¯ y, y − ¯ y (cid:105) Y + α (cid:104) ¯ u, u − ¯ u (cid:105) U d t = −(cid:104) Q ¯ y ( T ) + q, y ( T ) − ¯ y ( T ) (cid:105) Y + (cid:104) ¯ p (0) , y (0) − y (cid:105) Y . (3.6)Estimate (3.3) follows, by combining (3.4), (3.5), and (3.6). Corollary
The value function V T,Q,q ( · ) is differentiable. Moreover, (3.7) D y V T,Q,q ( y ) = Π( T, Q ) y + G ( T, Q ) q and Π( T, Q ) is self-adjoint and positive semi-definite.Proof. Take y ∈ Y and h ∈ Y . Denote by (¯ y, ¯ u, ¯ p ) and ( y, u, p ) the solutions to( OS ) with initial conditions y and y + h , respectively. Then, by Theorem 2.1,max (cid:0) (cid:107) y − ¯ y (cid:107) W (0 ,T ) , (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:1) ≤ (cid:107) ( y, u, p ) − (¯ y, ¯ u, ¯ p ) (cid:107) Υ T, ≤ M (cid:107) h (cid:107) Y . T. BREITEN AND L. PFEIFFER
Applying Lemma 3.2, we deduce that0 ≤ V T,Q,q ( y + h ) − V T,Q,q ( y ) − (cid:104) ¯ p (0) , h (cid:105) Y ≤ M (cid:107) h (cid:107) Y , which proves that V T,Q,q is differentiable with D y V T,Q,q ( y ) = ¯ p (0). Then (3.7)follows with (3.2).Let us take now q = 0. Then, the solution ( y, u, p ) to ( OS ) is a linear mapping of y . Since J T,Q, ( u, y ) is quadratic and convex, there exists a self-adjoint and positivesemi-definite operator ˆΠ( T ) such that V T,Q, ( y ) = (cid:104) y , ˆΠ( T ) y (cid:105) . Applying thefirst part of the lemma, we deduce that for all y ∈ Y , D y V T,Q, ( y ) = ˆΠ( T ) y =Π( T, Q ) y , which proves that ˆΠ( T ) = Π( T, Q ) and concludes the proof.
4. Linear-quadratic problems.4.1. Turnpike property.
We analyze now the class of problems ( P ) (definedin the introduction). By Lemma 2.8, ( P ) has a unique solution (¯ y, ¯ u ) with associatedcostate ¯ p , satisfying(4.1) ¯ y (0) = y , H ( y, u, p ) = ( f (cid:5) , g (cid:5) , − h (cid:5) ) , ¯ p ( ¯ T ) − Q ¯ y ( ¯ T ) = q. Note that the variables f (cid:5) , g (cid:5) , and h (cid:5) must be understood as constant time-functionsin the above optimality system. Let us first investigate the existence of a solution tothe static optimization problem. Lemma
The static optimization problem (1.1) has a unique solution ( y (cid:5) , u (cid:5) ) with unique associated Lagrange multiplier p (cid:5) ∈ V , i.e. p (cid:5) is such that (4.2) − ( Ay (cid:5) + Bu (cid:5) ) = f (cid:5) , − A ∗ p (cid:5) − C ∗ Cy (cid:5) = g (cid:5) , αu (cid:5) + B ∗ p (cid:5) = − h (cid:5) . Moreover, there exists a constant
M > , independent of ( f (cid:5) , g (cid:5) , h (cid:5) ) , such that (4.3) max (cid:0) (cid:107) y (cid:5) (cid:107) V , (cid:107) u (cid:5) (cid:107) U , (cid:107) p (cid:5) (cid:107) V (cid:1) ≤ M max (cid:0) (cid:107) f (cid:5) (cid:107) V ∗ , (cid:107) g (cid:5) (cid:107) V ∗ , (cid:107) h (cid:5) (cid:107) U (cid:1) . Proof.
Since by [5, page 207, equation 2.7] (with α = ) the operator A π isan isomorphism from V to V ∗ , we can define r (cid:5) = − A −∗ π (Π f (cid:5) − α Π Bh (cid:5) + g (cid:5) ) ∈ V .Similarly to the proof of Lemma 2.5 we next define y (cid:5) = A − π (cid:0) α Bh (cid:5) − f (cid:5) + α BB ∗ r (cid:5) (cid:1) ∈ V , p (cid:5) = Π y (cid:5) + r (cid:5) ∈ V , and u (cid:5) = − α ( h (cid:5) + B ∗ p (cid:5) ) ∈ U . It is easily verified that thetriplet ( y (cid:5) , p (cid:5) , u (cid:5) ) is a solution to (4.2) and that it satisfies (4.3).It remains to discuss the uniqueness of the solution to (1.1) and the uniquenessof the solution to (4.2). Let us first remark that if ( y, u, p ) is solution to (4.2), then( y, u ) is solution to (1.1) with associated Lagrange multiplier p , by convexity of theoptimization problem. Therefore, the uniqueness of the solution to (4.2) implies theuniqueness of the solution to (1.1).To prove the uniqueness of the solution to (4.2), it suffices to consider the case( f (cid:5) , g (cid:5) , h (cid:5) ) = (0 , , y, u, p ) be a solution to (4.2) with ( f (cid:5) , g (cid:5) , h (cid:5) ) = (0 , , r = p − Π y . It then follows that A ∗ π r = 0 and, hence, r = 0. Consequently,we have Π y = p and with Ay = − Bu we conclude that A π y = 0. This implies y = 0and p = Π y = 0. Since αu + B ∗ p = 0 , we finally obtain that u = 0, which concludesthe proof the lemma.From now on, we denote(4.4) ˜ q = q − p (cid:5) + Qy (cid:5) . HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS t is not too close to 0 and nottoo close to ¯ T , then ¯ y ( t ) and ¯ p ( t ) are close to y (cid:5) and p (cid:5) , respectively. Theorem
There exists a constant M , independent of the parameters ¯ T , Q ,and ( y , f (cid:5) , g (cid:5) , h (cid:5) , q ) such that for all t ∈ [0 , ¯ T ] , (4.5) max (cid:0) (cid:107) ¯ y ( t ) − y (cid:5) (cid:107) Y , (cid:107) ¯ p ( t ) − p (cid:5) (cid:107) Y (cid:1) ≤ M (cid:0) e − λt (cid:107) y − y (cid:5) (cid:107) Y + e − λ ( ¯ T − t ) (cid:107) ˜ q (cid:107) Y (cid:1) . Remark (cid:0) (cid:107) ¯ y ( t ) − y (cid:5) (cid:107) Y , (cid:107) ¯ p ( t ) − p (cid:5) (cid:107) Y (cid:1) ≤ M e − λt + M e − λ ( ¯ T − t ) , where theconstants M and M depend on all the data of the problem (except ¯ T ). Our estimateis thus more precise: It shows that these two constants are related to (cid:107) y − y (cid:5) (cid:107) Y and (cid:107) q − p (cid:5) + Qy (cid:5) (cid:107) Y , respectively. Proof of Theorem 4.2.
