Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain
aa r X i v : . [ m a t h . A P ] N ov Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X , Number , XX pp. X–XX
CONTROLLABILITY OF THE HEAT AND WAVE EQUATIONSAND THEIR FINITE DIFFERENCE APPROXIMATIONS BY THESHAPE OF THE DOMAIN
Jonathan Touboul
Mathematical Neuroscience Laboratory, CIRB-Coll`ege de France and BANG Laboratory, INRIA Paris-Rocquencourt.11, place Marcelin Berthelot75005 Paris, FRANCE (Communicated by the associate editor name)
Abstract.
In this article we study a controllability problem for a parabolicand a hyperbolic partial differential equations in which the control is the shapeof the domain where the equation holds. The quantity to be controlled isthe trace of the solution into an open subdomain and at a given time, whenthe right hand side source term is known. The mapping that associates thistrace to the shape of the domain is nonlinear. We show (i) an approximatecontrollability property for the linearized parabolic problem and (ii) an exactlocal controllability property for the linearized and the nonlinear equations inthe hyperbolic case. We then address the same questions in the context ofa finite difference spatial semi-discretization in both the parabolic and hyper-bolic problems. In this discretized case again we prove a local controllabilityresult for the parabolic problem, and an exact controllability for the hyperboliccase, applying a local surjectivity theorem together with a unique continuationproperty of the underlying adjoint discrete system.
Introduction.
The problem of characterizing the shape of a domain where a cer-tain dynamical phenomenon is partially observable is a model for a wide class ofapplications. A typical example is given by the identification of the shape of anhydrocarbon or water reservoir. Techniques used by geologists to tackle this prob-lem include sending shock waves into the ground or into a drill hole and measurethe reflected waves. The question we may address here is whether one can inferthe shape of the full reservoir by these partial and local informations. Chenaisand Zuazua [6] addressed this problem for the elliptic equation case dealing withthe Laplace equation with Dirichlet boundary conditions. They showed that thelinearized problem admits an approximate controllability property, and the finite-differences discretization presents a local controllability property.In the present manuscript, we extend their results by addressing the same ques-tions in a dynamical setting. In details, we shall consider in this manuscript evo-lution partial differential equations of parabolic and hyperbolic type equation onan open set Ω ⊂ R n , given an external source term. The domain Ω is assumedto be only partially known. It is potentially allowed to evolve with time, in whichcase it will be denoted Ω( t ), and is assumed to contain for all times a fixed simply Mathematics Subject Classification.
Primary: 35K05, 93B03; Secondary: 65M06.
Key words and phrases.
Heat equation, Wave equation, controllability, shape of the domain,semi-discrete controllability. connected open subset ω ⊂ R n . More precisely, we assume that the closure of ω isa subset of Ω( t ) for all times, a property we shall denote ω ⋐ Ω. We assume thatthe solution of the partial differential equation restricted to ω at time T is known,either because it is accessible or observable by means of measurement. The problemwe address is, loosely speaking, to recover the shape Ω( t ) from the knowledge of theexternal forcing term and the restriction of the solution to the subdomain ω at time T . This is the question we refer to as the controllability problem . It differs from [6]in that we consider evolution equations and allow the domain to dynamically evolvein time. These distinctions yield increased complexity of the functional setting thatnecessitate to partially modify the result obtained by Chenais and Zuazua.In this article, we only address the case of the heat and wave equations as paradig-matic examples of, respectively, parabolic and hyperbolic equations and will restrictthe study to the case of Dirichlet boundary conditions, although the problem mightarise for other differential operators and different boundary conditions and the meth-ods used here may apply in most regular cases.Returning to the mathematical formulation of the problem, we are given: • L a given parabolic or hyperbolic differential operator, i.e. either the heat orthe wave operator, • a source-term f ∈ L ( R + × R n ), • ω an open bounded subset of R n assumed regular, • y d ∈ H ( ω ) the observed solution of the PDE L y = f at time T restricted to ω .and aim at proving existence and uniqueness of a bounded, possibly time-varying,open set Ω( t ) ⊂ R n for t ∈ [0 , T ] with ω ⋐ Ω( t ) for all t and such that the solution y Ω( · ) of the equation: ( L y Ω = f on Ω( t ) y Ω ( t, x ) = 0 x ∈ ∂ Ω( t )satisfies y Ω ( T, · ) | ω = y d .It is therefore a shape identification problem, which can be seen as a control-lability problem in the sense that the domain Ω has to be determined so that y Ω ( T, · ) | ω = y d holds. Problems of this type, in different settings, have been ad-dressed using a variety of methods. Some authors have used optimization methods(e.g. [4, 7, 5, 15, 17, 18, 20]), writing the problem in the form:inf Ω ∈U k y Ω − y d k where U is the set of domains we take into account. Under suitable conditions onthe set of admissible domains U , they show that the existence of a minimizer canbe guaranteed. However, this existence result does not ensures that y Ω | ω = y d doesactually hold for a certain choice of the domain Ω, nor does it evaluate the minimaldistance between y Ω | ω and y d . Therefore, optimization techniques will not solve thecontrollability problem under consideration.It is important to note the fact that solutions to the controllability problem donot necessarily exist, and if they exist, these are not necessarily unique. In orderfor our controllability problem to be well posed, we will be interested in a localcontrollability property. More precisely, we consider a reference domain Ω forwhich the partial differential equation has a solution y . The question we ask isthe following: given y d a function close from y ( T ) | ω in a suitable topology, is therean open set Ω ∗ ( t ) close from Ω (in a sense that will be defined in the sequel) such HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 3 that y Ω ∗ ( T ) | ω = y d . This problem is referred to as the local controllability problem .It may also happen that we can approach y d arbitrarily close but cannot find anoptimal open set Ω ∗ ( t ) such that y Ω ∗ ( T ) | ω = y d . We will hence consider: • the exact controllability properties, where there exists a domain Ω ∗ ( t ) suchthat y Ω ∗ ( T, · ) | ω = y d • and the approximate controllability property, where for any ε > ε ( t ) such that (cid:13)(cid:13) y Ω ε ( T, · ) | ω − y d | ω k ≤ ε in a suitable space.There is an extensive literature on exact and approximate controllability problemsfor partial differential equations (see e.g. [12, 21, 25]), but very little has been donefor controls with respect to the shape of the domain.As discussed in detail in [6], the problem we consider presents various technicaldifficulties preventing the use of existing methods. The main problem is the non-linear nature of the mapping Ω y Ω , differing from existing results in nonlinearcontrollability problems often valid only for “mild” nonlinear perturbations, andwhich is not the case here since the trace of the solution depends on the shape ofthe domain in a genuinely nonlinear way.In order to achieve our program of controlling such equations, it is natural tostart by linearizing the problem, with the aim of applying the Inverse FunctionTheorem (IFT). This approach, developed in section 1, allows proving that thelinearized control problem presents an approximate local controllability property inthe parabolic case and a exact local controllability property in the hyperbolic case.The proof of this property consists in applying a duality argument together with aunique continuation result for the solutions of the adjoint system, which is obtainedas a consequence of Holmgren’s uniqueness theorem.However, even if we are able to prove the linearized problem is approximately orexactly controllable, this does not allow to conclude a controllability result aboutthe initial nonlinear system. The main limitations are related to the inherent com-plexity of the spaces in which the nonlinear problem holds. These limitations arereleased when considering spatially discretized versions of the problem, holding insimpler finite-dimensional spaces. This is what we show in section 2 in the contextof semi-discrete finite-difference approximations of the initial problem. By reduc-ing the problem into an ordinary differential equation in a finite-dimensional space,we will prove local controllability property for the semi-discrete parabolic problemand exact controllability result in the hyperbolic case. In the context of finite-dimensional systems, approximate and exact controllability are equivalent notions,and the problem may be reduced to an unique continuation issue for the adjoint sys-tem for the linearized system. The price to pay is that classical tools used for uniquecontinuation used in the domain of PDEs (like Holmgren Uniqueness Theorem orCarleman inequalities) do not seem to apply in spatially discrete systems. Thus, thefirst task that we undertake is to prove a new unique continuation result holding inour setting based on propagation properties the discrete scheme naturally induces.This allows to demonstrate that the linearized model is exactly controllable, whichimplies a local controllability result in virtue of the IFT. Note that these results areonly concerned with local controllability results for fixed discretization meshes. Avery interesting problem would be to address the convergence of the control shapesas the mesh-size h tends to zero. This is a problem of primary importance but itsanalysis is out of reach for the techniques developed here and not in the scope ofthis paper. At this respect it is important to note that very little is known about JONATHAN TOUBOUL the convergence of controls in the context of the controllability of numerical ap-proximations of PDEs too. For instance, in the case of controllability of the waveequation where the control is on the boundary condition, it is known that controlsdo not necessarily converge as the mesh size tends to zero because of high frequencyspurious oscillations (see [23]). However, in the context of the heat equation, atleast in one dimension, the controls driving solutions to rest do converge as themesh size tends to zero [22]. Similar results hold in the context of homogenizationfor wave and heat equations with rapidly oscillating coefficients [24]. It is highlynontrivial to extend these techniques when the control is the shape of the domain.1.
