Asymptotic Analysis of the Approximate Control for Parabolic Equations with Periodic Interface

Transcription

1 Asymptotic Analysis of the Approximate Control for Parabolic Equations with Periodic Interface Patrizia Donato Université de Rouen International Workshop on Calculus of Variations and its Applications on the occasion of Luísa Mascarenhas 65th birthday Patrizia Donato (University of Rouen) Lisbonne, december

2 PLAN 1 Introduction 2 The ε-problem with interface 3 Controllability for the ε-problem 4 Review of the homogenization results 5 Controllability for the homogenized problem 6 Convergence of the control problem 7 BON ANNIVERSAIRE, LUISA! Patrizia Donato (University of Rouen) Lisbonne, december

3 Introduction Control Theory Control Theory is a branch of mathematics which aims to find a control that will lead the given state of the system in a desirable situation. The system: An evolution system is considered either in terms of partial or ordinary differential equations. Patrizia Donato (University of Rouen) Introduction Lisbonne, december

4 Introduction The Controllability Problem for P.D.E. Then, given the initial state at time t = 0, we act on the system by a control, in order to reach a desired final state at time t = T (exact controllability) or to approach a desired final state (approximate controllability). Also, we want to act on an arbitrary zone (not everywhere..). The control can be the right-hand side of the equation (internal control) or the data in the boundary condition (boundary control). Patrizia Donato (University of Rouen) Introduction Lisbonne, december

5 Introduction The approximate controllability for parabolic problems Due to the regularizing effect of the heat equation, one cannot reach any given L 2 state. The approximate controllability problem One has approximate controllability if the set of reachable final states is dense in L 2 (Ω). The variational approach Following an idea by J.-L. Lions, the approximate control can be constructed as the solution of a related transposed (backward) problem having as final data the (unique) minimum point of a suitable functional. J.-L. Lions, Remarques sur la controlabilité approchée. in Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos, octubre 1990, Grupo de Análisis Matemático Aplicado de la University of Malaga, Spain (1991), Patrizia Donato (University of Rouen) Introduction Lisbonne, december

6 Introduction The approximate controllability problem for a model case Let Ω be a connected bounded open set of R n (n 2) and ω a given open non-empty subset of Ω. For the usual heat equation the problem reads Given w L 2 (Ω) and δ > 0 find ϕ L 2 (Ω) such that for a given u 0 L 2 (Ω) the solution u of u div (A u) = χ ω ϕ in Ω (0, T ), u = 0 on Ω (0, T ), u(x, 0) = u 0 in Ω, verifies the following approximate controllability: u(x, T ) w L 2 (Ω) δ. Patrizia Donato (University of Rouen) Introduction Lisbonne, december

7 Introduction Construction of the control for the model case In the variational approach of J.-L. Lions, ϕ is obtained as the solution of the following homogeneous transposed problem: ϕ div(a 0 ϕ) = 0 in Ω (0, T ), ϕ = 0 on Ω (0, T ), ϕ(x, T ) = ˆϕ 0 in Ω, the final data ˆϕ 0 being the (unique) minimum point of the functional J 0 on L 2 (Ω) given by J 0 (ψ 0 ) = 1 2 T 0 ω ψ 2 dx dt + δ ψ 0 L 2 (Ω) (w v(t ))(ψ 0 ) dx, Ω Patrizia Donato (University of Rouen) Introduction Lisbonne, december

8 Introduction where ψ is the solution of ψ div (A ψ) = 0 in Ω (0, T ), ψ = 0 on Ω (0, T ), ψ(x, T ) = ψ 0 in Ω and v is the solution of the problem v div (A v) = 0 in Ω (0, T ), v = 0 on Ω (0, T ), v(x, 0) = u 0 in Ω. Patrizia Donato (University of Rouen) Introduction Lisbonne, december

9 Introduction The case of oscillating coefficients In this case for every ε one can construct a control for the problem Given w ε L 2 (Ω) and δ > 0 find ϕ ε L 2 (Ω) such that for a given uε 0 L 2 (Ω) the solution u ε of u ε div (A ε u ε ) = χ ω ϕ ε in Ω (0, T ), u = 0 on Ω (0, T ), u ε (x, 0) = uε 0 in Ω, verifies the following approximate controllability: u ε (x, T ) w ε L 2 (Ω) δ. Suppose now that: { (i) u 0 ε u 0 for some u 0 and w in L 2 (Ω). strongly in L 2 (Ω), (ii) w ε w strongly in L 2 (Ω), Patrizia Donato (University of Rouen) Introduction Lisbonne, december