Let (˜ y, ˜ u, ˜ p ) = (¯ y, ¯ u, ¯ p ) − ( y (cid:5) , u (cid:5) , p (cid:5) ). We have˜ p ( ¯ T ) − Q ˜ y ( ¯ T ) = p ( ¯ T ) − p (cid:5) − Q ( y ( ¯ T ) − y (cid:5) ) = q + Qy (cid:5) − p (cid:5) = ˜ q. Then, by (4.1) and (4.2), ˜ y (0) = y − y (cid:5) , H ( y, u, p ) = (0 , , p ( ¯ T ) − Q ˜ y ( ¯ T ) = ˜ q , i.e.(˜ y, ˜ u, ˜ p ) is the solution to ( OS ), with parameters ( y − y (cid:5) , ¯ T , Q, ˜ q ). Let ( y (1) , u (1) , p (1) )and ( y (2) , u (2) , p (2) ) be the solutions to ( OS ), with parameters ( y − y (cid:5) , ¯ T , Q,
0) and(0 , ¯ T , Q, ˜ q ) respectively. Applying Theorem 2.1 to these systems with µ = λ and µ = − λ respectively, we obtain that (cid:107) ( y (1) , u (1) , p (1) ) (cid:107) Λ T,λ ≤ M (cid:107) y − y (cid:5) (cid:107) Y , (cid:107) ( y (2) , u (2) , p (2) ) (cid:107) Λ T, − λ ≤ M e − λ ¯ T (cid:107) ˜ q (cid:107) Y We immediately deduce that for all t ∈ [0 , ¯ T ]max (cid:0) (cid:107) y (1) ( t ) (cid:107) Y , (cid:107) p (1) ( t ) (cid:107) Y (cid:1) ≤ M e − λt (cid:107) y − y (cid:5) (cid:107) Y , max (cid:0) (cid:107) y (2) ( t ) (cid:107) Y , (cid:107) p (2) ( t ) (cid:107) Y (cid:1) ≤ M e − λ ( ¯ T − t ) (cid:107) ˜ q (cid:107) Y . Estimate 4.5 follows, since by linearity, (˜ y, ˜ u, ˜ p ) = ( y (1) , u (1) , p (1) ) + ( y (2) , u (2) , p (2) ). Remark B ∈ L ( U, Y ) (instead of simply B ∈ L ( U, V ∗ )),then a turnpike property can also be established for the control: (cid:107) u ( t ) − u (cid:5) (cid:107) U = α (cid:13)(cid:13) B ∗ ( p ( t ) − p (cid:5) ) (cid:13)(cid:13) U ≤ M (cid:0) e − λt (cid:107) y − y (cid:5) (cid:107) Y + e − λ ( ¯ T − t ) (cid:107) ˜ q (cid:107) Y (cid:1) . In this subsection, we analyze someproperties of the value function associated with Problem ( P ). For an initial time θ and an initial condition y θ , the value function is defined by( P ( θ )) V ¯ T ,Q,q ( θ, y θ ) = inf y ∈ W ( θ, ¯ T ) u ∈ L ( θ, ¯ T ; U ) (cid:90) ¯ Tθ (cid:96) ( y ( t ) , u ( t )) d t + (cid:104) y ( ¯ T ) , Qy ( ¯ T ) (cid:105) Y + (cid:104) q, y ( ¯ T ) (cid:105) Y , subject to: ˙ y ( t ) = Ay ( t ) + Bu ( t ) + f (cid:5) , y ( θ ) = y θ . The shifting realized in the proof of Theorem 4.2 shows that Problem ( P ) is equivalentto a problem of the same form as ( P ) (with a different value of q ). We compare thecorresponding value functions in the next lemma.8 T. BREITEN AND L. PFEIFFER
Lemma
The following relation holds true: V ¯ T ,Q,q ( θ, y θ ) = V T − θ,Q, ˜ q ( y θ − y (cid:5) ) + (cid:104) p (cid:5) , y θ (cid:105) Y + ( ¯ T − θ ) v (cid:5) + (cid:104) y (cid:5) , Qy (cid:5) (cid:105) Y + (cid:104) q − p (cid:5) , y (cid:5) (cid:105) , (4.6) where v (cid:5) := (cid:96) ( y (cid:5) , u (cid:5) ) is the value of the static optimization problem (1.1) .Proof. It is sufficient to prove the result for θ = 0. Let ( y, u ) be such that˙ y = Ay + Bu + f (cid:5) , y (0) = y . Let (˜ y, ˜ u ) = ( y, u ) − ( y (cid:5) , u (cid:5) ). Then, ˙˜ y = A ˜ y + B ˜ u ,˜ y (0) = y − y (cid:5) . We have J ¯ T ,Q,q ( u, y ) = (cid:90) ¯ T (cid:107) C ˜ y (cid:107) Z + (cid:104) C ∗ Cy (cid:5) + g (cid:5) , ˜ y (cid:105) V ∗ ,V + (cid:16) (cid:107) Cy (cid:5) (cid:107) Z + (cid:104) g (cid:5) , y (cid:5) (cid:105) Y (cid:17) d t + (cid:90) ¯ T α (cid:107) ˜ u ( t ) (cid:107) U + (cid:104) αu (cid:5) + h (cid:5) , ˜ u ( t ) (cid:105) U + (cid:16) α (cid:107) u (cid:5) (cid:107) U + (cid:104) h (cid:5) , u (cid:5) (cid:105) U (cid:17) d t + (cid:104) ˜ y ( T ) , Q ˜ y ( T ) (cid:105) Y + (cid:104) Qy (cid:5) + q, ˜ y ( T ) (cid:105) Y + (cid:104) y (cid:5) , Qy (cid:5) (cid:105) Y + (cid:104) q, y (cid:5) (cid:105) Y . (4.7)As in the proof of Lemma 3.2, the linear terms vanish. Using C ∗ Cy (cid:5) + g (cid:5) = − A ∗ p (cid:5) , αu (cid:5) + h (cid:5) = − B ∗ p (cid:5) , and integrating by parts, one indeed obtains that(4.8) (cid:90) ¯ T (cid:104) C ∗ Cy (cid:5) + g (cid:5) , ˜ y ( t ) (cid:105) V ∗ ,V + (cid:104) αu (cid:5) + h (cid:5) , ˜ u ( t ) (cid:105) U d t = −(cid:104) p (cid:5) , ˜ y ( ¯ T ) − ˜ y (0) (cid:105) Y . Combining (4.7) and (4.8), we obtain that J ¯ T ,Q,q ( u, y ) = (cid:90) ¯ T (cid:107) C ˜ y (cid:107) Z + α (cid:107) ˜ u ( t ) (cid:107) U d t + (cid:104) ˜ y ( ¯ T ) , Q ˜ y ( ¯ T ) (cid:105) Y + (cid:104) Qy (cid:5) + q − p (cid:5) , ˜ y ( ¯ T ) (cid:105) Y + K ( y ) , where K ( y ) = ¯ T v (cid:5) + (cid:104) y (cid:5) , Qy (cid:5) (cid:105) Y + (cid:104) p (cid:5) , y − y (cid:5) (cid:105) Y + (cid:104) q, y (cid:5) (cid:105) Y . We obtain with thedefinitions of J and ˜ q given in (3.1) and (4.4) that J ¯ T ,Q,q ( u, y ) = J T ,Q, ˜ q (˜ u, ˜ y )+ K ( y ).Therefore V ¯ T ,Q,q (0 , y ) = V T ,Q, ˜ q ( y − y (cid:5) ) + K ( y ) and the lemma is proved.We deduce from Lemma 4.5 some useful information on D y θ V ¯ T ,Q,q ( θ, y θ ). Moreprecisely, relation (4.6) below shows how the derivative of the value function deviatesfrom the equilibrium value p (cid:5) . Note that the first difference term, Π( ¯ T − θ, Q )( y θ − y (cid:5) ),vanishes when y θ = y (cid:5) and the second one, G ( ¯ T − θ, Q )˜ q , is very small for large valuesof ¯ T − θ . Corollary
The following relation holds true: (4.9) D y θ V ¯ T ,Q,q ( θ, y θ ) = Π( ¯ T − θ, Q )( y θ − y (cid:5) ) + G ( ¯ T − θ, Q )˜ q + p (cid:5) . Moreover, for all θ ∈ [0 , ¯ T ] , (4.10) ¯ p ( θ ) = Π( ¯ T − θ, Q )(¯ y ( θ ) − y (cid:5) ) + G ( ¯ T − θ, Q )˜ q + p (cid:5) . Proof.