Controllability of the heat and wave equations.
We consider here theproblem of the controllability of the heat and wave PDEs as a function of the shapeof the domain.1.1.
Controllability of the heat equation.
Let us start by introducing themathematical framework used throughout this section. We classically denote by W k, ∞ ( R n , R m ) the set of functions k times differentiable of R n (in the sense of dis-tributions), taking values in R m , with all differentials in L ∞ ( R n , R m ). We considerthe classical heat equation with a source term f ∈ L ( R + , R n ) holding on an openset Ω with smooth boundary ∂ Ω ∈ W , ∞ : ∂ t y ( t, x ) − ∆ y ( t, x ) = f ( t, x ) t > x ∈ Ω y (0 , x ) = 0 x ∈ Ω y ( t, x ) = 0 t > x ∈ ∂ Ω . (1)This equation defines a unique solution y . We are interested in possible valuesof y ( T ) | ω solutions of the heat equations holding in Ω( t ) a “perturbation” of Ω .We work in the standard setting for differentiation with respect to the domain(see for instance [15, 17, 19, 18, 20]). The admissible perturbed open sets are“small dynamical perturbations” of Ω . In details, we consider deformation func-tions ϕ : R + × R n R n that are L ([0 , T ] , W , ∞ ( R n , R n )). We moreover considertransformations such that for all t ∈ [0 , T ], the maps ( id + ϕ ( t )) and ( id + ϕ ( t )) − arehomemorphisms of W , ∞ ( R n , R n ) . Our choice will be, within this set, to consider ϕ such that k ϕ k ≤ ε for some ε >
0. This set is noted W .For ϕ ∈ W , we defineΩ ϕ ( t ) = ( id + ϕ ( t )) (Ω ) = { x + ϕ ( t )( x ); x ∈ Ω } . The perturbed equation is: ∂ t y ϕ ( t, x ) − ∆ y ϕ ( t, x ) = f ( t, x ) t > x ∈ Ω ϕ ( t ) y ϕ (0 , x ) = 0 x ∈ Ω ϕ (0) y ϕ ( t, x ) = 0 t > x ∈ ∂ Ω ϕ ( t ) (2)and function associating the trace of the solution on ω is noted Λ:Λ := (cid:26) W 7→ H ( ω ) ϕ y ϕ ( T, · ) | ω . (3)The range of Λ: R (Λ) = Λ( W ) = { y ϕ ( T, · ) | ω ; ϕ ∈ W} constitute the set of acces-sible states at time T . This is always the case if for all t ∈ [0 , T ], ϕ ( t ) is close from 0 in W , ∞ ( R n , R n ). HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 5
Existence and uniqueness of solutions for the perturbed problem.
In this sec-tion we use the variational formulation and show weak existence and uniqueness ofthe solution of the perturbed system.
Lemma 1.1.
The perturbed problem (2) is equivalent to the following partial dif-ferential equation on y ϕ ( t, x ) = y ϕ (cid:16) t, x + ϕ ( t )( x ) (cid:17) on R + × Ω : ∂ t y ϕ − | det ( id + ∇ ϕ ( t )) | div ( B ( ϕ ) ∇ y ϕ ) = f ◦ ( id + ϕ ) t > and x ∈ Ω y ϕ (0 , x ) = 0 x ∈ Ω y ϕ ( t, x ) = 0 t > and x ∈ ∂ Ω (4) with f ◦ ( id + ϕ )( t, x ) = f ( t, x + ϕ ( t )( x )) and B ( ϕ ) = (cid:12)(cid:12)(cid:12) det (cid:0) id + ∇ ϕ ( t ) (cid:1)(cid:12)(cid:12)(cid:12) (cid:18)h(cid:0) ∇ ( id + ϕ ( t )) (cid:1) ∗ i − (cid:19) ∗ h(cid:0) ∇ ( id + ϕ ( t )) (cid:1) ∗ i − . (5) Proof.
We use the variational formulation of equations (2) and show that it is equiv-alent to the variational formulation of the problem (4). Let φ ∈ H (cid:16) R + , H (Ω ϕ ( t )) (cid:17) a test function. We have: Z R + × Ω ϕ ( t ) f φdt dx = Z R + × Ω ϕ ( t ) ( ∂ t y ϕ φ − ∆ y ϕ φ ) dt dx = Z R + × Ω ϕ ( t ) ( ∂ t y ϕ φ + ∇ y ϕ ∇ φ ) dt dx. We introduce the function Ψ defined on R + × Ω by Ψ( t, x ) = φ ( t, x + ϕ ( t )( x )),and change variables in the above weak formulation defining r ∈ Ω such that x = r + ϕ ( r ). The determinant of the Jacobian matrix of the change of variable isgiven by det( id + ∇ ϕ ), yielding: Z R + × Ω { ∂ t y ϕ Ψ − ( ∇ y ϕ ) ◦ ( id + ϕ ) · ( ∇ φ ) ◦ ( id + ϕ ) } | det( id + ∇ ϕ ( t )) | dt dx = Z R + × Ω f ◦ ( id + ϕ )Ψ | det( id + ∇ ϕ ( t )) | dt dx (6)We therefore need to express ( ∇ Φ) ◦ ( id + ϕ ) as a function of ∇ (cid:16) Φ ◦ ( id + ϕ ) (cid:17) . Let H = Φ ◦ ( id + ϕ ) and T = ( id + ϕ ). We clearly have ∇ H = ( ∇ T ) ∗ ∇ Φ where thestar denotes the adjoint operator, yielding: Z R + × Ω ∂ t y ϕ Ψ − h(cid:0) ∇ ( id + ϕ ( t )) (cid:1) ∗ i − ∇ y ϕ h(cid:0) ∇ ( id + ϕ ( t )) (cid:1) ∗ i − ∇ Ψ | det( id + ∇ ϕ ( t )) | dt dx = Z R + × Ω f ◦ ( id + ϕ )Ψ | det( id + ∇ ϕ ( t )) | dt dx. (7)Using the operator B introduced in equation (5), it is easy to see that the initialvariational problem is equivalent to the variational problem: Z R + × Ω (cid:8) ∂ t y ϕ | det( id + ∇ ϕ ( t )) | − div (cid:0) B ( ϕ ) (cid:1) ∇ y ϕ (cid:9) Ψ dt dx = Z R + × Ω f ◦ ( id + ϕ )Ψ (cid:12)(cid:12) det (cid:0) id + ∇ ϕ ( t ) (cid:1)(cid:12)(cid:12) dt dx (8)whose solutions define the weak solutions of the partial differential equation (4). JONATHAN TOUBOUL
This formulation has the interest to hold on a fixed domain. Based on thisformulation, we show the well-posedness of the perturbed problem.
Proposition 1.
The PDE (2) has a unique solution in L (cid:16) , T ; H (Ω ) (cid:17) ∩ C (cid:16) [0 , T ] , L (Ω ) (cid:17) .Proof. Let H = L (Ω ) and V = H (Ω ), and recall that the dual space of V is,by definition, V ′ = H − (Ω ). Let ( f ∈ L (0 , T, V ′ ) u ∈ L (Ω ) = H A classical result due to Lions (see e.g. [2, theorem X.9]) ensures that, providedthat there exists a function a ( t ; u, v ) measurable as a function of time, bilinear in( u, v ) for all t ∈ [0 , T ] and such that there exist constants M, α and C such that: ( | a ( t ; u, v ) | ≤ M k u kk v k a ( t ; u, v ) ≥ α k v k V − C | v | H then there exists a unique function u such that: • u ∈ L (0 , T ; H (Ω )) ∩ C ([0 , T ] , L (Ω )) • ∂ t u ∈ L (0 , T ; V ′ ) • (cid:26) < ∂ t u, v > + a ( t ; u ( t ) , v ) = < f ( t ) , v > for a.e. t ∈ (0 , T ) , ∀ v ∈ H (Ω ) u (0) = u on Ω . This theorem readily applies to our case. Indeed, the bilinear form on L (Ω )given by: h f, g i = Z Ω f ( x ) g ( x ) | det( id + ∇ ϕ ) | dx. is positive definite because of the invertibility of id + ϕ ( t ) for ϕ ∈ W , hence definesa suitable scalar product on L (Ω ). Let us introduce the function: a ( t ; u, v ) = Z Ω B ( ϕ ) ∇ v ∇ (cid:18) u | det( id + ∇ ϕ ) | (cid:19) dx, which is well defined since we assumed ϕ ∈ W , ∞ ( R n , R n ). The function a is clearlymeasurable in t . Moreover, for ϕ close enough from 0 in W , we have B ( ϕ ) closefrom identity, and hence | k B ( ϕ ) k − | ≤ / ϕ small enough. We also have | det( id + ∇ ϕ ) | → ϕ → W .Therefore, by sufficiently restricting the space W , the following two inequalitiessimultaneously hold: • | | det( id + ∇ ϕ ) | − | ≤ /
2, and hence | det( id + ∇ ϕ ) | ≥ / • k B ( ϕ ) k ≤ / u | det ( id + ∇ ϕ ) | belongs to H (Ω ), implying: | a ( t ; u, v ) | ≤ Z Ω (cid:12)(cid:12)(cid:12)(cid:12) B ( ϕ ) ∇ v ∇ (cid:18) u | det( id + ∇ ϕ ) | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ / k∇ v k (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) u | det( id + ∇ ϕ ) | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k v k H k u k H HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 7 and eventually, a ( t ; v, v ) = Z Ω B ( ϕ ) ∇ v ∇ (cid:18) v | det( id + ∇ ϕ ) | (cid:19) ≥ Z Ω ∇ v ∇ (cid:18) v | det( id + ∇ ϕ ) | (cid:19) ≥ k v k H (Ω ) . Lions theorem hence applies, ensuring the existence and uniqueness of a solutionto the equations (4) and this solution belongs to the space L (0 , T ; H (Ω )) ∩ C ([0 , T ] , L (Ω )). This directly implies existence and uniqueness of a solution in L (0 , T ; H (Ω ϕ )) ∩ C ([0 , T ] , L (Ω ϕ )) for equations (2).1.1.2. Linearized problem.