10 Introduction An interesting question is Do the control and the corresponding solution of the ε-problem converge (as ε 0) to a control of the homogenized problem and to the corresponding solution, respectively? A positive answer is given in E. Zuazua, Approximate Controllability for Linear Parabolic Equations with Rapidly Oscillating Coefficients. Control Cybernet, 4 (1994), Patrizia Donato (University of Rouen) Introduction Lisbonne, december

11 Introduction The case of perforated domains Let Ω ε a domain perforated by a set S ε of ε-periodic holes of size ε. Then, for every ε one can construct a control for the problem Given w ε L 2 (Ω ε ) and δ > 0 find ϕ ε L 2 (Ω ε ) such that for a given uε 0 L 2 (Ω ε ) the solution u ε of u ε div (A ε u ε ) = χ ω ϕ ε in Ω ε (0, T ), u = 0 on Ω (0, T ), A ε u ε n = 0 on S ε (0, T ), u ε (x, 0) = uε 0 in Ω, verifies the following approximate controllability: u ε (x, T ) w ε L 2 (Ω ε) δ. Suppose that { (i) uε 0 u 0 strongly in L 2 (Ω), (ii) w ε w strongly in L 2 (Ω). Patrizia Donato (University of Rouen) Introduction Lisbonne, december

12 Introduction Again one can give a positive answer to the previous question, see P. Donato and A. Nabil, Approximate Controllability of Linear Parabolic Equations in Perforated Domains. ESAIM: Control, Optimization and Calculus of Variations, 6 (2001), In this case, in order to obtain the convergence result, the suitable functional in the construction of the control for the homogenized problem is J(ψ 0 ) = 1 T 2 θ ψ 2 dx dt + δ θ ψ 0 L 2 (ω) θ (w v(t )) ψ 0 dx, 0 ω Ω where θ is the proportion of material in the reference cell. Patrizia Donato (University of Rouen) Introduction Lisbonne, december

13 face A Heat Equation in a Composite with Interfacial Resistances Work in collaboration with Editha C. Jose (University of the Philippines Los Baños) We consider a more complicated case where the domain is a two-component domain, the holes being here replaced by a second material. On the periodic interface, a jump of the solution is prescribed, which is proportional to the conormal derivative via a parameter γ R, and a Dirichlet condition is imposed on the exterior boundary Ω. This problem models the heat diffusion in a two-component composite with an imperfect contact on the interface, see for a physical justification of the model H.S.Carslaw, J.C. Jaeger, Conduction of Heat in Solids. The Clarendon Press, Oxford, Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

14 face We describe here the case γ = 1, which is the most interesting case since the homogenized problem is a coupled system of a P.D.E. and a O.D.E., giving rise to a memory effect. However, our results concern all the value of γ R. Several questions must be addressed here : Can we construct an approximate control for the ε-problem? Can we construct an approximate control for the homogenized coupled problem? If such controls exist, do the control and the corresponding solution of ε-problem converge to a control of the homogenized problem and to the corresponding solution, respectively? We give here positive answers to all three questions. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

15 face The main difficulty To find suitable functionals for both problems, the oscillating problem and the homogenized one, which provide not only the approximate controls but also the desired convergences. Many functionals provide control, in particular we can change the constant in the different terms of the functional and still have controllability. But those providing the convergence of the problem have to be carefully chosen. Important tools The corrector results play an important role in the proofs. Unique continuation results are needed for the two problems, in particular for the homogenized coupled problem. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

16 face The Domain The domain in R n : Ω = Ω 1ε Ω 2ε, Y is the reference cell where Y = Y 1 Y 2, with Y 2 Y Γ := Y 2 Lipschitz continuous, where Ω 1ε is a connected union of ε n periodic translated sets of εy 1, Ω 2ε is a union of ε n periodic disjoint translated sets of εy 2, Γ ε := Ω 2ε is the interface between the two components, with Ω Γ ε =. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