Relation (4.9) is obtained by differentiating relation (4.6) and applyingCorollary 3.3. Using the same techniques as in Lemma 3.2 and Corollary 3.3, one canprove the following sensitivity relation: ¯ p ( θ ) = D y θ V ¯ T ,Q,q ( θ, ¯ y ( θ )). Applying (4.9),relation (4.10) follows. HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS
5. Error estimate for the RHC algorithm.
The receding-horizon algorithmfor solving ( P ) consists in solving a sequence of optimal control problems with smalltime-horizon T . A sampling time τ ≤ T is fixed. At iteration n of the algorithm, anoptimal control problem is solved on the interval ( nτ, nτ + T ) and only the restrictionto ( nτ, ( n + 1) τ ) of the solution is kept. The problem which is solved at the iteration n is of the following form:( P ( θ ; φ )) inf y ∈ W ( θ,θ + T ) u ∈ L ( θ,θ + T ; U ) (cid:90) θ + Tθ (cid:96) ( y ( t ) , u ( t )) d t + φ ( θ + T, y ( θ + T )) , subject to: ˙ y ( t ) = Ay ( t ) + Bu ( t ) + f (cid:5) , y ( θ ) = y θ , where θ and y θ are given. Let us describe the function φ used as final-time cost inthe above problem. We assume that two bounded mappings ˜Π : t ∈ [0 , ∞ ) → L ( Y )and ˜ G : t ∈ [0 , ∞ ) → L ( Y ) are given as well as an element ˜ p ∈ Y . For all t ≥
0, theoperator ˜Π( t ) is assumed to be self-adjoint and positive semi-definite. The function φ is defined by(5.1) φ ( t, y ) = (cid:104) y − y (cid:5) , ˜Π( ¯ T − t )( y − y (cid:5) ) (cid:105) Y + (cid:104) ˜ G ( ¯ T − t )˜ q, y (cid:105) + (cid:104) ˜ p, y (cid:105) Y . Observe that D y φ ( θ + T, y ) = ˜Π (cid:0) ¯ T − ( θ + T ) (cid:1) ( y − y (cid:5) ) + ˜ G (cid:0) ¯ T − ( θ + T ) (cid:1) ˜ q + ˜ p. This relation shows that φ ( θ + T, · ) can be viewed as an approximation of the valuefunction V ¯ T ,Q,q ( θ + T, · ) (up to an additive constant independent of the variable y ). If˜ p = p (cid:5) and if ˜Π and Π( · , Q ) as well as ˜ G and G ( · , Q ) coincide at time ¯ T − ( θ + T ), thenthe two problems ( P ( θ )) and ( P ( θ ; φ )) are equivalent, by the dynamic programmingprinciple.A third parameter N such that N τ ≤ ¯ T is also considered. At time N τ , Problem( P ( θ )) is solved (with θ = N τ ). We give now a precise description of the algorithm.
Algorithm 5.1
Receding-Horizon methodInput: τ ≥ T ≥ τ , and N such that N τ ≤ ¯ T ; for n = 0 , , , ..., N − do Find the solution ( y, u ) to ( P ( θ ; φ )) with θ = nτ , y θ = y n , and φ given by (5.1);Set y RH ( t ) = y ( t ) and u RH ( t ) = u ( t ) for a.e. t ∈ ( nτ, ( n + 1) τ );Set y n +1 = y ( τ ); end for Find the solution ( y, u ) to Problem ( P ( θ )) with θ = N τ and y θ = y N ;Set y RH ( t ) = y ( t ) and u RH ( t ) = u ( t ) for a.e. t ∈ ( N τ, ¯ T );We are now ready to state and prove the main result of the article. We make useof the following assumptions on ˜Π and ˜ G . Hypothesis t ≥
0, ˜Π( t ) is self-adjoint positive semi-definite. Thereexists a constant M > (cid:107) ˜ G ( t ) (cid:107) L ( Y ) ≤ M e − λt and (cid:107) ˜Π( t ) (cid:107) L ( Y ) ≤ M , ∀ t ≥ G = 0. In this situation,0 T. BREITEN AND L. PFEIFFER we then have φ ( t, y ) = (cid:104) ˜ p, y (cid:105) Y . We denote (cid:107) ˜Π − Π (cid:107) ∞ = sup T ∈ [0 , ∞ ) (cid:107) ˜Π( T ) − Π( T, Q ) (cid:107) L ( Y ) (cid:107) ˜ G − G (cid:107) ∞ ,λ = sup T ∈ [0 , ∞ ) (cid:107) e λT ( ˜ G ( T ) − G ( T, Q )) (cid:107) L ( Y ) . By Assumption 5.1 and Lemma 3.1, (cid:107) ˜Π − Π (cid:107) ∞ and (cid:107) ˜ G − G (cid:107) ∞ ,λ are finite. Theorem
There exist two constants τ > and M > such that for all τ and T with τ ≤ τ ≤ T ≤ ¯ T and for all N with N τ ≤ T , the following estimate holdstrue: max (cid:0) (cid:107) y RH − ¯ y (cid:107) W (0 , ¯ T ) , (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ; U ) (cid:1) ≤ M e − λ ( T − τ ) (cid:0) e − λT K + e − λ ( ¯ T − ( Nτ + T )) K + N (cid:107) ˜ p − p (cid:5) (cid:107) Y (cid:1) , (5.2) where K = (cid:107) ˜Π − Π (cid:107) ∞ (cid:107) y − y (cid:5) (cid:107) Y and K = (cid:0) (cid:107) ˜Π − Π (cid:107) ∞ + (cid:107) ˜ G − G (cid:107) ∞ ,λ (cid:1) (cid:107) ˜ q (cid:107) Y . Moreover, J ¯ T ,Q,q ( y RH , u RH ) − V ¯ T ,Q,q (0 , y ) ≤ M e − λ ( T − τ ) (cid:0) e − λT K + e − λ ( ¯ T − ( Nτ + T )) K + N (cid:107) ˜ p − p (cid:5) (cid:107) Y (cid:1) . (5.3) The constant M is independent of ( y , f (cid:5) , g (cid:5) , h (cid:5) , q ) , Q , ¯ T , τ , T , and N .Remark τ , by increasing T , or by reducing N , which is intuitive. Let us mention, however, that the constant τ constructed in the proof cannot be chosen arbitrarily small, therefore, our resultdoes not give information on the quality of the solution for arbitrarily small samplingtimes.The error estimate also suggests to choose ˜ p = p (cid:5) . In this case, one can recom-mend to choose N such that N ≈ ( ¯ T − T ) /τ , so that the two error terms e − λT K and e − λ ( ¯ T − ( Nτ + T )) K are of the same order (with respect to T ). Remark τ for the sampling time is revealedin the proof below; in a nutshell, this lower bound ultimately allows to sum up theerror terms accumulated at each iteration of the algorithm. Let us mention that thisbound is not necessary in other works based on a dynamic programming approachand dealing with continuous-time systems. Still in those works, a lower bound on theprediction horizon T , depending on τ , is needed (see [2, 3, 4]). Proof of Theorem 5.2.