Now that we have shown that the nonlinear problem waswell-posed, we introduce and analyze the linearized problem. The linearization isperformed with respect to the parameter ϕ . There are different notions of differ-entials with respect to the shape of the domain. In this article we are interestedin the Lagrangian differential, denoted y ′ ϕ , defined as the unique function (when itexists) satisfying the property: ∀ e Ω ⋐ Ω y ϕ | e Ω = y | e Ω + y ′ ϕ | e Ω + o ( ϕ ) . Proposition 2.
The Lagrangian shape differential y ′ ϕ satisfies the following equa-tion: ∂ t y ′ ϕ ( t, x ) − ∆ y ′ ϕ ( t, x ) = 0 t > and x ∈ Ω y ′ ϕ (0 , x ) = 0 x ∈ Ω y ′ ϕ ( t, x ) = − ϕ ( t ) .n ∂y ∂n t > and x ∈ ∂ Ω . (9) where the dot denotes the scalar product in R n .Proof. Let e Ω ⋐ Ω . We have in this open set the evolution equation ∂ t y ϕ − ∆ y ϕ = f ∀ ( t, x ) ∈ R + × e Ω . Since y satisfies the equation: ∂ t y − ∆ y = f on R + × e Ω , we necessarily have the following evolution equation for y ′ ϕ on R + × e Ω: ∂ t y ′ ϕ − ∆ y ′ ϕ = 0 on R + × e Ω . (10)The evolution equation of the Lagrangian shape differential being derived, we nowidentify the related boundary conditions. First of all, it is clear that y ′ ϕ ( t = 0 , x ) = 0 on e Ω . Let us now compute y ϕ | ∂ Ω . Considering, for some ψ ∈ C ∞ ( R n ) fixed, the integral R ∂ Ω y Ω ( t ) ψds and formally differentiating with respect to Ω, we obtain the following JONATHAN TOUBOUL formula (this calculation is rather classical, see for instance [1, 6.28]):0 = ∂∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) Ω=Ω Z ∂ Ω y Ω ( t ) ψds = Z ∂ Ω y ′ ϕ ( t ) ψds + Z ∂ Ω ϕ ( t ) .n ∂ ( y ψ ) ∂n ds = Z ∂ Ω y ′ ϕ ( t ) ψds + Z ∂ Ω ϕ ( t ) .n ( ∂y ∂n ψ + y ∂ψ∂n ) ds = Z ∂ Ω y ′ ϕ ( t ) ψds + Z ∂ Ω ϕ ( t ) .n ∂y ∂n ψds because y | ∂ Ω ≡
0. Hence we have: Z ∂ Ω ( y ′ ϕ ( t ) + ϕ ( t ) .n ∂y ∂n ) ψds = 0 ∀ ψ ∈ C ∞ ( R ) . The Lagrangian shape differential therefore satisfies the non homogeneous Dirichletboundary condition: y ′ ϕ ( t, x ) = − ϕ ( t ) · n ∂y ∂n ( t, x ) for ( t, x ) ∈ R + × ∂ Ω yielding our equation (9).1.1.3. Approximate controllability of the linearized problem. If d Λ(0) was an iso-morphism of W , ∞ ( R n , R n ) on H ( R n ), then the local inversion theorem wouldreadily imply exact local controllability. However we will see that this propertydoes not hold true, and a weaker property will be proved: we show that the lin-earized problem is approximately controllable. This property is demonstrated usingHolmgren’s theorem [9, 8] giving a uniqueness result for PDEs with real analyticcoefficients, and will be used here to show a propagation of zeros property from anon-characteristic surface. Lemma 1.2.
Let Ω an open subset of R n , with regular boundary ∂ Ω (e.g. W , ∞ as assumed for Ω ). Let γ a non-empty open subset of ∂ Ω . Any solution of theequations: ∂ t u ( t, x ) − ∆ u ( t, x ) = 0 t > , x ∈ Ω u (0 , x ) = 0 x ∈ Ω u ( t, x ) = 0 t > , x ∈ ∂ Ω ∂u∂n = 0 ∀ t > , x ∈ γ (11) is null on R + × Ω .Proof. Let D = ( ∂ t , ∂ x , . . . , ∂ x n ) and P ( D ) = − ∂ t − ∆ the heat differential op-erator. The associated characteristic polynomial is simply P ( T, X ) = − T − | X | , and its principal part is P ( T, X ) = −| X | = 0. The solution of this later equationis X = 0, and its direction (1 , , . . . , T = constant . The variational formulation satisfied by the function u ∈ C (0 , T ; H (Ω)) solution of (11) is given by: ∀ v ∈ H (Ω) , Z Ω ( ∂ t u v + ∇ u ∇ v ) dx − Z ∂ Ω ∂u∂n v ds = 0 . (12)Let now ˜ u be a continuation of u on an open subset e Ω = Ω ∪ Γ where Γ is an openset whose intersection with Ω is equal to γ . HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 9
Let ˜ u ∈ H ( e Ω) be the function defined by ˜ u ( t, x ) = u ( t, x ) x ∈ Ω . We have thefollowing relations: ∂ t ˜ u ( t, x ) = 0 on R + × Γ ∇ ˜ u = 0 on R + × Γ ∂ ˜ u∂n ( t, x ) = 0 on γ and hence we have : ∀ v ∈ H (Γ) , Z Γ ∂ t ˜ u v + ∇ ˜ u ∇ v − Z ∂ Γ ∂ ˜ u∂n vds = 0 . (13)Now, using (12) and (13), the condition ∂u∂n | γ ≡ ∪ ∂ Ω = γ , weobtain that ˜ u satisfies the following variational problem: ∀ v ∈ H ( e Ω) , Z e Ω ∂ t ˜ u v + ∇ ˜ u ∇ v − Z ∂ e Ω ∂ ˜ u∂n vds = 0 . (14)If both e Ω and Γ were convex sets, Holmgren’s theorem readily implies that ˜ u ≡ R + × Γ ⊂ R + × e Ω, P ( D ) is a differential operator with constant coefficientsand every plane which is characteristic with respect to P intersecting R + × e Ω inter-sects R + × Γ as well (since these are the planes of constant t ). Under these condi-tions, Holgren’s theorem [9, Theorem 5.3.3] implies that any solution of P ( D ) u = 0on e Ω vanishing on R + × Γ also vanishes on R + × e Ω, and in particular u ≡ e Ω are not convex. However, itis always possible to describe e Ω as the union of open balls (because of the regularityand connectedness of Ω). These discs are convex, and the intersection betweentwo discs is also convex, and the above argument applied on each element of thisdecomposition yield the desired result.Thanks to these results, we can prove the density result previously announcedfor the continuous problem:
Theorem 1.3.
