Controllability of linear PDEs (I): The wave equation

Controllability of linear PDEs (I): The wave equation M. González-Burgos IMUS, Universidad de Sevilla Doc Course, Course 2, Sevilla, 2018

Contents 1 Introduction. Statement of the problem 2 Distributed Approximate Controllability Existence and uniqueness of solution The adjoint problem. Approximate Controllability Holmgren s uniqueness theorem and consequences 3 An abstract exact controllability result. H.U.M. 4 Boundary Exact Controllability A fundamental identity The boundary inverse inequality Boundary exact controllability. H.U.M. 5 Some complementary results

1. Introduction. Statement of the problem

1. Introduction. Statement of the problem Let Ω R N be a bounded domain, N 1, with boundary Ω of class C 2. Let ω Ω be an open subset and let us fix T > 0. We consider the linear problem for the wave equation (hyperbolic system): tt 2y y = v1 ω in Q = Ω (0, T ), (1) y = 0 on Σ = Ω (0, T ), y(, 0) = y 0, t y(, 0) = y 1 in Ω. In (1) 1 ω represents the characteristic function of the set ω, y(x, t) is the state and v is the control function (which is localized in ω distributed control). 1 2 tt y(x, t) = 2 y t 2 (x, t), (x, t) Q. 2 y(x, t) = N i=1 2 y xi 2 (x, t), (x, t) Q. 3 (y 0, y 1 ) are the initial data and are given in appropriate spaces.

1. Introduction. Statement of the problem Remark Physical interpretation: The wave equation is a simplified model for a vibrating string (N = 1), membrane (N = 2), or elastic solid (N = 3). In these physical interpretations y(x, t) represents the displacement in some direction of the point x at time t > 0. Definition 1 We say that system (1) is exactly controllable at time T if y 1 d, y 2 d (in appropriate spaces), a control v s.t. the solution y v to (1) satisfies y v (, T ) = y 1 d, ty v (, T ) = y 2 d in Ω 2 We say that system (1) is null controllable at time T if in the previous definition we can take y 1 d y 2 d = 0. 3 We say that system (1) is approximately controllable at time T if yd 1, y d 2, ε > 0, control v S.T. (y v (, T ), t y v (, T )) (yd 1, y d 2 ) H < ε.

1. Introduction. Statement of the problem Remark 1 Due to the finite speed of propagation of the solutions of the wave equation, in order to have a controllability result for system (1), the control time T > 0 must be large enough (if T is short, the action over ω cannot reach points far from ω). 2 Linearity and Reversibility: Null controllability at time T if and only if exact controllability at time T. 3 Infinite dimensional space: Exact controllability at time T = approximate controllability at time T. However, we will see that system (1) can be approximately controllable at time T but not exactly controllable at T. 4 We have to specify the functional setting in which system (1) is well posed. The positive controllability results for this system depend on these function spaces.

1. Introduction. Statement of the problem In this session we will also consider the boundary controllability problem for the wave equation : tt 2 y y = 0 in Q = Ω (0, T ), (2) y = v1 γ on Σ = Ω (0, T ), y(, 0) = y 0, t y(, 0) = y 1 in Ω, where γ Ω is an open subset of the boundary. Again, In (1) 1 γ represents the characteristic function of the set γ, y(x, t) is the state and v is the control function (which is localized in γ: boundary control). In the problem (2) we are exerting a control on the system by means of the boundary Dirichlet condition and this control only acts on γ, a part of the boundary. Remark The problem of the existence and uniqueness of solution of system (2) is technically more complex than the case of system (1).

2. Distributed Approximate Controllability Existence and uniqueness of solution

2. Distributed Approximate Controllability 2.1. Existence and uniqueness of solution First, we deal with the problem of existence and uniqueness of solution for system (1): (1) Theorem 2 tt y y = v1 ω in Q, y = 0 on Σ y(, 0) = y 0, t y(, 0) = y 1 in Ω. Let us assume that Ω is a C 2 bounded domain. Then, for every v L 1 (0, T ; L 2 (Ω)), y 0 H 1 0 (Ω) and y 1 L 2 (Ω) system (1) has a unique solution y v C([0, T ]; H 1 0 (Ω)) C1 ([0, T ]; L 2 (Ω)). Moreover, C > 0 s.t. a max ( y v(t) H 1 + t y v (t) ) C( y 0 + y 1 + v t [0,T ] 0 L 1 (L 2 )). a stands for the norm in L 2 (Ω).