Let us set define, for n ∈ { , ..., N − } , a n = max (cid:0) (cid:107) y RH − ¯ y (cid:107) W ( nτ, ( n +1) τ ) , (cid:107) u RH − ¯ u (cid:107) L ( nτ, ( n +1) τ ; U ) (cid:1) b n = (cid:107) y RH ( nτ ) − ¯ y ( nτ ) (cid:107) Y . We also define a N = max (cid:0) (cid:107) y RH − ¯ y (cid:107) W ( Nτ, ¯ T ) , (cid:107) u RH − ¯ u (cid:107) L ( Nτ, ¯ T ; U ) (cid:1) . Let M be theconstant involved in Theorem 2.1, for µ = λ and for Q = { ˜Π( t ) | t ≥ } . Necessarily, M ≥
1. Let r ∈ (0 ,
1) be a fixed real number and let the constant τ > e − λτ ≤ M e − λτ < r < Step 1: proof of estimates on a n and b n .The first part of the proof consists in proving the following three estimates. a n ≤ M b n + M e − λ ( T − τ ) (cid:0) e − λ ( nτ + T ) K + e − λ ( ¯ T − ( nτ + T )) K + (cid:107) ˜ p − p (cid:5) (cid:107) Y (cid:1) , (5.4) b n +1 ≤ rb n + M e − λ ( T − τ ) (cid:0) e − λ ( nτ + T ) K + e − λ ( ¯ T − ( nτ + T )) K + (cid:107) ˜ p − p (cid:5) (cid:107) Y (cid:1) , (5.5) a N ≤ M b N , (5.6) HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS n = 0 , ..., N −
1. Let us set t n = nτ and t (cid:48) n = nτ + T , for all n = 0 , ..., N . Wealso set ¯ y n = ¯ y ( nτ ) and recall that y n = y RH ( nτ ). Let us denote by ( y, u ) the solutionto problem ( P ( θ ; φ )) with θ = nτ and y θ = y n . Let p be the associated costate. Byconstruction, ( y RH , u RH ) and ( y, u ) coincide on the interval ( t n , t n +1 ). Let us writethe optimality conditions satisfied by (¯ y, ¯ u, ¯ p ) and ( y, u, p ) on the interval ( t n , t (cid:48) n ). ByCorollary 4.6, we have ¯ y ( t n ) = ¯ y n H (¯ y, ¯ u, ¯ p ) = ( f (cid:5) , g (cid:5) , − h (cid:5) )¯ p ( t (cid:48) n ) − Π( ¯ T − t (cid:48) n , Q )(¯ y ( t (cid:48) n ) − y (cid:5) ) = G ( t (cid:48) n )˜ q + p (cid:5) . The optimality conditions associated with ( y, u, p ) write y ( t n ) = y n H ( y, u, p ) = ( f (cid:5) , g (cid:5) , − h (cid:5) ) p ( t (cid:48) n ) − ˜Π( ¯ T − t (cid:48) n )( y ( t (cid:48) n ) − y (cid:5) ) = ˜ G ( t (cid:48) n )˜ q + ˜ p. Thus, the triple (ˆ y, ˆ u, ˆ p )( t ) := ( y, u, p )( t n + t ) − (¯ y, ¯ u, ¯ p )( t n + t ) satisfies(5.7) ˆ y (0) = y n − ¯ y n H (ˆ y, ˆ u, ˆ p ) = (0 , , p ( T ) − ˜Π( ¯ T − t (cid:48) n )ˆ y ( T ) = w, where w = (cid:0) ˜Π( ¯ T − t (cid:48) n ) − Π( ¯ T − t (cid:48) n , Q ) (cid:1) (¯ y ( t (cid:48) n ) − y (cid:5) )+ (cid:0) ˜ G ( ¯ T − t (cid:48) n ) − G ( ¯ T − t (cid:48) n ) (cid:1) ˜ q + (˜ p − p (cid:5) ) . (5.8)The triple (ˆ y, ˆ u, ˆ p ) is the solution to ( OS ) with parameters ( y n − ¯ y n , T, ˜Π( ¯ T − t (cid:48) n ) , w ).Let us estimate (cid:107) w (cid:107) Y . By Theorem 4.2, we have(5.9) (cid:107) ¯ y ( t (cid:48) n ) − y (cid:5) (cid:107) Y ≤ M (cid:0) e − λ ( nτ + T ) (cid:107) y − y (cid:5) (cid:107) Y + e − λ ( ¯ T − ( nτ + T )) (cid:107) ˜ q (cid:107) (cid:1) . By assumption,(5.10) (cid:107) ˜ G ( ¯ T − t (cid:48) n ) − G ( ¯ T − t (cid:48) n , Q ) (cid:107) L ( Y ) ≤ e − λ ( ¯ T − ( nτ + T )) (cid:107) ˜ G − G (cid:107) ∞ ,λ . Combining (5.8), (5.9), and (5.10), and using the definitions of K and K , we obtain(5.11) (cid:107) w (cid:107) Y ≤ e − λ ( nτ + T ) K + e − λ ( ¯ T − ( nτ + T )) K + (cid:107) ˜ p − p (cid:5) (cid:107) Y . Let us find now some estimates for (ˆ y, ˆ u, ˆ p ). To this end, we proceed as in the proofof Theorem 4.2. We consider the solutions (ˆ y (1) , ˆ u (1) , ˆ p (1) ) and (ˆ y (2) , ˆ u (2) , ˆ p (2) ) to thelinear system ( OS ), with parameters ( y n − ¯ y n , T, ˜Π( ¯ T − t (cid:48) n ) ,
0) and (0 , T, ˜Π( ¯ T − t (cid:48) n ) , w ),respectively, so that (ˆ y, ˆ u, ˆ p ) = (ˆ y (1) , ˆ u (1) , ˆ p (1) ) + (ˆ y (2) , ˆ u (2) , ˆ p (2) ). Let us first applyTheorem 2.1 to the first system (with µ = 0). We obtain(5.12) (cid:107) (ˆ y (1) , ˆ u (1) , ˆ p (1) ) (cid:107) Λ T, ≤ M (cid:107) y n − ¯ y n (cid:107) Y = M b n . Lemma 1.1 and Theorem 2.1, applied to (ˆ y (2) , ˆ u (2) , ˆ p (2) ) with µ = − λ , yield (cid:107) (ˆ y (2) , ˆ u (2) , ˆ p (2) ) (cid:107) Λ τ, ≤ M e λτ (cid:107) (ˆ y (2) , ˆ u (2) , ˆ p (2) ) (cid:107) Λ τ, − λ ≤ M e λτ (cid:107) (ˆ y (2) , ˆ u (2) , ˆ p (2) ) (cid:107) Λ T, − λ ≤ M e − λ ( T − λ ) (cid:107) w (cid:107) Y . (5.13)2 T. BREITEN AND L. PFEIFFER