17 face The ε-problem in the two-component domain Consider for γ R the following parabolic system of equations: u 1ε div(a( x ε ) u 1ε) = χ ω1ε ϕ 1ε in Ω 1ε (0, T ), u 2ε div(a( x ε ) u 2ε) = χ ω2ε ϕ 2ε in Ω 2ε (0, T ), A( x ε ) u 1ε n 1ε = A( x ε ) u 2ε n 2ε on Γ ε (0, T ), A( x ε ) u 1ε n 1ε = ε γ h( x ε )(u 1ε u 2ε ) on Γ ε (0, T ), u ε = 0 on Ω (0, T ), u 1ε (x, 0) = U1ε 0 in Ω 1ε, u 2ε (x, 0) = U2ε 0 in Ω 2ε, where n iε is the unitary outward normal to Ω iε (i = 1, 2), ω is a given open non-empty subset of Ω, and we set ω iε = ω Ω iε, i = 1, 2. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

18 face Assumptions A is an n n matrix field which is Y periodic, symmetric and of class C 1 (Ȳ ) such that for some 0 < α < β, one has { (A(y)λ, λ) α λ 2, λ R n and a.e. in Y. A(y)λ β λ. h is a Y - periodic function in L (Γ) such that h 0 R with 0 < h 0 < h(y), y a.e. in Γ. (U 0 1ε, U0 2ε ) L2 (Ω 1ε ) L 2 (Ω 2ε ), (ϕ 1ε, ϕ 2ε ) L 2 (0, T ; L 2 (Ω)) L 2 (0, T ; L 2 (Ω)). Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

19 face Remark Observe that for any v L 2 (Ω) v = vχ + vχ. Ω1ε Ω2ε Then the map Φ : v L 2 (Ω) (v Ω1ε, v Ω2ε ) L 2 (Ω 1ε ) L 2 (Ω 2ε ) is a bijective isometry since v 2 L 2 (Ω) = v 2 L 2 (Ω 1ε ) + v 2 L 2 (Ω 2ε ), for every v L2 (Ω). In the sequel, when needed we identify v L 2 (Ω) with (v Ω1ε, v Ω2ε ) L 2 (Ω 1ε ) L 2 (Ω 2ε ). Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

20 face The variational formulation We set ( x ) ( x ) A ε (x) = A, h ε (x) = h. ε ε and we introduce the functional spaces V ε := {v 1 H 1 (Ω 1ε ) v 1 = 0 on Ω} equipped with the norm v 1 V ε := v 1 L 2 (Ω 1ε) and v =(v W ε. 1, v 2 ) L 2 (0, T ; V ε ) L 2 (0, T ; H 1 (Ω 2ε )) s.t. = v L 2 (0, T ; (V ε ) ) L 2 (0, T ; (H 1 (Ω 2ε )) )}, equipped with the norm v W ε = v 1 L 2 (0,T ;V ε ) + v 2 L 2 (0,T ;H 1 (Ω 2ε)) + v 1 L 2 (0,T ;(V ε ) ) + v 2 L 2 (0,T ;(H 1 (Ω 2ε)) ). Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

21 face Then, the variational formulation associated to the problem is Find u ε = (u 1ε, u 2ε ) in W ε such that T 0 T u 1ε, v 1 (V ε ),V ε dt + u 2ε, v 2 (H 1 (Ω 2ε )),H 1 (Ω 2ε ) dt T + A ε u 1ε v 1 dx dt + 0 Ω 1ε T + ε γ = T 0 0 for every (v 1, v 2 ) L 2 (0, T ; V ε ) L 2 (0, T ; H 1 (Ω 2ε )) u 1ε (x, 0) = U1ε 0 in Ω 1ε and u 2ε (x, 0) = U2ε 0 in Ω 2ε. 0 T Γ ε h ε (u 1ε u 2ε )(v 1 v 2 ) dσ x dt ω 1ε ϕ 1ε v 1 dx dt + T 0 0 A ε u 2ε v 2 dx dt Ω 2ε ϕ 2ε v 2 dx dt ω 2ε Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

22 face The related Controllability Problem Given w iε L 2 (Ω iε ), i = 1, 2 δ 1 > 0 and δ 2 > 0, find a control ϕ ε = ( ϕ 1ε, ϕ 2ε ) such that the solution u ε = (u 1ε, u 2ε ) of the above problem verifies the estimates (i) u 1ε (T ) w 1ε L 2 (Ω 1ε ) δ 1 (ii) u 2ε (T ) w 2ε L 2 (Ω 2ε ) δ 2. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