2.1. Existence and uniqueness of solution We are going to take U = L 2 (Q) (control space) and H H 1 0 (Ω) L2 (Ω) (state space). Thanks to the previous theorem, for every (y 0, y 1 ) H 1 0 (Ω) L2 (Ω) we can also introduce the reachable state set from (y 0, y 1 ) at time T : R(T ; y 0, y 1 ) = {(y v (T ), t y v (T )) : v L 2 (Q)} H 1 0 (Ω) L2 (Ω). Again, it is a linear variety of H H 1 0 (Ω) L2 (Ω), and System (1) is exactly controllable in H at time T for every (y 0, y 1 ) H 1 0 (Ω) L2 (Ω) one has R(T ; y 0, y 1 ) H 1 0 (Ω) L2 (Ω). System (1) is null controllable in H at time T for every (y 0, y 1 ) H 1 0 (Ω) L2 (Ω) one has (0, 0) R(T ; y 0, y 1 ). System (1) is approximately controllable in H at time T for every (y 0, y 1 ) H 1 0 (Ω) L2 (Ω) one has that R(T ; y 0, y 1 ) is dense in H 1 0 (Ω) L2 (Ω).

2. Distributed Approximate Controllability The adjoint problem. Approximate Controllability

2.2 The adjoint problem. Approximate Controllability For ϕ 0 L 2 (Ω) and ϕ 1 H 1 (Ω) (final data), let us consider the adjoint problem tt 2 ϕ ϕ = 0 in Q, (3) ϕ = 0 on Σ ϕ(, T ) = ϕ 0, t ϕ(, T ) = ϕ 1 in Ω. One has Theorem Let us assume that Ω is a C 2 bounded domain. Then, for every ϕ 0 L 2 (Ω) and ϕ 1 H 1 (Ω), problem (3) has a unique solution ϕ C([0, T ]; L 2 (Ω)) C 1 ([0, T ]; H 1 (Ω)). Moreover, C > 0 s.t. max ( ϕ(t) + tϕ(t) H 1) C( ϕ 0 + ϕ 1 H 1). t [0,T ] Proof: It is a consequence of Theorem 2 and the transposition method.

2.2 The adjoint problem. Approximate Controllability Theorem Let us fix (y 0, y 1 ) H 1 0 (Ω) L2 (Ω), (ϕ 0, ϕ 1 ) L 2 (Ω) H 1 (Ω) and v L 2 (Ω) and let y v and ϕ be the solutions to (1) and (3) associated, resp., to (y 0, y 1 ) and v and (ϕ 0, ϕ 1 ). Then, ( t y v (T ), ϕ 0 ) L 2 (Ω) ϕ 1, y v (T ) (y 1, ϕ(0)) L 2 (Ω) + t ϕ(0), y 0 = vϕ1 ω dx dt. Proof: See J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, 1988. Q

2.2 The adjoint problem. Approximate Controllability Approximate controllability As in the case of linear o. d. systems, linearity of (1) implies R(T ; y 0, y 1 ) = (Y (T ), t Y (T )) + R(T ; 0, 0), with Y the solution to (1) for v 0. Then, approximate controllability at time T of system (1) is equivalent to the identity R(T ; 0, 0) = H 1 0 (Ω) L2 (Ω), and we can take (y 0, y 1 ) 0, i.e., tt 2y y = v1 ω in Q, (4) y = 0 on Σ y(, 0) = 0, t y(, 0) = 0 in Ω. Let us introduce the operator L T : v L 2 (Q) L T v = ( t y v (T ), y v (T )) L 2 (Ω) H 1 0 (Ω), (y v solution to (4)). From Theorem 2, L T L(L 2 (Q); L 2 (Ω) H 1 0 (Ω)).