Assume that there exist a non-empty subset γ of ∂ω on which ∀ ( t, x ) ∈ [0 , T ] × γ, ∂y ∂n ( t, x ) = 0 Then R = { y ′ ϕ ( T ) | ω ; ϕ ∈ W} is dense in L ( ω ) .Proof. We prove that R ⊥ = { } . Indeed, any g ∈ L (Ω ) belonging to R ⊥ is suchthat: Z ω g h = 0, ∀ h ∈ R . By definition, R = { y ′ ϕ ( T ) | ω ; ϕ ∈ W} so the later condition is equivalent to : ∀ ϕ ∈ W Z Ω gy ′ ϕ ( T ) = 0 . (15)Let us define the adjoint state φ ∈ H (Ω) associated to g : − ∂ t φ ( t, x ) − ∆ φ ( t, x ) = g ⊗ δ t = T t ≥ , x ∈ ωφ ( t = 0 , x ) = 0 x ∈ ωφ ( t, x ) = 0 t ≥ , x ∈ ∂ω where we denote: ∀ v ∈ C (0 , T ; H ( ω )) h g ⊗ δ t = T , v i = Z ω gv ( T ) dx. We now show that φ = 0, that is g = 0. Equation (15) implies that for any ϕ ∈ W , h− ∂ t φ − ∆ φ, y ′ ϕ i = Z ω gy ′ ϕ ( T ) = 0 . Integrating by parts yields:0 = Z R + × ω ( − ∂ t φ − ∆ φ ) y ′ ϕ = Z R + × ω φ∂ t y ′ ϕ − Z ω φ (0) y ′ ϕ (0) dx + Z R + × ω ∇ φ ∇ y ′ ϕ − Z R + × ∂ω y ′ ϕ ∂φ∂n = Z R + × ω ( ∂ t y ′ ϕ − ∆ y ′ ϕ ) φ + Z R + × ∂ω − y ′ ϕ ∂φ∂n + φ ∂y ′ ϕ ∂n = Z R + × ∂ω ϕ.n ∂y ∂n ∂φ∂n dtds. We deduce that for all t in [0 , T ], we have ∂y ∂n ∂φ∂n = 0 on ∂ω .Since on the non-negligible subset γ ∈ ∂ω we have ∂y ∂n = 0 ∀ t ∈ [0 , T ], we deducethat necessarily ∂φ∂n ( t, x ) = 0 on R + × γ . Lemma 1.2 implies that φ ( t, x ) ≡ t, x ) ∈ (0 , T ) × ω or ( T, ∞ ) × ω . The equation − ∂ t φ − ∆ φ = g ⊗ δ t = T imposes a jump condition on φ at T :[ | φ | ] = φ ( T + ) − φ ( T − ) = g. But since here, φ ( T − , x ) = φ ( T + , x ) = 0 for every x ∈ ω , we necessarily have g = 0.We therefore conclude that the range R of the operator d Λ(0) is dense in L ( ω ).We emphasize the fact that the control can be chosen constant in time, i.e. thatthe heat equation is approximately controllable by a rigid deformation of the openset Ω . In other words, we showed the existence of approximate controls throughtime invariant sets Ω.Note also that the condition of theorem 1.3, namely the fact that the normaldifferential of y is not vanishing on a non-empty subset of the boundary of ω for alltimes, can appear relatively strong. This condition is however necessary in order forthe observation to be performed at a given time T . If we are interested in the traceof the solutions on ω depending on time, y ϕ ( t, x ) | x ∈ ω and ϕ ∈ W , ∞ ( R n , R n ) (i.e. ϕ does not depend on time), then the controllability property can be proved, usingthe same techniques, under the weaker assumption that ∃ δ > ∀ t ∈ [0 , δ ]we have the non-degeneracy condition ∂y ∂n = 0.We hence proved that the linearized heat equation is approximately controllable,i.e. that it has a dense range on H (Ω ), and no exact controllability property wasproved. And for good reason: the regularity introduced by the parabolic form ofthe equation prevents from such an exact controllability property to hold. Thisweaker form of controllability prevents us from using this result to address thecontrollability of the nonlinear problem. HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 11
The hyperbolic operators do not enjoy the same regularization properties asparabolic operators. We now treat the problem in the hyperbolic setting and showthat, in contrast, we obtain an exact controllability property.1.2.
Controllability of the wave equation.
We now address the same problemin the case of the wave equation. The setting and intermediate results are similaras those of the parabolic case, and will be presented in less detail. It is howeverimportant to note that the main argument ensuring controllability is completelydistinct: Holmgren’s uniqueness theorem was used in the parabolic case to showapproximate controllability, and here we will use results on the controllability of thewave equations with respect to boundary conditions.We consider admissible domains U as dynamical perturbations of the original do-main Ω ϕ ( t ) = ( id + ϕ ( t ))(Ω ) where ϕ ( t ) are L ((0 , T ) , W , ∞ ( R n , R n )) such that forall t ∈ [0 , T ], ϕ ( t ) belongs to be the ball of center 0 and radius α of W , ∞ ( R n , R n ).This set is denoted M . We fix an observation time T ≥ )).Let f ∈ L ( R + , H − ( R n )), y ∈ H (Ω ϕ (0)) and y ∈ L (Ω ϕ (0)). We areinterested in y ϕ the solution of the equations: ∂ t y ϕ ( t, x ) − ∆ y ϕ ( t, x ) = f ( t, x ) ∀ t > , x ∈ Ω ϕ ( t ) y ϕ (0 , x ) = y ( x ) ∀ x ∈ Ω ϕ (0) ∂ t y ϕ (0 , x ) = y ( x ) ∀ x ∈ Ω ϕ (0) y ϕ ( t, x ) = 0 ∀ t > , x ∈ ∂ Ω ϕ ( t ) . (16)We denote by y be the solution associated to the trivial perturbation ϕ ≡
0, andΛ the map: Λ : ( M → H ( ω ) × L ( ω ) ϕ → ( y ϕ ( T, · ) | ω , ∂ t y ϕ ( T, · ) | ω ) . (17)The question we address is to characterize the set of traces at t = T and on ω ofthe solutions of this problem when ϕ is an admissible transformation, i.e. the rangeof the operator Λ: R ( M ) = { ( y ϕ ( T, · ) , ∂ t y ϕ ( T, · )) | ω ; ϕ ∈ M} . The same method as the one we used for the heat equation yields to the followingequivalent variational formulation of equations (16) holding on R + × Ω : Z R + × Ω { ∂ t ¯ y ϕ | det( id + ∇ ϕ ( t )) | − div( B ( ϕ ( t ))) ∇ ¯ y ϕ } Ψ dt dx = Z R + × Ω f ◦ ( id + ϕ ( t ))Ψ | det( id + ∇ ϕ ( t )) | dt dx (18)where B ( ϕ ( t )) = (cid:12)(cid:12)(cid:12) det (cid:0) id + ∇ ϕ ( t ) (cid:1)(cid:12)(cid:12)(cid:12) (cid:18)h(cid:0) ∇ ( id + ϕ ( t ) ) (cid:1) ∗ i − (cid:19) ∗ h(cid:16) ∇ (cid:0) id + ϕ ( t ) (cid:1)(cid:17) ∗ i − . We observe that B ( ϕ ) is symmetrical, positive because it is a perturbation of iden-tity (this condition constrains our choice of α the maximal norm of ϕ in M ). Wenow consider L (Ω ) equipped with the dot product : h ( a, b ) , ( c, d ) i H (Ω ) × L (Ω ) = Z Ω B ( ϕ ( t )) ∇ a. ∇ c + Z Ω b d | det( id + ∇ ϕ ( t )) | dx. A proof analogous to the one performed in parabolic case, proposition 1 (based onan application of a theorem due to Lions [2]) ensures existence and uniqueness ofsolution for the perturbed system.The Lagragian shape derivative for the wave equation satisfies the equations: ∂ t y ′ ϕ − ∆ y ′ ϕ = 0 y ′ ϕ ( t = 0) = 0 ∂ t y ′ ϕ ( t = 0) = 0 y ′ ϕ ( t ) | ∂ Ω = − ϕ ( t ) .n ∂y ( t ) ∂n . (19)We now show that d Λ(0) is surjective on H ( ω ) × L ( ω ), ensuring an exactcontrollability property of the wave equation using a local surjectivity theorem.Note that the proof we provide here differs significantly from the proof providedin the parabolic case. It is based on a controllability result due to Lions [13] andmakes use of the reversibility of the wave equation. Theorem 1.4.
Assume that there exists γ ∈ ∂ω such that ∂y ∂n ≥ ε > for all times t ∈ [0 , T ] . Then the linearized function d Λ(0) is surjective from L ((0 , T ) × Ω ) onto H ( ω ) × L ( ω ) .Proof. The proof of this theorem is based on classical results on the null boundarycontrollability of the wave equation, proved by Lions in [11, 13]. Using HilbertUniqueness Method (HUM), Lions considers an open bounded subset Ω of R n witha smooth boundary ∂ Ω, a control time
T > γ ⊂ ∂ Ω.Lions shows that for any initial condition ( u , u ) ∈ L (Ω) × H − (Ω), there existsa function v ∈ L ((0 , T ) × γ ) such that the solution of the equation ∂ t u − ∆ u = 0 on Ω × (0 , T ) u (0) = u on Ω ∂ t u (0) = u on Ω u ( t, x ) = v ( t, x ) on γ × (0 , T )is such that u ( T, x ) = ∂ t u ( T, x ) = 0. The function z ( t, x ) = u ( T − t, x ) satisfiesthe heat equation with zero boundary conditions and control v ( T − t, x ) on γ ,and has the property that z ( T, x ) = u and ∂ t z ( T, x ) = u on γ . Let us alsoremark that under the assumptions of the theorem, there exist several functions ψ ∈ L ((0 , T ) , W , ∞ ( R n , R n )) such that − ψ ( t, x ) .n ∂y ( t,x ) ∂n = v ( t, x ) x ∈ γ . For anyof these ψ , we have z = y ′ ψ solution of equation (19). Therefore, the linearizedfunction d Λ(0) defined by: d Λ(0) : ( L ([0 , T ] , W , ∞ ( R n , R n )) −→ H ( ω ) × L ( ω ) ψ −→ ( y ′ ψ ( T, · ) | ω , ∂ t y ′ ψ ( T, · ) | ω )is surjective.The surjectivity property on the differential of Λ directly implies the exact con-trollability of the original wave equation with respect to the shape of the domain,stated in the following: Theorem 1.5.