23 face Some References E. Jose, Homogenization of a Parabolic Problem with an Imperfect Interface. Rev. Roumaine Math. Pures Appl., 54 (2009) (3), P. Donato, E. Jose, Corrector Results for a Parabolic Problem with a Memory Effect. ESAIM: M2AN 44 (2010), P. Donato, E. Jose, Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance. ESAIM: Control, Optimisation and Calculus of Variations, 21 (1), (2015), , DOI /cocv/ P. Donato, E. Jose, Approximate Controllability of a Parabolic System with Imperfect Interfaces, to appear in Philippine Journal of Sciences. F. Ammar Khodja, personal communication, Appendix. Patrizia Donato (University of Rouen) The ε-problem with interface Lisbonne, december

24 problem The Variational Approach to the Controllability Problem Let (w 1ε, w 2ε ) L 2 (Ω 1ε ) L 2 (Ω 2ε ) and ϕ 0 L 2 (Ω) and define the functional J ε by J ε (ϕ 0 ) = 1 2 where θ i = Y i Y ( T 0 ω 1ε ϕ 1ε 2 dx dt T 0 ω 2ε ϕ 2ε 2 dx dt ) + +δ 1 ϕ 0 L 2 (Ω 1ε ) + δ 2 ϕ 0 L 2 (Ω 2ε ) (w 1ε v 1ε (T ))ϕ 0 dx Ω 1ε (w 2ε v 2ε (T ))ϕ 0 dx, Ω 2ε for i = 1, 2. Observe that χ converges to θ i in L only weakly *. Then, one Ωiε difficulty in this study is that the L 2 -weak convergence of a function v ε to some v do not imply the convergence of χ v ε to θ i v. Ωiε Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

25 problem In the functional defined above, ϕ ε = (ϕ 1ε, ϕ 2ε ) is the solution of the transposed problem of the system given by ϕ 1ε div(a ε ϕ 1ε ) = 0 in Ω 1ε (0, T ), ϕ 2ε div(a ε ϕ 2ε ) = 0 in Ω 2ε (0, T ), A ε ϕ 1ε n 1ε = A ε ϕ 2ε n 2ε on Γ ε (0, T ), A ε ϕ 1ε n 1ε = ε γ h ε (ϕ 1ε ϕ 2ε ) on Γ ε (0, T ), ϕ ε = 0 on Ω (0, T ), ϕ 1ε (x, T ) = ϕ 0 Ω1ε in Ω 1ε, ϕ 2ε (x, T ) = ϕ 0 Ω2ε in Ω 2ε. Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

26 problem On the other hand, v ε = (v 1ε, v 2ε ) is the solution of the auxiliary problem v 1ε div(a ε v 1ε ) = 0 in Ω 1ε (0, T ), v 2ε div(a ε v 2ε ) = 0 in Ω 2ε (0, T ), A ε v 1ε n 1ε = A ε v 2ε n 2ε on Γ ε (0, T ), A ε v 1ε n 1ε = ε γ h ε (v 1ε v 2ε ) on Γ ε (0, T ), v ε = 0 on Ω (0, T ), v 1ε (x, 0) = U1ε 0 in Ω 1ε, v 2ε (x, 0) = U2ε 0 in Ω 2ε, where n i is the unitary outward normal to Ω iε (i = 1, 2) and (U1ε 0, U0 2ε ) L2 (Ω 1ε ) L 2 (Ω 2ε ). Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

27 problem The Controllability Result for Fixed ε Theorem 1 [D-Jose] Let T > 0, δ 1 > 0, δ 2 > 0 be given real numbers and U 0 ε be in L 2 (Ω). Fix w ε = (w 1ε, w 2ε ) L 2 (Ω 1ε ) L 2 (Ω 2ε ). Let ϕ 0 ε be the unique minimum point of the functional J ε and ϕ ε = ( ϕ 1ε, ϕ 2ε ) the solution of the transposed problem with final data ϕ 0 ε. Then the solution u ε = (u 1ε, u 2ε ) of the following system: u 1ε div(a ε u 1ε ) = χ ω1ε ϕ 1ε in Ω 1ε (0, T ), u 2ε div(a ε u 2ε ) = χ ω2ε ϕ 2ε in Ω 2ε (0, T ), A ε u 1ε n 1ε = A ε u 2ε n 2 on Γ ε (0, T ), A ε u 1ε n 1 = ε γ h ε (u 1ε u 2ε ) on Γ ε (0, T ), u ε = 0 on Ω (0, T ), u 1ε (x, 0) = U1ε 0 in Ω 1ε, u 2ε (x, 0) = U2ε 0 in Ω 2ε, satisfies the following estimate: (i) u 1ε (T ) w ε L 2 (Ω 1ε) δ 1 (ii) u 2ε (T ) w ε L2 (Ω 2ε) δ 2. Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