2.2 The adjoint problem. Approximate Controllability Remark Problem (1) is approximately controllable at time T R(L T ) = L 2 (Ω) H 1 0 (Ω). Let us point out that, in general, R(L T ) R(L T ) (infinite dimensional space). But, R(L T ) = (ker L T ), where L T L(L2 (Ω) H 1 (Ω); L 2 (Q)) is the adjoint operator (which is defined (ϕ 0, ϕ 1 ), L T v = (L T (ϕ 0, ϕ 1 ), v) L 2 (Q), (ϕ 0, ϕ 1 ) L 2 (Ω) H 1 (Ω)). As a consequence of Theorem 4, L T (ϕ 0, ϕ 1 ) = ϕ1 ω L 2 (Q).

2.2 The adjoint problem. Approximate Controllability Theorem SUMMARIZING Let us fix T > 0. Then, system (1) is approximately controllable in H 1 0 (Ω) L2 (Ω) at time T the adjoint problem satisfies the property (unique continuation property) If ϕ C([0, T ]; L 2 (Ω)) C 1 ([0, T ]; H 1 (Ω)) is the solution to (3) associated to ϕ 0 L 2 (Ω) and ϕ 1 H 1 (Ω) and ϕ 0 in ω (0, T ), then ϕ 0 ϕ 1 0 (i.e. ϕ 0 in Q). Remark We have obtained the same result than in the case of o.d. systems. But, is the unique continuation property for the problem (3) true?? It depends on Ω, ω and T.

2. Distributed Approximate Controllability Holmgren s uniqueness theorem and consequences

2.3 Holmgren s uniqueness theorem. Consequences In order to prove Theorem 5 we are going to use a consequence of the Holmgren s uniqueness theorem: Theorem Let P(D) be a differential operator in R k with constant coefficients. Let O 1 and O 2 two convex open sets in R k such that O 1 O 2 and every hyperplane Π which is characteristic with respect to P(D) and satisfies Π O 2 also satisfies Π O 1. Let ϕ D (O 2 ) be a distribution satisfying P(D)ϕ = 0 in D (O 2 ) and ϕ 0 in O 1. Then, ϕ 0 in O 2. For a proof of this result see L. Hörmander, Linear Partial Differential Operators, 1970. We are going to apply this result to the wave operator P(D)ϕ = 2 ttϕ ϕ.

2.3 Holmgren s uniqueness theorem. Consequences Remark Given the hyperplane Π = {(x, t) R N R : N i=1 a ix i + bt = c}, with a i, b, c R (1 i N), then, Π is characteristic with respect to P(D) N a 2 i = b 2. i=1 For a proof of this result, see M. Renardy, R.C. Rogers, An Introduction to PDE, 1993. Now, it is easy to prove,

2.3 Holmgren s uniqueness theorem. Consequences Corollary Let us fix z 1, z 2 R N, δ > 0 y τ > 2 z 1 z 2 and let us take O 1 = B(z 1 ; δ) (0, τ), O 2 = B ((1 λ)z 1 + λz 2 ; δ) (λ z 1 z 2, τ λ z 1 z 2 ). λ [0,1] Then, if ϕ D (O 2 ) and satisfies 2 tt ϕ ϕ = 0 in O 2 and ϕ = 0 in O 1, one has ϕ = 0 in O 2. Proof: The characteristic hyperplanes are given by N a i x i + t = c, i=1 N a 2 i = 1. i=1

2.3 Holmgren s uniqueness theorem. Consequences Let us introduce the quantity δ(ω, ω) = sup inf δ(x, y), x Ω y ω with δ(x, y) = inf{ γ : γ C([0, 1]; Ω) and γ(0) = x, γ(1) = y}. Then, one has the unique continuation property for the adjoint problem. Theorem Let us assume that T > 2δ(Ω, ω) = T 0 (Ω, ω) and let ϕ C 0 ([0, T ]; L 2 (Ω)) a solution to (3) s.t. Then, ϕ 0 in Q = Ω (0, T ). ϕ 0 in ω (0, T ).