We deduce from (5.12) and (5.13) that a n = max (cid:0) (cid:107) ˆ y (cid:107) W (0 ,τ ) , (cid:107) ˆ u (cid:107) L (0 ,τ ) (cid:1) ≤ (cid:107) (ˆ y, ˆ u, ˆ p ) (cid:107) Λ τ, ≤ (cid:107) (ˆ y (1) , ˆ u (1) , ˆ p (1) ) (cid:107) Λ τ, + (cid:107) (ˆ y (2) , ˆ u (2) , ˆ p (2) ) (cid:107) Λ τ, ≤ M (cid:0) b n + e − λ ( T − λ ) (cid:107) w (cid:107) Y (cid:1) . (5.14)Estimate (5.4) follows from (5.11) and (5.14). Let us apply again Theorem 2.1 to(ˆ y (1) , ˆ u (1) , ˆ p (1) ), now with µ = λ . We obtain(5.15) (cid:107) (ˆ y (1) , ˆ u (1) , ˆ p (1) ) (cid:107) Λ T,λ ≤ M (cid:107) y n − ¯ y n (cid:107) Y = M b n . It follows that (cid:107) ˆ y (1) ( τ ) (cid:107) Y ≤ M e λτ b n ≤ M e λτ b n = rb n . As a direct consequence of(5.13), we have (cid:107) ˆ y (2) ( τ ) (cid:107) Y ≤ M e − λ ( T − τ ) (cid:107) w (cid:107) Y . It follows that(5.16) b n +1 = (cid:107) ˆ y ( τ ) (cid:107) Y ≤ (cid:107) ˆ y (1) ( τ ) (cid:107) Y + (cid:107) ˆ y (2) ( τ ) (cid:107) Y ≤ rb n + M e − λ ( T − τ ) (cid:107) w (cid:107) Y . Estimate (5.5) follows from (5.11) and (5.16).Let us prove the estimate on a N . Denoting by ( y, u, p ) the solution to ( P ( θ ))with θ = N τ and y θ = y N , we obtain that (ˆ y, ˆ u, ˆ p )( t ) := ( y, u, p ) − (¯ y, ¯ u, ¯ p )( t N + t )is the solution to ( OS ), with parameters ( y N − ¯ y N , ¯ T − t N , Q, µ = 0, we obtain a N ≤ (cid:107) (ˆ y, ˆ u, ˆ p ) (cid:107) Λ ¯ T − tN , ≤ M (cid:107) y N − ¯ y N (cid:107) Y ≤ M b N , as wasto be proved. Step 2: proof of the general estimates.In order to prove the result, we need to find an estimate for (cid:80) Nn =0 a n . We start byestimating b n . Re-arranging (5.5), we obtain that b n +1 ≤ rb n + (cid:0) M e − λ ( T − τ ) − λT K (cid:1) e − nλτ + (cid:0) M e − λ ( T − τ ) − λ ( ¯ T − T ) K (cid:1) e nλτ + (cid:0) M e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) Y (cid:1) . Let us introduce three sequences ( c n ) n =0 ,...,N , ( d n ) n =0 ,...,N , and ( e n ) n =0 ,...,N definedby c = 0, d = 0, e = 0, and c n +1 = rc n + e − nλτ , d n +1 = rd n + e nλτ , e n +1 = re n + 1 . It is easy to check by induction that(5.17) b n ≤ M (cid:2)(cid:0) e − λT + λτ K (cid:1) c n + (cid:0) e − λ ( T − τ ) − λ ( ¯ T − T ) K (cid:1) d n + (cid:0) e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) e n (cid:3) . Lemma 5.5 below allows to estimate ( c n ) n =0 ,...,N , ( d n ) n =0 ,...,N , and ( e n ) n =0 ,...,N . Wehave r > e − λτ ≥ e − λτ , thus(5.18) c n ≤ (cid:16) − e − λτ r (cid:17) − r n − ≤ (cid:16) − e − λτ r (cid:17) − r n − ≤ M r n − . Moreover, r < ≤ e λτ , therefore(5.19) d n ≤ (cid:16) − re λτ (cid:17) − e ( n − λτ ≤ (1 − r ) − e ( n − λτ ≤ M e ( n − λτ . We also have (cid:107) e n (cid:107) Y ≤ M . Combining (5.17), (5.18), and (5.19), we obtain that(5.20) b n ≤ M (cid:0) K e − λT + λτ r n − + K e − λ ( T − τ ) − λ ( ¯ T − T ) e ( n − λτ + e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) . HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS a n ≤ M K e − λ ( T − τ ) − λT (cid:0) r n − + e − λnτ (cid:1) + M K e − λ ( T − τ ) − λ ( ¯ T − T ) (cid:0) e ( n − λτ + e nλτ (cid:1) + M e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) . (5.21)We have e − nλτ ≤ e − ( n − λτ ≤ e − ( n − λτ ≤ r n − as well as e ( n − λτ ≤ e − λτ e nλτ ,which allows to simplify (5.21) as follows:(5.22) a n ≤ M (cid:0) K e − λT + λτ r n − + K e − λ ( T − τ ) − λ ( ¯ T − T ) e nλτ + e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) . We have(5.23) N − (cid:88) n =0 r n − ≤ r ∞ (cid:88) n =0 r n = 1 r (1 − r ) ≤ M N − (cid:88) n =0 e nλτ = e Nλτ − e λτ − ≤ e Nλτ e λτ − ≤ M e
Nλτ . Combining (5.22) and (5.23), we obtain that(5.24) N − (cid:88) n =0 a n ≤ M (cid:0) K e − λT + λτ + K e − λ ( T − τ ) − λ ( ¯ T − ( Nτ + T )) + N e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) . We obtain with (5.6) and (5.20) that a N ≤ M b N ≤ M (cid:0) K e − λT + λτ r N − + K e − λ ( T − τ ) − λ ( ¯ T − T ) e ( N − λτ + e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) ≤ M (cid:0) K e − λT + λτ + K e − λ ( T − τ ) − λ ( ¯ T − ( Nτ + T )) + e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) . (5.25)Finally, (5.24) and (5.25) yieldmax (cid:0) (cid:107) y RH − ¯ y (cid:107) W (0 , ¯ T ) , (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ; U ) (cid:1) ≤ (cid:80) Nn =0 a n ≤ M (cid:0) K e − λT + λτ + K e − λ ( T − τ ) − λ ( ¯ T − ( Nτ + T )) + N e − λ ( T − τ ) (cid:107) ˜ p − p (cid:5) (cid:107) (cid:1) , which proves (5.2). Using the same techniques as in Lemma 3.2, one can show theexistence of M such that J T,Q,q ( u RH , y RH ) − V ¯ T ,Q,q (0 , y ) ≤ M max (cid:0) (cid:107) y RH − ¯ y (cid:107) W (0 , ¯ T ) , (cid:107) u RH − ¯ u (cid:107) L (0 ,T ; U ) (cid:1) . Using then (5.2), we obtain (5.3). The theorem is proved.The following lemma is an independent technical result, used only in the aboveproof.