28 problem Sketch of the proof Existence of the minimum By standard arguments one can prove that the functional J ε is continuous and strictly convex. Then, for every fixed ε, we prove the coerciveness, i.e. J ε (ϕ 0 lim inf ε) ϕ 0 ε L 2 (Ω) ϕ 0 ε L 2 (Ω) min{δ 1, δ 2 }. To do that it is essential to use the unique-continuation property of Saut-Scheurer. Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

29 problem Proof of the control result Let ψε 0 L 2 (Ω) and i = 1, 2. If ϕ 0 ε is the minimum point of the functional J ε we have 2 ( T ) ϕ iε ψ iε dx dt (w ε v iε (T ))ψε 0 dx i=1 0 ω i Ω iε δ 1 ψε 0 L 2 (Ω 1ε ) + δ 2 ψε 0 L 2 (Ω 2ε ), where ϕ ε = ( ϕ 1ε, ϕ 2ε ) is the solution of the transposed problem with the corresponding final data ϕ 0 ε. Then we prove that (u iε (T ) w ε )ψ 0 ε dx δ i ψε 0 L 2 (Ω 1ε ), ψε 0 L 2 (Ω iε ), i = 1, 2, Ω iε which implies the controllability result. Patrizia Donato (University of Rouen) Controllability for the ε-problem Lisbonne, december

30 sults A review of the homogenization and correctors results From now on, we consider γ = 1, the most interesting case. The homogenization result [Jose] Let A ε and h ε be as before and z ε = (z 1ε, z 2ε ) be the solution of z iε div(a ε z iε ) = g iε in Ω iε (0, T ), i = 1, 2, A ε z 1ε n 1ε = A ε z 2ε n 2ε on Γ ε (0, T ), A ε z 1ε n 1ε = ε h ε (z 1ε z 2ε ) on Γ ε (0, T ), z ε = 0 on Ω (0, T ), z iε (x, 0) = Zε 0 Ωiε in Ω iε, i = 1, 2, where Zε 0 L 2 (Ω) and (g 1ε, g 2ε ) [L 2 (0, T ; L 2 (Ω))] 2. Suppose that (i) (χ Z Ω1ε ε 0, χ Z Ω2ε ε 0 ) (θ 1 Z1 0, θ 2Z2 0) weakly in [L2 (Ω)] 2, (ii) (χ g 1ε, χ g 2ε ) (θ 1 g 1, θ 2 g 2 ) weakly in [L 2 (0, T ; L 2 (Ω))] 2, Ω1ε Ω1ε Patrizia Donato (University of Rouen) Review of the homogenization results Lisbonne, december

31 sults Then there exists a linear continuous extension operator P1 ε L(L2 (0, T ; V ε ); L 2 (0, T ; H0 1(Ω))) L(L2 (0, T ; L 2 (Ω 1ε )); L 2 (0, T ; L 2 (Ω))) such that (i) P1 ε z 1ε z 1 weakly in L 2 (0, T ; H0 1(Ω)), (ii) z 1ε θ 1 z 1 weakly* in L (0, T ; L 2 (Ω)), (iii) z 2ε z 2 weakly* in L (0, T ; L 2 (Ω)), (iv) ε 1 2 z 1ε z 2ε L 2 (0,T ;L 2 (Γ ε)) < c, where denotes the zero extension to the whole of Ω. Furthermore, { (i) A ε z 1ε A 0 z 1 weakly in L 2 (0, T ; [L 2 (Ω)] n ), (ii) A ε z 2ε 0 weakly in L 2 (0, T ; [L 2 (Ω)] n ), where A 0 λ := m Y (A ŵ λ ), the function ŵ λ H 1 (Y 1 ) being for any λ R n, the unique solution of the problem div (A ŵ λ ) = 0 in Y 1, (A ŵ λ ) n 1 = 0 in Γ, ŵ λ λ y Y -periodic and m Y1 (ŵ λ λ y) = 0. Patrizia Donato (University of Rouen) Review of the homogenization results Lisbonne, december