2.3 Holmgren s uniqueness theorem. Consequences Proof: Assume Ω is convex: x Ω, z 1 ω and ε > 0 s.t. T > 2 z 1 x, B(z 1 ; ε) ω and O 2 = λ [0,1] B ((1 λ)z 1 + λx; ε) (λ z 1 x, T λ z 1 x ). We can apply Corollary 7 with O 1 = B(z 1 ; ε) (0, T ) (ϕ 0 in O 1 ), deducing ϕ = 0 in O 2. In particular (λ = 1) ϕ = 0 in B(x, ε) ( z 1 x, T z 1 x ). The hypothesis over T gives T /2 ( z 1 x, T z 1 x ) = ϕ(x, T /2) = t ϕ(x, T /2) = 0, for every x Ω. Then, ϕ 0 in Q. Remark Observe that δ(ω, ω) < δ(ω), with δ(ω) = sup x,y Ω δ(x, y). Then, if T > 2δ(Ω) T 1 (Ω), we can apply Theorem 8 for every ω.

3. An abstract exact controllability result. H.U.M.

3. An abstract exact controllability result. H.U.M. Remark Linearity and Reversibility: System (1) is null controllable at time T if and only if system (1) is exactly controllable at time T. Reformulation of the exact controllability: System (1) is exactly controllable in F at time T (y 0, y 1 ) F there exists a control v L 2 (Q) such that the solution y v to tt 2y y = v1 ω in Q = Ω (0, T ), (5) y = 0 on Σ = Ω (0, T ), y(, T ) = 0, t y(, T ) = 0 in Ω satisfies (y v (0), t y v (0)) = (y 0, y 1 ). We can solve this problem using the so-called Hilbert Uniqueness Method (H.U.M. for short).

3. An abstract exact controllability result. H.U.M. Step 1: Let us fix (ϕ 0, ϕ 1 ) L 2 (Ω) H 1 (Ω) and consider tt 2 ϕ ϕ = 0 in Q, (6) ϕ = 0 on Σ ϕ(, 0) = ϕ 0, t ϕ(, 0) = ϕ 1 in Ω, (adjoint problem to (5)). We know this problem has a unique solution (Theorem 3) ϕ C 0 ([0, T ]; L 2 (Ω)) C 1 ([0, T ]; H 1 (Ω)) and, for C > 0, (direct inequality) (7) ϕ L 2 (Q) C( ϕ 0 + ϕ 1 H 1). Step 2: Consider the backward problem tt 2ψ ψ = ϕ1 ω in Q, (8) ψ = 0 on Σ ψ(, T ) = 0, t ψ(, T ) = 0 in Ω

3. An abstract exact controllability result. H.U.M. Again this problem has a unique solution (Theorem 2) ψ C([0, T ]; H 1 0 (Ω)) C1 ([0, T ]; L 2 (Ω)). Thus, we can define the operator { Λ(ϕ0, ϕ 1 ) = ( t ψ(0), ψ(0)) L 2 (Ω) H 1 0 (Ω), Remark Λ L(L 2 (Ω) H 1 (Ω); L 2 (Ω) H 1 0 (Ω)) If (y 1, y 0 ) R(Λ), i.e, if Λ(ϕ 0, ϕ 1 ) = (y 1, y 0 ) then, taking v ϕ1 ω with ϕ solution to (6), one has y v ψ (y v and ϕ, resp., the solutions to (5) and (6)) and y v (T ) = t y v (T ) = 0 in Ω. Step 3: If ζ is the solution to (6) for (ζ 0, ζ 1 ) L 2 (Ω) H 1 (Ω), then (Theorem 4) Λ(ϕ 0, ϕ 1 ), (ζ 0, ζ 1 ) = ( t ψ(0), ζ 0 ) ζ 1, ψ(0) = ϕζ1 ω dx dt. Q