Let y the solution of the unperturbed problem: ∂ t y − ∆ y = f on Ω y ( t = 0) = y on Ω ∂ t y ( t = 0) = y on Ω y | ∂ Ω( t ) ( t ) = 0 ∀ t > . HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 13
There exists a neighborhood of ( y ( T ) | ω , ∂ t y ( T ) | ω ) in H ( ω ) × L ( ω ) , denoted N such that for all A ∈ N , there exists ϕ ∈ M such that A = Λ( ϕ ) .Proof. We proved that Λ was differentiable and that its differential at 0 is surjective.So the local surjectivity theorem (see e.g. Luenberger [14] )proves theorem 1.5.We emphasize on the fact that the controllability property, in that case, holdsboth for the linearized and the original non-linear (with respect to the control ϕ )problem. Let us emphasize the fact that in the present case, the control cannot beconsidered constant in time in contrast with the case of the heat equation.2. Control of the semi-discrete wave and heat equations.
In the previ-ous section, we addressed the problem of the controllability of the heat and waveequations with respect to the shape of the domain and proved that the hyperbolicequation was locally exactly controllable with respect to the shape of the domain,the parabolic problem was not exactly controllable and its linearization was approx-imately controllable.The approximate controllability property of the parabolic equation did not implyan analogous property for the nonlinear problem: though the range of the traceoperator was dense, this did not imply a local inversion property because of theproblem holds in an infinite-dimensional space. We shall now turn our attentionto the discretized problem. This study has two main interests. First, it is relevantfrom a computational point of view and, second, from the mathematical point ofview, since the discretized version of the Laplacian operator is finite dimensional,the density of the range of the linearized operator will imply surjectivity of thenonlinear operator.For simplicity, this section is restricted to the analysis in two dimensions, wherethe open set Ω is a square. Using the classical methods as proposed in [6], itwould be possible to extend these results to general domains, with an importantincrease of complexity in the notations, but no profound change in the mathematicalarguments. Moreover, note that we are interested here in a fixed discretization ofthe open set Ω . In other words, the approach does not address the convergence ofthis control as the stepsize of the mesh tends to zero.In details, we are interested in the semi-discrete heat and wave equations on arectangle [0 , a ] × [0 , b ] ∈ R discretized this set with a step h . The infinite continuousspace system is therefore replaced by a finite-dimensional evolution problem on thediscrete points ofΩ h = { m = ( ih, jh ); ( i, j ) ∈ { , . . . , M } × { , . . . , N }} . Both the heat and wave equations make use of the spatial Laplacian operator, whichwe now define in the discretized setting. To this purpose, we introduce the followingdefinitions:
Definition 2.1.
Let m = ( ih, jh ) ∈ ( Z h ) and the discrete neighborhood of m defined: B ( m ) = { ( kh, lh ); ( k, l ) = ( i, j ) , ( i − , j ) , ( i + 1 , j ) , ( i, j − , ( i, j + 1) } The set of strict neighbors of m is B ( m ) = B ( m ) \ { m } . Definition 2.2.
The discrete interior of Ω h is defined by o Ω h = { m ∈ Ω h ; B ( m ) ⊂ Ω h } . The discrete boundary of Ω h is defined by Γ h = Ω h \ o Ω h and the exterior of Ω h as: o F h = ( Z h ) \ Ω h .These sets form a partition of ( Z h ) . We will assume for simplicity that a singleedge of the boundary is moving, for instance { ( i, j ); i = 0 } . This means that theonly moving part of this set is our control. Furthermore, the free points of theboundary will move only along the normal to this boundary. Definition 2.3 (Functional spaces) . We denote F ( X ) the set of real-valued map-pings from a space X and F ( X ) those vanishing on the boundary of X . We considerin this paper time dependent maps, taking values in F (Ω h ). In particular we willuse C ([0 , T ] , F (Ω h )), the set of continuous functions [0 , T ]
7→ F (Ω h ) and the spaces L p ([0 , T ] , F (Ω h )). Remark 1.
The set F (Ω h ) is isomorphic and identified to R MN .We are now in a position to define a discretized version of the Laplacian operator,as follows. Definition 2.4.
Let A be the finite difference operator with Dirichlet boundaryconditions defined by: ( F (Ω h ) −→ F ( o Ω h ) φ −→ Aφ where ∀ m ∈ o Ω h , [ Aφ ] m = 1 h [4 φ ( m ) − X p ∈ B ( m ) ,p = m φ ( p )] (20)2.1. Semi-discrete heat equation in a square.
We now turn our attentionspecifically to the case of the semi-discretized heat equation. The reference domainΩ h being fixed, we consider a reference state u as the solution of the equation u ∈ F (Ω h ) such that (cid:26) ∂ t u + Au = Fu ( t = 0) = u (21)where the source term F ∈ F ( o Ω h ) is deduced from the source term f of the contin-uous initial problem by a simple discretization .2.1.1. The perturbed problem.
As in the continuous case, we are interested in smallperturbations of the shape of the domain Ω h . Changing the shape of Ω h consists inmoving continuously the nodes of the mesh corresponding to x = 0. On this newsubset, the finite difference Laplace operator is modified as follows: Definition 2.5.
We consider the set { V j ; j = 1 ..N − } of vector fields Ω h R by: ∀ j ∈ { ..N − } , ( V j ( m ) = (0 ,
0) if m = (0 , jh ) V j ( m ) = (1 ,
0) if m = (0 , jh ) . If f is continuous, then F is defined by F m = f ( m ) for m ∈ o Ω h . If f is not continuous, forinstance is f ∈ L or H , then F m will be a mean value of f on a neighborhood of m . HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 15
Let W h be the vector space spanned by the family ( V j ) j ∈{ ...N − } . The perturba-tions we consider in this problem are in the set: W h = { N − X j =1 h λ j ( t ) V j ; t → λ j ( t ) ∈ L ∞ ( R + ; W h ) ∩ C ( R + ; W h )and such that sup j =1 ..N − k λ j k ∞ < / } (22) Remark 2.
Note that the perturbation has the same magnitude as h . This doesnot allow us even in the better cases to have the continuous case as a limit casesince the perturbation tends to the trivial condition as the mesh becomes finer. Definition 2.6.
Let us define Γ the first layer of interior nodes:Γ = { (1 , j ) ; j ∈ { , ..N − }} . The perturbed heat operator on this new discretized set is defined as:
Definition 2.7.
Let ϕ ( t ) = P N − j =1 λ j ( t ) hV j ∈ W h . The operator A ( ϕ ) : F (Ω ϕh ) ( o Ω h ) is defined as: h [4 φ ( m ) − P p ∈ B ( m ) , p = m φ ( p )] ∀ m ∈ o Ω h \ Γ h [2(1 + λ j ( t ) ) φ (1 ,j ) − λ j ( t ) φ (2 ,j ) − φ (1 ,j +1) − φ (1 ,j − ] for m = (1 , j ) ∈ Γ (23) Proposition 3.
The operator A ( ϕ ) is bounded for all ϕ ∈ W h .Proof. It is easy to show that: • For m ∈ o Ω h \ Γ , we have [ A ( ϕ ) φ ] m = [ Aφ ] m = h [4 φ ( m ) − P p ∈ B ( m ) ,p = m φ ( p )],and hence: | [ A ( ϕ ) φ ] m | ≤ h k φ k ∞ + X p ∈ B ( m ) ,p = m k φ k ∞ ≤ h k φ k ∞ . • For m = (1 , j ) ∈ Γ , we have[ A ( ϕ ) φ ] m = 1 h (2(1 + 11 + λ j ( t ) ) φ (1 ,j ) −
22 + λ j ( t ) φ (2 ,j ) − φ (1 ,j +1) − φ (1 ,j − ) , and hence we have: | [ A ( ϕ ) φ ] m | ≤ h (cid:18) k φ k ∞ + 22 − / k φ k ∞ + 2 k φ k ∞ (cid:19) ≤ h k φ k ∞ . Since we are in F ( o Ω h ) which is a finite dimension vector space, all the norms areequivalent so the operator A ( ϕ ) is bounded on W h . Definition 2.8.
The perturbed state −→ u ϕ ( x, t ) ∈ F ( o Ω h ) is the unique solution in C ([0 , T ] , F ( o Ω h )) of the semi-discrete problem: ( ∂ t −→ u ϕ + A ( ϕ ) −→ u ϕ = F −→ u ϕ ( t = 0) = −→ u . (24)This solution exists and is unique, and defined for all time t > Proof.
Equation (24) is a linear ordinary differential equation with t -measurable vec-tor field, hence classical theory (Cauchy-Lipschitz theorem) implies local existenceand uniqueness of the perturbed state, which will be continuous in its definition do-main. Non-explosion property is a classical application of Gronwall’s lemma basedon the boundedness of the perturbed operator and of the source term F (proposition3).2.1.2. Controllability of the semi-discrete heat equation.