32 sults The pair (z 1, z 2 ) C 0 ([0, T ]; L 2 (Ω)) L 2 (0, T ; H 1 0 (Ω)) C 0 ([0, T ]; L 2 (Ω)) with z 1 L2 (0, T ; H 1 (Ω)) is the unique solution of the homogenized coupled system θ 1 z 1 div (A0 z 1 ) + c h (θ 2 z 1 z 2 ) = θ 1 g 1 in Ω (0, T ), z 2 c h(θ 2 z 1 z 2 ) = θ 2 g 2 in Ω (0, T ), z 1 = 0 on Ω (0, T ), z 1 (0) = Z1 0, z 2(0) = θ 2 Z2 0 in Ω, where c h = 1 Y 2 Γ h(y) dσ y. This result can be interpretated as a memory effect. Indeed, solving the EDO and replacing in the PDE gives a PDE with a memory term. Patrizia Donato (University of Rouen) Review of the homogenization results Lisbonne, december

33 sults The corrector result [D-Jose] Under the assumption of the homogenization theorem, suppose further that for Z 0 ε L 2 (Ω) and g iε L 2 (0, T ; L 2 (Ω)), i = 1, 2 one has (χ Ω1ε Z 0 ε, χ Ω2ε Z 0 ε ) (θ 1 Z 0 1, θ 2 Z 0 2 ) weakly in [L 2 (Ω)] 2 and (i) g iε g i strongly in L 2 (0, T ; L 2 (Ω)), (ii) Zε 0 2 L 2 (Ω 1ε ) + Z ε 0 2 L 2 (Ω 2ε ) θ 1 Z1 0 2 L 2 (Ω) + θ 2 Z2 0 2 L 2 (Ω). Remark In particular, assumption (i) holds if for i = 1, 2, g iε = g ε Ωiε g ε g strongly in L 2 (0, T ; L 2 (Ω)). On the other hand, the assumptions on the initial data hold if for i = 1, 2, one has for instance χ Ωiε Z 0 ε = Z 0 iε Ω iε for some Z 0 iε L2 (Ω) such that Ziε 0 Z i 0 strongly in L 2 (Ω) (as it will be true in the control problem). Patrizia Donato (University of Rouen) Review of the homogenization results Lisbonne, december and

34 sults If (e j ) j=1,...,n is the canonical basis of R n and ŵ j is the solution of the cell problem written for λ = e j, j = 1,..., n, let C ε = (Cij ε) 1 i,j n be the corrector matrix defined, for i, j = 1,..., n, by C ij (y) := ŵ j (y), a.e. on Y 1, Cij ε (x) = y C ( x ) ij a.e. on Ω. i ε Assuming that Γ is of class C 2, the following corrector results hold true: (i) lim z 1ε z 1 C 0 (0,T ;L 2 (Ω ε 0 1ε )) = 0, (ii) lim z 2ε θ 1 ε 0 2 z 2 C 0 (0,T ;L 2 (Ω 1ε )) = 0, (iii) (iv) lim z 1ε C ε z 1 L 2 (0,T ;[L 1 (Ω ε 0 1ε )] n ) = 0, lim z 2ε L 2 (0,T ;[L 2 (Ω ε 0 2ε )] n ) = 0. Patrizia Donato (University of Rouen) Review of the homogenization results Lisbonne, december

35 lem Construction of the Control for the Homogenized Problem Let T > 0, δ 1 > 0, δ 2 > 0 be given, w be given in L 2 (Ω) and U 0 1 and U0 2 be in L 2 (Ω). For a given w L 2 (Ω), we define the functional J 0 on L 2 (Ω) L 2 (Ω) by J 0 (Φ 0, Ψ 0 ) = 1 2 θ 1 T Ω 0 ω ϕ 1 2 dx dt θ 1 2 T Ω 0 ω ϕ 2 2 dx dt +δ 1 θ1 Φ 0 L 2 (Ω) + δ 2 θ2 Ψ 0 L 2 (Ω) θ 1 (w v 1 (T )) Φ 0 ( dx θ 2 w θ 1 2 v 2(T ) ) Ψ 0 dx, Patrizia Donato (University of Rouen) Controllability for the homogenized problem Lisbonne, december

36 lem where (ϕ 1, ϕ 2 ) is the solution of the following homogeneous transposed problem: θ 1 ϕ 1 div(a 0 ϕ 1 ) + c h (θ 2 ϕ 1 ϕ 2 ) = 0 in Ω (0, T ), ϕ 2 c h (θ 2 ϕ 1 ϕ 2 ) = 0 in Ω (0, T ), ϕ 1 = 0 on Ω (0, T ), ϕ 1 (x, T ) = Φ 0, ϕ 2 (x, T ) = θ 2 Ψ 0 in Ω and (v 1, v 2 ) is the solution of the problem θ 1 v 1 div(a 0 v 1 ) + c h (θ 2 v 1 v 2 ) = 0 in Ω (0, T ), v 2 c h (θ 2 v 1 v 2 ) = 0 in Ω (0, T ), v 1 = 0 on Ω (0, T ), v 1 (x, 0) = U1 0, v 2(x, 0) = θ 2 U2 0 in Ω. Patrizia Donato (University of Rouen) Controllability for the homogenized problem Lisbonne, december