Lemma
Let r > and r > be two positive real numbers. Consider thesequence ( ξ n ) n ∈ N defined by ξ = 0 , ξ n +1 = r ξ n + r n , ∀ n ∈ N . If r < r , then ξ n ≤ − r /r r n − , for all n ∈ N . If r < r , then ξ n ≤ − r /r r n − ,for all n ∈ N . T. BREITEN AND L. PFEIFFER
Proof.
One can easily check by induction that ξ n = r n − n − (cid:88) i =0 (cid:16) r r (cid:17) i = r n − n − (cid:88) i =0 (cid:16) r r (cid:17) i , ∀ n ∈ N . If r < r , then (cid:80) n − i =0 (cid:0) r r (cid:1) i ≤ (cid:0) − r r (cid:1) − , which proves the first estimate. If r < r ,then (cid:80) n − i =0 (cid:0) r r (cid:1) i ≤ (cid:0) − r r (cid:1) − and the second estimate follows.
6. Infinite-horizon problems.6.1. Formulation of the problem and overtaking optimality.
In this sub-section we investigate the case of linear-quadratic optimal control problems with aninfinite horizon. The investigated problem can be seen as a limit problem of ( P ) when¯ T goes to ∞ . For this purpose, we introduce the space L (0 , ∞ ) of locally squareintegrable functions and the space W loc (0 , ∞ ) of functions y : (0 , ∞ ) → V such thatfor all T > y | (0 ,T ) ∈ W (0 , T ). Consider the problem( P ( ∞ )) inf y ∈ W loc (0 , ∞ ) u ∈ L (0 , ∞ ; U ) (cid:90) ∞ (cid:96) ( y ( t ) , u ( t )) d t subject to: ˙ y ( t ) = Ay ( t ) + Bu ( t ) + f (cid:5) , y (0) = y . In general, the above integral is not proper and one needs to use an appropriate notionof optimality. Let us mention that this difficulty would also arise if we chose W (0 , ∞ )and L (0 , ∞ ; U ) as function spaces. We call a pair ( y, u ) ∈ W loc (0 , ∞ ) × L (0 , ∞ ; U ) feasible pair if ˙ y = Ay + Bu + f and y (0) = y . Definition
A feasible pair (¯ y, ¯ u ) ∈ W loc (0 , ∞ ) × L (0 , ∞ ; U ) is said to be overtaking optimal for Problem ( P ( ∞ )) if for all feasible pairs ( y, u ) ∈ W loc (0 , ∞ ) × L (0 , ∞ ; U ) , lim inf T →∞ (cid:0) J T, , ( u, y ) − J T, , (¯ u, ¯ y ) (cid:1) ≥ . The notion of overtaking optimality is rather classical in the literature, see forexample [30], where some existence results are established. We construct now a pair(¯ y, ¯ u ) which will be the unique overtaking optimal solution to problem ( P ( ∞ )). Let˜ y ∈ W (0 , ∞ ), ˜ p ∈ W (0 , ∞ ), and ˜ u ∈ L (0 , ∞ ; U ) be defined by ˙˜ y = A π ˜ y , ˜ y (0) = y − y (cid:5) , ˜ p = Π˜ y , ˜ u = − α B ∗ p . Using the same arguments as in Lemma 2.5, we cancheck that ˜ p ∈ W (0 , ∞ ) with − ˙˜ p = A ∗ ˜ p + C ∗ C ˜ y . We finally set(¯ y, ¯ u, ¯ p )( t ) = ( y (cid:5) , u (cid:5) , p (cid:5) ) + (˜ y, ˜ u, ˜ p )( t ) . We have (¯ y, ¯ u, ¯ p ) ∈ W loc (0 , ∞ ) × L (0 , ∞ ; U ) × W loc (0 , ∞ ). A key point in ouranalysis is that for all T >
0, the triplet (¯ y, ¯ u, ¯ p ) is the unique solution to the followingoptimality system:(6.1) ¯ y (0) = y , H ( y, u, p ) = ( f (cid:5) , g (cid:5) , − h (cid:5) ) , ¯ p ( T ) − Π¯ y ( T ) = p (cid:5) − Π y (cid:5) . One can prove with standard arguments (¯ y, ¯ u, ¯ p ) is the unique overtaking optimalsolution. We refer the reader to [23], where a more general class of linear-quadraticproblems is investigated. Proposition
The pair (¯ y, ¯ u ) is the unique overtaking optimal solution to ( P ( ∞ )) . More precisely, we have (6.2) lim inf T →∞ (cid:16) J ( u, y, T ) − J (¯ u, ¯ y, T ) − (cid:16) α − ε (cid:17) (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:17) ≥ , HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS for all ε > and for all feasible ( y, u ) .Proof. Let us first prove that(6.3) J ( u, y, T ) − J (¯ u, ¯ y, T ) − α (cid:90) T (cid:107) u − ¯ u (cid:107) U d t = (cid:90) T (cid:107) C ( y − ¯ y ) (cid:107) Z −(cid:104) ¯ p ( T ) , y ( T ) − ¯ y ( T ) (cid:105) Y . The calculations are very similar to those of the proof of Lemma 3.2. We have J ( u, y, T ) − J (¯ u, ¯ y, T ) = (cid:90) T (cid:107) C ( y − ¯ y ) (cid:107) Z + α (cid:107) u − ¯ u (cid:107) U d t + (cid:90) T (cid:104) C ∗ C ¯ y + g, y − ¯ y (cid:105) Y + (cid:104) α ¯ u + h, u − ¯ u (cid:105) U d t. (6.4)Using C ∗ C ¯ y + g (cid:5) = − ˙¯ p − A ∗ ¯ p , α ¯ u + h (cid:5) = − B ∗ ¯ p and integrating by parts, we obtainthat(6.5) (cid:90) T (cid:104) C ∗ C ¯ y + g (cid:5) , y − ¯ y (cid:105) Y + (cid:104) α ¯ u + h (cid:5) , u − ¯ u (cid:105) d t = −(cid:104) ¯ p ( T ) , y ( T ) − ¯ y ( T ) (cid:105) Y . Combining (6.4) and (6.5), we obtain (6.3).Let ˆ y = y − ¯ y . We have ˙ˆ y = A ˆ y + B ( u − ¯ u ), ˆ y (0) = 0. Therefore, by Lemma 2.7,there exists a constant M independent of T such that (cid:107) y ( T ) − ¯ y (cid:107) Y = (cid:107) ˆ y ( T ) (cid:107) ≤ M (cid:0) (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) + (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) (cid:1) . The adjoint ¯ p is bounded, since ¯ p = p (cid:5) + ˜ p , where ˜ p ∈ W (0 , ∞ ). Therefore, |(cid:104) ¯ p ( T ) , y ( T ) − ¯ y ( T ) (cid:105) Y | ≤ M (cid:0) (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) + (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) (cid:1) , where again M does not depend on T . We deduce that J ( u, y, T ) − J (¯ u, ¯ y, T ) − (cid:16) α − ε (cid:17) (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) ≥ (cid:0) ε (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) − M (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:1) + (cid:16) (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) − M (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) (cid:17) . The two terms on the r.h.s. in the above inequality are bounded from below. Thus,if one of them tends to infinity (which is the case if (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) −→ T →∞ ∞ or (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) −→ T →∞ ∞ ), then (cid:16) J ( u, y, T ) − J (¯ u, ¯ y, T ) − (cid:16) α − ε (cid:17) (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:17) −→ T →∞ ∞ and therefore, (6.2) holds true. Otherwise, if (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) and (cid:107) C ( y − ¯ y ) (cid:107) L (0 ,T ; Z ) are both bounded, then y − ¯ y ∈ W (0 , ∞ ) (by Lemma 2.7) and therefore y ( T ) − ¯ y ( T ) −→ T →∞ (cid:104) ¯ p ( T ) , y ( T ) − ¯ y ( T ) (cid:105) Y −→ T →∞