As in the continuous case,we consider the map Λ h : ( W h −→ F ( o Ω h ) ϕ −→ −→ u ϕ ( T ) (25)where −→ u ϕ ( T ) is solution of equation (24). Let Z d = u ( T ), where u is the referencestate (21). The problem we address is to find a neighborhood V (0) ∈ V of thereference domain and V ( Z d ) ∈ F ( o Ω h ) of the trace at t = T of the reference solutionsuch that V ( Z d ) ⊂ Λ h ( V (0)).In the finite-dimension spaces where the problem is now set, we use the localinversion theorem to demonstrate this property. First of all we will prove that Λ h isdifferentiable in the neighborhood of the origin, and that d Λ h (0) is surjective. Thenwe will use the adjoint state technique to prove that the surjectivity of d Λ h (0) isequivalent to a pool of conditions the semi-discrete adjoint should state satisfies.Finally, we will prove the controllability property proving those conditions on theadjoint state, which happen to be a property of discrete unique continuation. (i). Differentiability Let us denote tr the trace operator: tr : ( C ([0 , T ] , F (Ω h )) R φ φ ( T )The map Λ h is the composition of the trace operator and the map U : ϕ −→ −→ u ϕ .The trace function is linear and continuous. So we only need to prove that U isFr´echet-differentiable in 0. Proposition 4. ∀ ϕ ∈ W h , ∀ ψ ∈ W h , the map ϕ
7→ −→ u ϕ is differentiable in ϕ inthe direction of ψ , and the Gˆateaux differential h D G Λ h ( ϕ ) , ψ i , denoted −→ v ϕ ( ψ ) issolution of the differential equation: ( ∂ t −→ v ϕ ( ψ ) + A ( ϕ ) −→ v ϕ ( ψ ) = −h ˙ A ϕ , ψ i−→ u ϕ −→ v ϕ ( ψ )( t = 0) = −→ . (26) Proof.
First let W λ = u ϕ + λψ − u ϕ λ . The function D G Λ h is the limit, when it exists,of W λ when λ tends to 0.(i). Necessary condition : We assume that this limit exists, denote it v , and wecompute the equation this limit satisfies. The function u ϕ + λψ satisfies theequations: ( ∂ t −−−−→ u ϕ + λψ + A ( ϕ + λψ ) −−−−→ u ϕ + λψ = F ( t ) −−−−→ u ϕ + λψ ( t = 0) = −→ u . HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 17
We differentiate this equation with respect to λ at λ = 0, and we obtainthe equation v , ( ∂ t v + h ˙ A ( ϕ ) , ψ i−→ u ϕ + A ( ϕ ) −→ v = 0 −→ v ( t = 0) = 0 . So we deduce that if the differential of W λ exists, then it is the solution v ϕ ( ψ ) of the ordinary differential equation (26): ( ∂ t −→ v ϕ ( ψ ) + A ( ϕ ) −→ v ϕ ( ψ ) = −h ˙ A ( ϕ ) , ψ i−→ u ϕ −→ v ϕ ( ψ )( t = 0) = −→ . (ii). Sufficient condition : We show that the solution of the ordinary differentialequation (26) is the limit of W λ . Indeed, the map ( λ j ) j =1 ...N − A ( φ )is C ∞ , so the differential h ˙ A ϕ , ψ i is defined, and furthermore h ˙ A ϕ , ψ i−→ u ϕ is L (0 , T ). So Cauchy-Lipschitz theorem ensures existence and uniqueness ofsolution, and we can prove that it is defined for all time using Gronwall’slemma. Let us now show that the function −→ v ( t ) defined is indeed the limitof W λ when λ →
0. Let W λ − v := ε λ . We have: ε λ = ∂ t ε λ + A ( ϕ + λψ ) W λ − A ( ϕ ) v = (cid:20) A ( ϕ + λψ ) − A ( ϕ ) λ − h ˙ A ( ϕ ) , ψ i (cid:21) u ϕ . Since A ∈ C in ϕ , the right hand of the equality is O ( λ ), as well as( A ( ϕ + λψ ) − A ( ϕ )) v which comes from the left hand of the equality. Soeventually, ε λ satisfies the equation: ( ∂ t ε λ + A ( ϕ ) ε λ = o ( λ ) ε λ ( t = 0) = −→ k ∂ t ε λ k ≤ α k ε λ k + M λ and hence k ε λ ( t ) k ≤ λ Mα e αT .Therefore, ε λ converges uniformly to 0 when λ → W λ exists and is indeed the function v solution of (26). Proposition 5.
The differential of u φ with respect to φ , ϕ −→ D G Λ h ( ϕ ) is continuous.Proof. D G Λ h ( ϕ ) : ψ −→ −→ v where −→ v is solution of: (cid:26) ∂ t −→ v + A ϕ −→ v = −h ˙ A ϕ , ψ i−→ u ϕ −→ v ( t = 0) = 0 . Hence −→ v is solution of ∂ t −→ v = F ϕ,ψ ( t, −→ v ) with F ϕ,ψ ( t, −→ x ) = − A ϕ −→ x − h ˙ A ϕ , ψ i−→ y ϕ . We note that the function ϕ → F ϕ,ψ is continuous, since ϕ → A ϕ is continuous. Cauchy-Lipschitz’ theorem with param-eters gives us that ϕ → y ϕ is also continuous.Moreover, the differential h ˙ A ϕ , ψ i is continuous in ϕ since A ϕ has a rationalvariation in ϕ , and by definition of ϕ , these rational fractions have no singularpoint on φ , implying that the dependence in ϕ remains continuous. So Cauchy-Lipschitz’ theorem with parameters gives us the continuity of y ′ ϕ in ϕ . Theorem 2.9. Λ h is Fr´echet-differentiable at .Proof. We already proved that: • the Gˆateaux differentials in all directions of V exist (Prop.4) • these differentials are continuous (Prop. 5)Using the property that a function Gˆateaux differentiable in all directions and thedifferential of which being continuous is differentiable, we conclude on the Fr´echetdifferentiability of Λ h at 0. (ii). Adjoint state technique We recall that ϕ ∈ L (0 , T ; W h ). We have: ϕ ( t ) = N − X j =0 hλ j ( t ) V j . Λ h , defined in (25) is the composed application of the trace function at t = T withthe map U defined by: U : ( C ([0 , T ]; W h ) −→ C ([0 , T ]; F ( o Ω h )) ϕ −→ u ϕ where u ϕ is solution of the equations: (cid:26) ∂ t u ϕ + A ( ϕ ) u ϕ = Fu ϕ ( t = 0) = 0 . Furthermore, recall that d Λ h (0) : ( C ([0 , T ] , W h ) −→ F ( o Ω h ) ϕ −→ y ϕ ( T )and y ϕ is solution of (cid:26) ∂ t y ϕ + A (0) y ϕ = −h ˙ A (0) , ϕ i u y ϕ ( t = 0) = 0 . Eventually, we denote Y : ϕ −→ y ϕ (so we have d Λ h (0) = tr | t = T ◦ Y ) and remarkthat the adjoint of the trace map is given, for any c ∈ F ( o Ω h ), by tr ∗ c = cδ t = T . Wenow prove that the map d Λ h (0) is surjective. To this purpose, we use the adjointstate technique.The surjectivity of d Λ h (0) is equivalent to the fact that: { c ∈ F ( o Ω h ); ∀ ϕ ∈ W h h d Λ h (0) ϕ, c i = 0 } = { } . Any c ∈ F ( o Ω h ) such that ∀ ϕ ∈ W h h d Λ h (0) ϕ, c i = 0 is such that h tr | t = T ( Y ( ϕ )) , c i =0 for any ϕ ∈ W h , which is equivalent to the property: ∀ ϕ ∈ W h h Y ( ϕ ) , tr | ∗ t = T ( c ) i = 0 (27) HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 19
Definition 2.10.
The adjoint state associated to y ϕ and tr | ∗ t = T ( c ) is the uniquesolution X of the equations: ( − ∂ t X + AX = tr | ∗ t = T ( c ) X ( t = 0) = 0 (28) Remark 3.
Using the fact that A is self-adjoint, we clearly have: h ∂ t y ϕ + Ay ϕ , T i = h ∂ t y ϕ , T i + h Ay ϕ , T i = −h y ϕ , ∂ t T i + h y ϕ , A ∗ T i = h y ϕ , − ∂ t T + AT i Using this definition we replace in (27) tr | ∗ t = T ( c ) by its expression in function ofthe adjoint state X defined in (28) and obtain the set of equivalent statements: ∀ ϕ ∈ W h h Y ( ϕ ) , tr | ∗ t = T ( c ) i = 0 ⇔ ∀ ϕ ∈ W h h Y ( ϕ ) , − ∂ t X + AX i = 0 ⇔ ∀ ϕ ∈ W h h ∂ t y ϕ + Ay ϕ , X i = 0 ⇔ ∀ ϕ ∈ W h h A ′ ϕ y , X i = 0 . This proves the following:
Theorem 2.11.
The differential d Λ h (0) of Λ h at ϕ = 0 is surjective if and only ifwe have the following uniqueness property: If c ∈ F ( o Ω h ) is such that h X, A ′ ϕ y i L (0 ,T ; F ( o Ω h )) = 0 , ∀ ϕ ∈ W h (29) where X is solution of : (cid:26) − ∂ t X + AX = tr | ∗ t = T ( c ) X ( t = 0) = 0 , then necessarily c = 0 . (iii). Calculation of the differential of A at Proposition 6.