3. An abstract exact controllability result. H.U.M. Assuming T > T 0 (Ω, ω), then ((ϕ 0, ϕ 1 ), (ζ 0, ζ 1 )) F Q ϕζ1 ω dx dt and ( (ϕ 0, ϕ 1 ) F ϕ 2 dx dt ω (0,T ) ) 1/2 are, resp., a scalar product and a norm in L 2 (Ω) H 1 (Ω), not necessarily equivalent to the usual norm. In fact, (direct inequality (7)) (ϕ 0, ϕ 1 ) F C( ϕ 0 + ϕ 1 H 1). Let F be the completion of L 2 (Ω) H 1 (Ω) with respect the norm F (then, F is a Hilbert space which contains the space L 2 (Ω) H 1 (Ω) as a dense subspace).

3. An abstract exact controllability result. H.U.M. Step 4: Conclusion For every (ϕ 0, ϕ 1 ), (ζ 0, ζ 1 ) L 2 (Ω) H 1 (Ω), Λ(ϕ 0, ϕ 1 ), (ζ 0, ζ 1 ) = ((ϕ 0, ϕ 1 ), (ζ 0, ζ 1 )) F. The density implies that the previous equality is valid (ϕ 0, ϕ 1 ), (ζ 0, ζ 1 ) F and Λ : F F is an isomorphism. From Remark 9, if (y 1, y 0 ) F, then we have solved the null controllability problem at time T for system (1). Equivalently, we have solved the exact controllability problem at time T for system (1) in F.

3. An abstract exact controllability result. H.U.M. We have proved Theorem Assume that T > T 0 (Ω, ω) and let F be the completion of L 2 (Ω) H 1 (Ω) with respect to the norm F. Then, (y 0, y 1 ) s.t. (y 1, y 0 ) F, there exists v L 2 (Q) s.t. the solution y v to (1) satisfies y v (T ) = 0, t y v (T ) = 0. Remark From the direct inequality (7) we have deduced that L 2 (Ω) H 1 (Ω) F with dense continuous embedding. Then, F L 2 (Ω) H0 1 (Ω) also with dense continuous embedding. F depends on Ω, ω and T.

3. An abstract exact controllability result. H.U.M. Remark 1 F is an abstract Hilbert space. It would be very interesting to give an explicit expression of F in terms of known Sobolev spaces. 2 H.U. M. is a constructive method: Let us fix (y 0, y 1 ) s.t. (y 1, y 0 ) F. Then, the previous proof provides us a control v which drives the system to the rest. Indeed, let us consider (ϕ 0, ϕ 1 ) F solution to Λ(ϕ 0, ϕ 1 ) = (y 1, y 0 ) in F. Then, v = ϕ1 ω, with ϕ the solution to (1) solves the problem. Moreover, (ϕ 0, ϕ 1 ) is the solution of the problem Minimize 1 ϕ 2 dx dt (y 1, y 0 ), (ϕ 0, ϕ 1 ) F 2,F ω (0,T ) Subject to (ϕ 0, ϕ 1 ) F.

3. An abstract exact controllability result. H.U.M. Remark Let us assume that there is C T > 0 s.t. the inequality (inverse inequality for the adjoint problem) (9) ϕ 0 2 + ϕ 1 2 H C 1 T ϕ 2 dx dt ω (0,T ) holds, with ϕ the solution to (3) associated to (ϕ 0, ϕ 1 ) L 2 (Ω) H 1 (Ω). Then the norm F is equivalent to the usual norm in L 2 (Ω) H 1 (Ω) and F L 2 (Ω) H 1 (Ω) (or equivalently, F L 2 (Ω) H0 1(Ω)). Using the reversibility of problem (3), the inequality (9) is equivalent to the observability inequality for the adjoint problem (10) ϕ(0) 2 + ϕ(0) 2 H C 1 T ϕ 2 dx dt. ω (0,T )

4. Boundary Exact Controllability

4 Boundary Exact Controllability We go back to the previous boundary controllability problem tt 2 y y = 0 in Q = Ω (0, T ), (2) y = v1 γ on Σ = Ω (0, T ), y(, 0) = y 0, t y(, 0) = y 1 in Ω. Our goal is to prove an exact controllability result for this problem. To this end, let us take U = L 2 (Σ) (control space) and H L 2 (Ω) H 1 (Ω) (state space). We also consider the associated adjoint problem tt 2 ϕ ϕ = 0 in Q, (3) ϕ = 0 on Σ ϕ(, T ) = ϕ 0, t ϕ(, T ) = ϕ 1 in Ω, where (ϕ 0, ϕ 1 ) H 1 0 (Ω) L2 (Ω).