0. Wededuce then from (6.3) thatlim inf T →∞ J ( u, y, T ) − J (¯ u, ¯ y, T ) − α (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) ≥ , which proves (6.2) and that (¯ y, ¯ u ) is overtaking optimal.6 T. BREITEN AND L. PFEIFFER
Let us prove uniqueness. Let ( y, u ) be overtaking optimal. Then, by definition,0 ≤ lim inf T →∞ (cid:0) J (¯ u, ¯ y, T ) − J ( u, y, T ) (cid:1) ≤ lim sup T →∞ (cid:0) J (¯ u, ¯ y, T ) − J ( u, y, T ) (cid:1) . Therefore, using (6.2) with ε = α ,0 ≥ − lim sup T →∞ (cid:0) J (¯ u, ¯ y, T ) − J ( u, y, T ) (cid:1) = lim inf T →∞ (cid:0) J ( u, y, T ) − J (¯ u, ¯ y, T ) (cid:1) ≥ lim inf T →∞ (cid:16) J ( u, y, T ) − J (¯ u, ¯ y, T ) − α (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ≥ + α T →∞ (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) ≥ α T →∞ (cid:107) u − ¯ u (cid:107) L (0 ,T ; U ) . We immediately deduce that u = ¯ u . Thus y = ¯ y , which concludes the proof ofuniqueness.The next lemma deals with the asymptotic analysis of J (¯ u, ¯ y, T ). Lemma
For all
T > , the following equality holds true: J (¯ u, ¯ y, T ) = T v (cid:5) + 12 (cid:104) y − y (cid:5) , Π( y − y (cid:5) ) (cid:105) Y − (cid:104) p (cid:5) , ¯ y ( T ) − y (cid:5) (cid:105)− (cid:104) ¯ y ( T ) − y (cid:5) , Π(¯ y ( T ) − y (cid:5) ) (cid:105) Y , (6.6) where v (cid:5) is the value of problem (1.1) . A direct consequence is the following relation: (6.7) lim T →∞ J (¯ u, ¯ y, T ) T = v (cid:5) . Proof.
A direct consequence of (6.1) is that (¯ y, ¯ u ) | (0 ,T ) is the unique solution to P ( y , T, Π , q ), where q = p (cid:5) − Π y (cid:5) . The corresponding ˜ q (defined by (4.4)) is then˜ q = q − p (cid:5) + Π y (cid:5) = 0 . By Corollary 4.6, we have(6.8) V ( y , T, Π , q ) = V ( y − y (cid:5) , T, Π ,
0) + (cid:104) p (cid:5) , y (cid:105) Y + T v (cid:5) + 12 (cid:104) y (cid:5) , Π y (cid:5) (cid:105) Y + (cid:104) q − p (cid:5) , y (cid:5) (cid:105) Y . As was explained in the proof of Corollary 3.3, V ( y, T, Π ,
0) = (cid:104) y, Π( T, Π) y (cid:105) Y . ByLemma 3.1, Π( T, Π) = Π. Therefore, (6.8) becomes(6.9) V ( y , T, Π , q ) = 12 (cid:104) y − y (cid:5) , Π( y − y (cid:5) ) (cid:105) Y + (cid:104) p (cid:5) , y (cid:105) Y + T v (cid:5) + 12 (cid:104) y (cid:5) , Π y (cid:5) (cid:105) Y − (cid:104) Π y (cid:5) , y (cid:5) (cid:105) Y . We also have J (¯ u, ¯ y, T ) = J (¯ u, ¯ y, T, Π , q ) − (cid:104) ¯ y ( T ) , Π¯ y ( T ) − (cid:104) q, ¯ y ( T ) (cid:105) Y = V ( y , T, Π , q ) − (cid:104) ¯ y ( T ) , Π¯ y ( T ) − (cid:104) q, ¯ y ( T ) (cid:105) Y . (6.10)Formula (6.6) can be obtained by combining (6.9) and (6.10). Formula (6.7) followsfrom the fact that ¯ y − y (cid:5) converges exponentially to 0. HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS As before, one can find an approxi-mation of (¯ y, ¯ u ) by using the RHC algorithm. We have ¯ p ( T ) = Π(¯ y ( T ) − y (cid:5) ) + p (cid:5) .Therefore, a good choice of a terminal cost function in the receding horizon algorithmis a function whose derivative (w.r.t. y ) is an approximation of Π( y − y (cid:5) ) + p (cid:5) . Wetherefore consider(6.11) φ ( t, y ) = (cid:104) y − y (cid:5) , ˆΠ( y − y (cid:5) (cid:105) Y + (cid:104) ˆ p, y (cid:105) Y , where ˆ p ∈ Y and ˆΠ ∈ L ( Y ) is self-adjoint and positive semi-definite. If one choosesˆΠ = Π and ˆ p = p (cid:5) , then the Receding-Horizon algorithm provides the exact overtakingoptimal solution to the problem. Let us mention that the function φ that we pro-pose for the infinite-horizon problem is independent of time. The Receding-Horizonalgorithm is now very similar to Algorithm 5.1. Algorithm 6.1
Receding-Horizon methodInput: τ ≥ T ≥ τ , and N ∈ N ;Set n = 0 and y n = y ; for n = 0 , , , ..., N − do Find the solution ( y, u ) to ( P ( θ ; φ )) with θ = nτ , y θ = y n , and φ given by (6.11);Set y RH ( t ) = y ( t ) and u RH ( t ) = u ( t ) for a.e. t ∈ ( nτ, ( n + 1) τ );Set y n +1 = y ( τ ); end for Theorem
There exist two constants τ > and M > such that for all ( y , f (cid:5) , g (cid:5) , h (cid:5) ) , for all τ ≤ τ ≤ T , the following estimate holds true: max (cid:0) (cid:107) y RH − ¯ y (cid:107) W (0 ,Nτ ) , (cid:107) u RH − ¯ u (cid:107) L (0 ,Nτ ; U ) (cid:1) ≤ M e − λ ( T − τ ) (cid:0) e − λT (cid:107) ˆΠ − Π (cid:107) L ( Y ) (cid:107) y (cid:107) Y + N (cid:107) ˆ p − p (cid:5) (cid:107) Y (cid:1) . (6.12) Remark τ and increasing T should improvethe quality of the solution obtained with the Receding-Horizon algorithm (still, thecase of arbitrarily small values of τ is not covered). Also, one should choose ˆ p = p (cid:5) since in this case the error estimate becomes independent of N . Proof of Theorem 6.4.