Let j ∈ { ...N − } and φ ∈ F ( o Ω h ) . For all µ ∈ C ([0 , T ] , R ) , wedenote h A ′ , µ ( t ) V j i the differential of A at in the direction µ ( t ) V j . We have: [ h A ′ , µ ( t ) V j i φ ] m = ( ∀ m ∈ o Ω h \ Γ µ ( t ) h (cid:0) φ (2 ,j ) − φ (1 ,j ) (cid:1) if m = (1 , j ) (30) Proof.
For all j ∈ { ...N − } , the point (0 , j ) of the boundary has a unique neigh-bor in o Ω h , which is (1 , j ). From the definition 2.7 of A ( ϕ ), for all µ : t → µ ( t ) C ([0 , T ]; R ), we have : ∀ m ∈ o Ω h , m = (1 , j ) , [ A µ ( t ) V j φ ] m = [ Aφ ] m and we have also[ A ( µ ( t ) V j ) φ ] (1 ,j ) = 1 h (cid:18) µ ( t ) ) φ (1 ,j ) −
22 + µ ( t ) φ (2 ,j ) − φ (1 ,j +1) − φ (1 ,j − (cid:19) . Hence we get[ A ( µ ( t ) V j ) φ ] (1 ,j ) − [ Aφ ] (1 ,j ) = 1 h [2(1 + 11 + µ ( t ) − φ (1 ,j ) − ( 22 + µ ( t ) − φ (2 ,j ) ] . = 1 h [2(1 + 1 − µ ( t ) + o ( µ ) − φ (1 ,j ) − (1 − µ ( t ) − o ( µ )) φ (2 ,j ) ] . (iv). The condition X | Γ = 0 Definition 2.12.
The function Y ∈ C ([0 , T ] , F ( o Ω h )) satisfies the discrete non-degeneracy condition if and only if: ∀ t > , ∀ j ∈ { , ..., N − } , Y (2 ,j ) − Y (1 ,j ) = 0 . Remark 4.
This condition can be seen as an finite difference approximation of thecondition ∂y∂n = 0. Indeed, let y ∈ C ( V ( x )) where V ( x ) is a neighborhood of x .Assume that y satisfies y ( x ) = 0. Then performing a Taylor expansion, we get: y ′ ( x ) = 1 h [2 y ( x + h ) − y ( x + 2 h )] + o ( h ) . Note also that in the continuous case, we only need to assume that ∂y∂n does notvanish on an open set of the boundary, and not all along the boundary. Here weneed to assume the non-degeneracy condition all along the boundary of Ω h to provethe discrete unique continuation. Proposition 7.
Under the discrete non-degeneracy condition (definition 2.12) onthe reference state, we have: X | Γ ≡ . Proof.
The relation (29) gives us : h X, A ′ ( µ ( t ) V j ) y i L (0 ,T ); F ( o Ω h ) = 0 , ∀ j ∈ { , .., N − } , ∀ µ ( t ) C ∞ ( R + , R ) . Proposition 6 gives us therefore that ∀ j ∈ { , .., N − } , • ∀ m ∈ o Ω h , m = (1 , j ) , [ A ′ ( µ ( t ) V j ) φ ] m = 0 , • [ A ′ ( µ ( t ) V j ) φ ] (1 ,j ) = h [ φ (2 ,j ) − φ (1 ,j ) ] µ ( t )so we eventually have : h X, A ′ ( µ ( t ) V j ) y i L (0 ,T ); F ( o Ω h ) = 0 , Thus Z T X m ∈ o Ω h µ ( t )[ A ′ ( µ ( t ) V j ) y ] m ( t ) X m ( t ) dt = Z T µ ( t ) 1 h [ 12 y (2 , j ) − y (1 , j )] X (1 ,j ) ( t ) dt = 0 . Therefore, for all t > h [ y (2 , j ) − y (1 , j )]( t ) X (1 ,j ) ( t ) = 0Since y (2 , j ) − y (1 , j ) never vanishes, we have: X (1 ,j ) ( t ) = 0 ∀ t > ∀ j ∈ { , ..., N − } . (v). Unique discrete continuation The aim of this section is to prove that the uniqueness condition appearing intheorem 29 (equation (29)) is valid. This uniqueness condition is proved using thefact that under the discrete non-degeneracy condition on the reference state y , X | Γ ≡ X given by equations (2.10) identically null, so tr ∗ t = T ( c ) = 0 and c = 0. HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 21
The method we use is based on the study of the propagation of the zeros of X onΩ h from its boundary, analogous to the approach developed in the continuous caseusing Holmgren’s theorem. The main difference is that the propagation of zerosin the continuous case is a global property, whereas it is a local property in thediscrete case . Theorem 2.13.
The unique solution of the equations − ∂ t X + AX = tr | ∗ t = T ( c ) X ( t = 0) = 0 X | Γ = 0 X | Γ = 0 (31) is X ≡ , and so c = 0 .Proof. (i). First we are interested in the equation − ∂ t X + AX = tr | ∗ t = T ( c ). Onthe sets [0 , T [ and ] T, ∞ [, the equation simply reads − ∂ t X + AX = 0 so wehave existence, uniqueness and continuity of the solution X in these domains.The right hand term can be interpreted as an imposed jump condition at time t = T . Indeed, let us write the variational formulation of the problem(31):Let v ∈ C ∞ ( R + , F ( o Ω h )). The variational formulation reads: Z ∞ X m ∈ o Ω h − ∂ t X m v m + ( AX ) m .v m dt = X m ∈ o Ω h v m ( T ) c m i.e : Z ∞ X m ∈ o Ω h ∂ t X m v m + ( AX ) m .v m dt = X m ∈ o Ω h v m ( T ) c m The imposed jump at t = T reads: − ∂ t X + AX = 0 X ( t = 0) = 0[ | X | ]( T ) = cX | Γ = 0 (32) This is why we need to assume the non-degeneracy condition all along the the boundary ofΩ h . where we denoted [ | X | ]( T ) = X ( T + ) − X ( T − ) the jump of X at t = T . Thevariational formulation of the problem reads: ∀ v ∈ C ∞ ( R + , F ( o Ω h ))0 = Z ∞ X M ∈ o Ω h − ∂ t X m v m + AX m .v m dt = Z T X M ∈ o Ω h − ∂ t X m v m + AX m .v m dt + Z ∞ T X M ∈ o Ω h − ∂ t X m v m + AX m .v m dt = Z T X M ∈ o Ω h X m ∂ t v m + AX m .v m dt + Z ∞ T X M ∈ o Ω h X m ∂ t v m + AX m .v m dt − X M ∈ o Ω h ( X m ( T + ) − X m ( T − )) v m ( T ) + X m (0) v m (0)= Z ∞ X M ∈ o Ω h X m ∂ t v m + AX m .v m dt − X M ∈ o Ω h [ | X m | ]( T ) v m ( T ) . We have the same variational formulations, so the solution are identical.(ii). Calculation of the solution X : we have ∀ j ∈ { ...N } , X ,j = X ,j = 0. Wereason by induction on k . Assume that on the column k − { ( i, j ); i = k − } )and the column k ( { ( i, j ); i = k } ) we had X = 0. In this case, X also vanisheson the column k + 1.Indeed, let j ∈ { , .., N } (a) If j = 0 or j = N then we have indeed X ( k +1 ,j ) = 0 because X vanisheson Γ h , the boundary of Ω h .(b) If j ∈ { , .., N − } . Let us write the equation satisfied by X ( k,j ) : ( ∂ t X ( k,j ) + ( AX ) ( k,j ) = 0 ∂ t X ( k,j ) + h (cid:2) X ( k,j ) − X ( k +1 ,j ) − X ( k − ,j ) − X ( k,j +1) − X ( k,j − (cid:3) = 0 . (33)Since we assumed that: X ( k,i ) ( t ) = X ( k − ,i ) ( t ) ≡
0. We reinject thiscondition in (33) and we get: − X ( k +1 ,j ) ( t ) = 0 ∀ t. So we prove that if the solution vanishes on two consecutive columns, thenthe solution vanishes on all the other columns. The hypothesis being thatthe solution X vanishes on the column i = 0 and i = 1, we indeed provedthat X is identically null.(iii). We conclude that the solution X of this problem is time continuous, (it isconstant equal to 0) and that the jump of the solution at t = T is null, so c = 0. (vi). Discrete controllability resultTheorem 2.14. Assume that the reference state y defined by (24) satisfies thenon-degeneracy discrete condition (2.12) . Let y ϕ be the solution of the perturbedstate (24) and Z d = y ( T ) . HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 23
There exist neighborhoods V (0) ⊂ C ([0 , T ]; W h ) and V ( Z d ) ⊂ F ( o Ω h ) such thatfor all Z ∈ V ( Z d ) there exists ϕ ∈ V (0) such that y ϕ ( T ) = Z .Proof. This is a consequence of the local surjectivity property of the map Λ h (definedin (25)). Proposition 7 shows that when the reference state satisfies the discretenon-degeneracy condition, then X | Γ = 0. In theorem 2.13 we proved that thissecond relation implies that X = 0 and that c = 0. This readily implies that d Λ h (0) is surjective using theorem 29, which completes the proof.Therefore, we have proved that the semi-discrete heat equation was locally ex-actly controllable, which was not the case of the continuous-space equation. Wenow turn our attention to the case of the wave equation. We realize again that acontrol independent of the time can be found in that case.2.2. Semi-discrete wave equation in a square.