37 lem The Controllability Result for the coupled system Theorem 2 [D-Jose] Let ( Φ 0, Ψ 0 ) be the unique minimum point of the functional J 0 and ( ϕ 1, ϕ 2 ) the solution of (36) with final data ( Φ 0, θ Ψ0 2 ). Then if (u 1, u 2 ) is the solution of θ 1 u 1 div(a 0 u 1 ) + c h (θ 2 u 1 u 2 ) = χ ω θ 1 ϕ 1 in Ω (0, T ), u 2 c h (θ 2 u 1 u 2 ) = χ ω ϕ 2 in Ω (0, T ), u 1 = 0 on Ω (0, T ), u 1 (x, 0) = U1 0, u 2(x, 0) = θ 2 U2 0 in Ω, we have the following approximate controllability: θ 1 u 1 (x, T ) + u 2 (x, T ) w L 2 (Ω) δ 1 θ1 + δ 2 θ2. Patrizia Donato (University of Rouen) Controllability for the homogenized problem Lisbonne, december

38 lem Sketch of the proof By standard arguments one can prove that the functional J 0 is continuous and strictly convex. Then, we have to prove the coerciveness, i.e. as before, that for any sequence {(Φ 0 n, Ψ 0 n)} in [L 2 (Ω)] 2 such that (Φ 0 n, Ψ 0 n) [L 2 (Ω)] 2, one has lim inf n J 0 (Φ 0 n, Ψ 0 n) (Φ 0 n, Ψ 0 n) [L 2 (Ω)] 2 min{δ 1 θ1, δ 2 θ2 }. Here the proof is more delicate, in particular the unique-continuation property of Saut-Scheurer cannot be applied to the EDP-EDO coupled system. We use a non trivial result proved ad hoc by F. Ammar Khodja (University of Besançon), written in Appendix of our paper. Then, J 0 has a unique minimum point. Again, we are able to derive the controllability result. Patrizia Donato (University of Rouen) Controllability for the homogenized problem Lisbonne, december

39 lem Asymptotic Behaviour of the Control Problem Theorem 3 [D-Jose] Suppose that T, δ 1, δ 2 > 0 and that Γ is of class C 2. Let w ε and Uε 0 be given in L 2 (Ω). Let u ε = (u 1ε, u 2ε ) and ϕ ε = ( ϕ 1ε, ϕ 2ε ) the related solution and the approximates control given by Theorem 2, respectively. For {w ε } ε L 2 (Ω) and {Uε 0 } ε L 2 (Ω), we suppose that for some Ui 0, i = 1, 2 and w in L2 (Ω), they satisfy the following assumptions: (i) χ U Ωiε ε 0 θ i Ui 0 weakly in L 2 (Ω), (ii) Uε 0 2 L 2 (Ω 1ε ) + U0 ε 2 L 2 (Ω 2ε ) θ 1 U1 0 2 L 2 (Ω) + θ 2 U2 0 2 L 2 (Ω), (iii) w ε w strongly in L 2 (Ω). (Recall that in particular we can suppose that χ Ωiε U 0 ε = U 0 iε Ω iε U 0 iε U0 i strongly in L 2 (Ω).) with Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

40 lem Then as ε 0, one has (i) χ ω1ε ϕ1ε χ ω θ 1 ϕ 1 weakly in L 2 (0, T ; L 2 (Ω)), (ii) χ ω2ε ϕ2ε χ ω ϕ 2 weakly in L 2 (0, T ; L 2 (Ω)), (iii) (χ ϕ 0 Ω1ε ε, χ ϕ 0 Ω2ε ε) (θ Φ0 1, θ Ψ0 2 ) weakly in [L 2 (Ω)] 2, where ( ϕ 1, ϕ 2 ) is the solution of the trasposed problem with final data ( Φ 0, θ Ψ0 2 ) and ( Φ 0, Ψ 0 ) is the unique minimum point of the functional J 0. Moreover, (i) P1 εu 1ε u 1 weakly in L 2 (0, T ; H0 1(Ω)), (ii) ũ 1ε θ 1 u 1 weakly* in L (0, T ; L 2 (Ω)), (iii) ũ 2ε u 2 weakly* in L (0, T ; L 2 (Ω)), Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