4 Boundary Exact Controllability From Theorem 2, ϕ C 0 ([0, T ]; H 1 0 (Ω)) C1 ([0, T ]; L 2 (Ω)) and for a constant C > 0. Remark max ( ϕ(t) H t [0,T ] 0 1 + t ϕ(t) ) C( ϕ 0 H 1 + ϕ 1 ) 0 In this section we will present the method of multipliers which will allow us to prove that system (2) is well posed. Also from this method we will be able to prove the exact controllability result under appropriate assumptions on γ and T.

4. Boundary Exact Controllability A fundamental identity

4.1 A fundamental identity We consider the non-homogenous wave equation tt 2 θ θ = f in Q = Ω (0, T ), (11) θ = 0 on Σ = Ω (0, T ), θ(, 0) = θ 0, t θ(, 0) = θ 1 in Ω, with (θ 0, θ 1 ) H 1 0 (Ω) L2 (Ω) and f L 1 (0, T ; L 2 (Ω)). From Theorem 2, system (11) has a unique solution θ C 0 ([0, T ]; H 1 0 (Ω)) C1 ([0, T ]; L 2 (Ω)) and for a constant C > 0 one has max ( θ(t) H t [0,T ] 0 1 + t θ(t) ) C( θ 0 H 1 + θ 1 + f 0 L 1 (L 2 )).

4.1 A fundamental identity Lemma Let Ω R N be a bouded domain with Ω C 2. Let q C 1 (Ω) N be a vector-valued function. Then 1 q n θ 2 2 Σ n ds dt = ( t θ(t), q θ(t)) T 0 + ( θ) q θ Q + 1 q( t θ 2 θ 2 ) f q θ dx dt 2 Q Q with ϕ the solution to (11) associated to (θ 0, θ 1 ) H 1 0 (Ω) L2 (Ω) and f L 1 (0, T ; L 2 (Ω)). n = n(x): the outward unit normal to Ω at point x Ω. Proof: It can be obtained by multiplying the equation of θ by q θ and integrating by parts several times. J.-L. LIONS, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, 1988.

4.1 A fundamental identity Corollary Let θ be the solution to (11) associated to (θ 0, θ 1 ) H0 1(Ω) L2 (Ω) and f L 1 (0, T ; L 2 (Ω)). Then, θ n L2 (Σ) and, for C > 0, the inequality θ 2 n C(T + 1)( θ 0 2 + θ H 1 1 2 + f 2 L L 2 0 1 (L 2 ) ). (Σ) Proof: Ω C 2, then, n C 1 ( Ω) N and there exists h C 1 (Ω) N s.t. h(x) = n(x) x Ω. We apply Lemma 10 to θ with q h. Application to the adjoint problem (3): From this Corollary we deduce ϕ n L2 (Σ) and the direct inequality for the adjoint equation (12) ϕ 2 n C(T + 1)( ϕ 0 2 + ϕ 1 2 ). Σ

4.1 A fundamental identity Theorem Given (y 0, y 1 ) L 2 (Ω) H 1 (Ω) and v L 2 (Σ), there exists a unique solution y v C 0 ([0, T ]; L 2 (Ω)) C 1 ([0, T ]; H 1 (Ω)) to the direct problem (2). Moreover, C > 0 s.t. max ( y v(t) + t y v (t) H 1) C( y 0 + y 1 H 1). t [0,T ] Moreover, if ϕ is the solution to the adjoint problem (3) corresponding to (ϕ 0, ϕ 1 ) H 1 0 (Ω) L2 (Ω), then ϕ 0, t y v (T ) (ϕ 1, y v (T )) ϕ(0), y 1 + ( t ϕ(0), y 0 ) = v ϕ n 1 γ ds dt. Proof: It is a consequence of Corollary 11 and the transposition method (for more details, see J.-L. LIONS). Σ