The controllability of the wavesequations is demonstrated in an analogous manner. Similarly to the paraboliccase, the wave equation unperturbed state u is solution of the ordinary differentialequation: u ∈ F (Ω h ) ∂ t u + Au = Fu ( t = 0) = u ∂ t u ( t = 0) = u (34)Where the discrete Laplace operator A and the function F are defined as in sec-tion 2.1.2.2.2.1. The perturbed state.
As in the continuous state, we are interested in smallperturbations of the shape of the domain Ω h . We are quite free in the choice ofthe admissible transformations, and only look for sufficient conditions for the exactcontrollability. The first assumption we make on the perturbation is that the shapeof the domain will be modified only moving nodes of the mesh using C transfor-mations in time. Moreover, the only moving nodes are located on the line x = 0,and will move along the normal to the boundary. Because of the finite propagationspeed of information in the wave equation, the problem will be well posed if theboundary does not moves faster than the information, i.e. the differential of thedeformation should not have a module greater than the information propagationspeed, in our case 1.On those perturbed open sets, the operator approximating the Dirichlet Lapla-cian is identical to the one defined for the heat equation. We recall that W h isthe real vector space spanned by ( V j ) j ∈{ ...N − } defined in 2.5. The admissibletransformations we consider belong to the space: V = n N − X j =1 hλ j ( t ) V j ; t → λ j ( t ) ∈ W , ∞ ( R + ; W h ) ∩ C ( R + ; W h )and such that sup j =1 ..N − k λ j k ∞ < / , k ∂ t λ j k ∞ < o . We denote by M ( ϕ ) denote the 2 n × n matrix: (cid:18) − idA ( ϕ ) 0 (cid:19) The perturbed state, denoted −→ u ϕ ( x, t ) ∈ F ( o Ω h ), is the unique solution in C ([0 , T ] , F ( o Ω h )) of the semi-discrete problem: ∂ t −→ u ϕ + A ( ϕ ) −→ u ϕ = F −→ u ϕ ( t = 0) = −→ u ∂ t −→ u ϕ ( t = 0) = −→ u (35)A direct application of standard theory of ordinary differential equations ensuresthat: Proposition 8.
We define U ϕ := (cid:18) u ϕ ∂ t u ϕ (cid:19) . For all ϕ ∈ V , U ( ϕ ) is well definedand bounded in C ([0 , T ] , F ( o Ω h )) Controllability of the semi-discrete wave equation.
We show a surjectivityproperty of the map: Λ h : ( V 7→ F ( o Ω h ) ϕ U ϕ ( T ) (36)where U ϕ ( T ) is solution of the equation (35). Let Z d = u ( T ), where u is thereference state defined in (34). (i). Differentiability We denote here again tr the trace function at t = T . The map Λ h is thecomposition of the function S : ϕ −→ U ϕ and the trace function. The Fr´echet-differentiability of Λ h is equivalent to the Fr´echet-differentiability of S at 0.This differentiability is an immediate consequence of the differentiability of A ( ϕ ).Indeed, by Cauchy-Lipschitz’ theorem with parameters, if A is Fr´echet-differentiablein ϕ , then the matrix M ( ϕ ) = (cid:18) − idA ( ϕ ) 0 (cid:19) is differentiable in ϕ which gives the Fr´echet-differentiability of U ϕ in ϕ . Proposition 9. φ → U φ is differentiable at in the direction ψ , and the Fr´echet-differential h d Λ h (0) , ψ i , denoted Y ψ , is solution of the differential equation: (cid:26) ∂ t Y ψ + M (0) Y ψ = −h ˙ M , ψ i U Y ψ ( t = 0) = 0 . (ii). Adjoint state technique We have Λ h = tr t = T ◦ S . Moreover, recall that d Λ h (0) : ( C ([0 , T ] , W h ) −→ F ( o Ω h ) ϕ −→ Y ϕ ( T )and Y ϕ is solution of (cid:26) ∂ t Y ϕ + M (0) y ϕ = −h ˙ M (0) , ϕ i U Y ϕ ( t = 0) = 0 . Eventually, we denote L : ϕ −→ Y ϕ . We clearly have d Λ h (0) = tr | t = T ◦ L . Let usstart by proving that the differential of Λ h at 0 is surjective, using the adjoint statemethod. d Λ h (0) is surjective ⇔ { c ∈ F ( o Ω h ) ; ∀ ϕ ∈ Vh d Λ h (0) ϕ, c i = 0 } = { } HAPE-CONTROLLABILITY OF EVOLUTION EQUATIONS 25 and moreover, ∀ ϕ ∈ V h d Λ h (0) ϕ, c i = 0 ⇔∀ ϕ ∈ V h tr | t = T ( S ( ϕ )) , c i = 0 ⇔∀ ϕ ∈ V h S ( ϕ ) , tr | ∗ t = T ( c ) i = 0 (37)Furthermore, we have S ( ϕ ) = Y ϕ is solution of the differential equation: ∂ t Y ϕ + M (0) y ϕ = − M ′ ϕ y Definition 2.15.
The adjoint state Y ϕ and tr | ∗ t = T ( c ) is the unique solution X ofthe equations: (cid:26) − ∂ t X + M ∗ X = tr | ∗ t = T ( c ) X ( t = 0) = 0 (38)With this definition, we replace (37) tr | ∗ t = T ( c ) by its expression in function ofthe adjoint state X (38).We deduce the following theorem: Theorem 2.16.
The differential d Λ h (0) of Λ h at ϕ = 0 is surjective if and only ifwe have the following uniqueness property:If c ∈ F ( o Ω h ) is such that h X, M ′ ϕ y i L (0 ,T ; F ( o Ω h )) = 0 , ∀ ϕ ∈ V (39) with (cid:26) − ∂ t X + M ∗ (0) X = tr | ∗ t = T ( c ) X ( t = 0) = 0 (40) Then necessarily c = 0 . We now turn to compute the differential of M at 0 Proposition 10. M ′ (0) = (cid:18) A ′ (0) 0 (cid:19) Where A ′ (0) is defined in 6. A simple corollary of proposition 7 ensures that:
Proposition 11.
Assume that the reference state satisfies the discrete non-degeneracycondition on the whole boundary of Ω h . Then the relation (39) implies that X | Γ ≡ . We are in a position to show the uniqueness property (39). From proposition11, under the discrete non-degeneracy condition on the reference state y , we have X | Γ ≡
0. We now show that this condition, together with the definition of theadjoint state (2.15): (cid:26) − ∂ t X + M ∗ (0) X = tr | ∗ t = T ( c ) X ( t = 0) = 0implies that X is identically vanishing, so tr ∗ t = T = 0 and c = 0.To this purpose, we will study the zeros propagation of X on Ω h from its boundaries. Theorem 2.17.
The relations: − ∂ t X + M ∗ (0) X = tr | ∗ t = T ( c ) X ( t = 0) = 0 X | Γ = 0 X | Γ = 0 (41) implies that X ≡ and c = 0 . The proof uses exactly the same approach as the heat equation case.This results allows to prove the following controllability theorem.
Theorem 2.18.
Assume that the reference state y defined in (21) satisfies thediscrete non degeneracy condition 2.12. Let Y ϕ be the solution of the perturbedequation (24) . Let finally Z d = Y ( T ) . Then there exist neighborhoods V (0) ⊂ C ([0 , T ]; W h ) and V ( Z d ) ⊂ F ( o Ω h ) such that ∀ Z ∈ V ( Z d ) ∃ ϕ ∈ V (0) such that Y ϕ ( T ) = Z . Conclusion.
In this paper we proved that the linearized heat equation was ap-proximately controllable with respect to the shape of the domain, while the waveequation is locally exactly controllable. We addressed the same questions in the caseof the semi-discrete equations in two dimensions in a square and we proved thatthe two types of equations are exactly controllable. Nevertheless, the methods wedeveloped in this paper do not allow us to see the discrete control as an approxima-tion of the continuous control in the wave equation. Another discretization methodshould be used to address this question, the mixed finite elements method. Indeed,we claim that one of the main obstacle to this interesting issue is the discretizationmethod used, which does not behaves smoothly in the limit h →
0. For instance weknow (see [16, 10]) that in boundary control problem of the unidimensional waveequation, spurious modes with high frequency numerical oscillations appear and theobservability constant tends to infinity when h tends to 0. It has been proved alsothat this semi-discrete model is not uniformly controllable in the limit h → L (0 , T ) and in such a way that they converge to the HUM control of the continuouswave equation, i.e. the minimal L -norm control. This result motivates to studyin contrast to the classical finite element semi-discretization a mixed finite elementscheme. Acknowledgments:
The author warmly acknowledge Enrique Zuazua for hostingthe author in his laboratory at the Universidad Aut´onoma de Madrid, proposingthe study and for insightful and helpful discussions. The author also warmly ac-knowledges Gr´egoire Allaire for great and fruitful scientific discussions.
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