41 lem where the couple (u 1, u 2 ) satisfies θ 1 u 1 div(a 0 u 1 ) + c h (θ 2 u 1 u 2 ) = θ 1 χ ω ϕ 1 in Ω (0, T ), u 2 c h (θ 2 u 1 u 2 ) = χ ω ϕ 2 in Ω (0, T ), u 1 = 0 on Ω (0, T ), u 1 (x, 0) = U1 0, u 2(x, 0) = θ 2 U2 0 in Ω. The couple ( ϕ 1, ϕ 2 ) is an approximate control for the homogenized problem (41) corresponding to w and the constants δ 1 and δ 2, that is θ 1 u 1 (x, T ) + u 2 (x, T ) w L 2 (Ω) δ 1 θ1 + δ 2 θ2. Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

42 lem Sketch of the proof The proof is long and lies on several propositions. Proposition 1 The functionals J ε are uniformly coercive, that is, lim inf ϕ 0 ε L 2 (Ω) ε 0 J ε (ϕ 0 ε) ϕ 0 ε L 2 (Ω) min{δ 1, δ 2 }. Corollary Let ϕ 0 ε the unique minimum point of J ε. Then, there exists a constant C independent of ε such that ϕ 0 ε L 2 (Ω) C. Hence, there exists (ξ 0, ν 0 ) [L 2 (Ω)] 2 such that (up to a subsequence) ( ) χ ϕ 0 Ω1ε ε, χ ϕ 0 Ω2ε ε ( θ 1 ξ 0, θ 2 ν 0) weakly in [L 2 (Ω)] 2. Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

43 lem Then, we prove the following two essential results (in the spirit of the Γ-convergence), whose proof is rather technical: Proposition 2 The functional J ε satisfies ) lim J ε (χ Φ 0 + χ Ψ 0 ( = J 0 Φ 0, Ψ 0), ε 0 Ω1ε Ω2ε for every (Φ 0, Ψ 0 ) in [L 2 (Ω)] 2. Remark In the proof we use the corrector results for the transposed problem with final data χ Ω1ε Φ 0 + χ Ω2ε Ψ 0. Proposition 3 For any sequence {ψε} 0 ε L 2 (Ω) such that as ε 0, ( ) χ ψ 0 Ω1ε ε, χ ψ 0 Ω2ε ε (θ 1 Φ 0, θ 2 Ψ 0 ) weakly in [L 2 (Ω)] 2, for some (Φ 0, Ψ 0 ) in [L 2 (Ω)] 2, we have lim inf ε 0 J ε(ψ 0 ε) J 0 (Φ 0, Ψ 0 ). Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

44 lem This allows to prove Theorem 4 Let U 0 and w be given in L 2 (Ω). Let ϕ 0 ε be the minimum point of J ε, and ( Φ 0, Ψ 0 ) the unique minimum point of the functional J 0. Then, as ε 0, ( ) ( ) χ ϕ 0 Ω1ε ε, χ ϕ 0 Ω2ε ε θ Φ0 1, θ 2 Ψ0 weakly in [L 2 (Ω)] 2. Proof From the Corollary and Proposition 3 we have J 0 (ξ 0, ν 0 ) lim inf ε 0 J ε( ϕ 0 ε). Since ϕ 0 ε is the minimum point of the functional J ε, for any (Φ 0, Ψ 0 ) [L 2 (Ω)] 2, using Proposition 2 we have, lim sup J ε ( ϕ 0 ( ε) lim sup J ε χω1ε Φ 0 + χ Ψ 0) = J 0 (Φ 0, Ψ 0 ). (1) ε 0 Ω2ε Then, we get (ξ 0, ν 0 ) = ( Φ 0, Ψ 0 ) where ( Φ 0, Ψ 0 ) is the unique minimum point of the functional J 0 and consequently, the whole sequence in the Corollary converges. Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

45 lem Proof of Theorem 3 Once proved Theorem 4, using the homogenization and correctors results we can pass to the limit in all the problems which complete the proof! Patrizia Donato (University of Rouen) Convergence of the control problem Lisbonne, december

46 Patrizia Donato (University of Rouen) BON ANNIVERSAIRE, LUISA! Lisbonne, december