Let us fix ¯
T > N τ . As a direct consequence of (6.1),(¯ y, ¯ u ) | (0 , ¯ T ) is the unique solution to ( P ) with initial condition y , horizon ¯ T , Q = Π,and q = p (cid:5) − Π y (cid:5) . The corresponding ˜ q is null. Consider now the pair (˜ y RH , ˜ u RH )obtained when solving this problem with the same values of the parameters τ , T ,and N and with ˜Π( T ) = ˆΠ and ˜ G ( T ) = 0. By construction, ( y RH , u RH ) and(˜ y RH , ˜ u RH ) coincide on (0 , N τ ). Estimate (6.12) is directly obtained by applyingTheorem 5.2. Indeed, the constant K involved in (5.2) is null, since ˜ q = 0 and sincesup T ∈ [0 , ∞ ) (cid:107) ˜Π( T ) − Π( T, Π) (cid:107) L ( Y ) = (cid:107) ˆΠ − Π (cid:107) L ( Y ) , by Lemma 3.1.
7. Numerical verification.
In this section we aim at measuring the tightnessof our estimate. Our focus is the dependence of (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ; U ) with respect to τ and T . We consider for this purpose an optimal control problem with state variable8 T. BREITEN AND L. PFEIFFER of dimension 2 and scalar control, described by the following data: A = (cid:18) . . − . (cid:19) , B = (cid:18) (cid:19) , C = (cid:18) (cid:19) , α = 0 . ,y = (cid:18) (cid:19) , Q = (cid:18) (cid:19) , q = (cid:18) (cid:19) , ¯ T = 30 . Observe that the matrix A is not stable. The optimal control and the associatedtrajectory are represented on the graphs of Figure 1. The dashed lines correspond tothe values of u (cid:5) and y (cid:5) , respectively. TurnpikeOptimal trajectory y Turnpike
Fig. 1 . Optimal control and optimal trajectory
We have generated different controls with the RHC algorithm, for values of τ and T ranging from 0 . . , ˜ G = 0 , ˜ p = p (cid:5) , N = (cid:98) ( ¯ T − T ) /τ (cid:99) . All optimal control problems have been solved with the limited-memory BFGS method,with a tolerance of 10 − for the L -norm of the gradient of the reduced cost function.For the discretization of the state equation, we have used the implicit Euler schemewith time-step equal to 5 × − . As a consequence of Theorem 5.2, there exist τ > M >
0, both independent of τ and T , such that (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) ≤ M e − λT + λτ ,for τ ≤ τ ≤ T ≤ ¯ T . Thus the quantity ρ ( τ, T ) := ln( (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) ) + 2 λT − λτ is bounded from above, for sufficiently large values of τ . The results obtained for (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) and 100 ρ ( τ, T ) are shown in Figures 2 and 3, where λ = 0 .
36 is theopposite of the spectral absicissa of A π . A first observation is that (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) isdecreasing with respect to T and increasing with respect to τ . Moreover, the number ρ ( τ, T ) takes values between 0 .
40 and 0 .
73. The variation of ρ ( τ, T ) can be regardedas small, in comparison with the variation of 2 λT − λτ (approximately equal to 5,comparing T = 0 . T = 7 . ρ is constant andconclude that our error estimate gives an accurate description of the dependence of (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ; U ) with respect to τ and T . Conclusion.
New error bounds for linear optimality systems associated withoptimal control problems have been obtained in weighted spaces. They have enabled
HC METHOD FOR LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS Tτ . . . . . . . . − . − . − . − . −
21 1 . . − . − . − . − . − . . . − . − . − . −
22 8 . − . − . − . − . . − . − . −
23 2 . − . − . . − Tτ . . . . . . − . − . − . − . − . − . − . −
21 1 . − . − . − . − . − . − . − . − . . − . − . − . − . − . − . − . −
22 1 . − . − . − . − . − . − . − . − . . − . − . − . − . − . − . − . −
23 2 . − . − . − . − . − . − . − . − . . − . − . − . − . − . − . − . −
14 3 . − . − . − . − . − . − . − . − . . − . − . − . − . − . − . −
15 2 . − . − . − . − . − . − . . − . − . − . − . −
16 1 . − . − . − . − . . − . − . −
27 1 . − . − . . − Fig. 2 . (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) for different values of τ and T . Tτ . . . . . . . . . . . . . . . . Fig. 3 . 100(ln( (cid:107) u RH − ¯ u (cid:107) L (0 , ¯ T ) ) + 2 λT − λτ ) , for different values of τ and T . us to improve the exponential turnpike property for linear-quadratic problems and toobtain a precise error estimate for the control generated by the RHC algorithm.Future research will be dedicated to the extension of our results to non-linearsystems. Let us mention that an error estimate for the RHC method has been obtainedfor stabilization problems of bilinear systems in [17], by application of the inversemapping theorem in weighted spaces. Another axis of research will focus on theextension of our results to the wave equation. Acknowledgements.
This project has received funding from the European Re-search Council (ERC) under the European Unions Horizon 2020 research and inno-0
T. BREITEN AND L. PFEIFFER vation programme (grant agreement No 668998).
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