4. Boundary Exact Controllability The boundary inverse inequality

4.2 The boundary inverse inequality Let us fix x 0 R N and introduce { Γ(x 0 ) {x Ω : (x x 0 ) n(x) > 0}, R(x 0 ) = max x Ω x x 0, T (x 0 ) 2R(x 0 ), and let us take γ Γ(x 0 ) in the controllability problem (2). Theorem Let us fix T > T (x 0 ). Then, for every (ϕ 0, ϕ 1 ) H0 1(Ω) L2 (Ω), one has ( (13) [T T (x 0 )] ϕ 0 2 + ϕ 1 2) R(x 0 ) ϕ 2 n, Γ(x 0 ) (0,T ) with ϕ the solution to the adjoint problem (3) associated to (ϕ 0, ϕ 1 ). Proof: It is a consequence of Lemma 10: q x x 0 (the details are in J.-L. LIONS).

4.2 The boundary inverse inequality Remark 1 The inequality (13) is called the inverse inequality for the adjoint problem (3). In particular, we will see that it solves the boundary controllability problem for system (2) when γ Γ(x 0 ) and T is large enough (with an explicit estimate of T ). 2 The previous result gives the following unique continuation property for the adjoint problem (3) If ϕ C([0, T ]; H 1 0 (Ω)) C1 ([0, T ]; L 2 (Ω)) is the solution to (3) associated to ϕ 0 H 1 0 (Ω) and ϕ 1 L 2 (Ω) and ϕ n 0 on Γ(x 0 ) (0, T ), then ϕ 0 ϕ 1 0. Following the ideas developed for the distributed approximate controllability problem, from this property we deduce the boundary approximate controllability result at time T for system (2).

4.3 Boundary exact controllability. H.U.M. GOAL: Prove the boundary controllability result for system (2) when γ Γ(x 0 ). We follow the same approach as before: Hilbert Uniqueness Method. The boundary controllability problem is tt 2 y y = 0 in Q = Ω (0, T ), (2) y = v1 Γ(x0 ) on Σ = Ω (0, T ), y(, 0) = y 0, t y(, 0) = y 1 in Ω. This problem is well posed for every (y 0, y 1 ) L 2 (Ω) H 1 (Ω) and v L 2 (Σ) (U L 2 (Σ) is the control space and H L 2 (Ω) H 1 (Ω) is the state space).

4.3 Boundary exact controllability. H.U.M. Again, linearity and reversibility allow us to reformulate the controllability problem: System (2) is exactly controllable at time T (y 0, y 1 ) L 2 (Ω) H 1 (Ω) there exists a control v L 2 (Σ) such that the solution y v to (14) tt 2 y y = 0 in Q = Ω (0, T ), y = v1 Γ(x0 ) on Σ = Ω (0, T ), y(, T ) = 0, t y(, T ) = 0 in Ω satisfies (y v (0), t y v (0)) = (y 0, y 1 ).

4.3 Boundary exact controllability. H.U.M. Hilbert Uniqueness Method: For (ϕ 0, ϕ 1 ) H0 1(Ω) L2 (Ω) we consider tt 2 ϕ ϕ = 0 in Q, ϕ = 0 on Σ ϕ(, 0) = ϕ 0, t ϕ(, 0) = ϕ 1 in Ω, tt 2 ψ ψ = 0 in Q, ψ = ϕ n 1 Γ(x 0 ) on Σ ψ(, T ) = 0, t ψ(, T ) = 0 in Ω, and Λ : (ϕ 0, ϕ 1 ) H 1 0 (Ω) L2 (Ω) ( t ψ(0), ψ(0)) H 1 (Ω) L 2 (Ω). Then Λ L(H0 1(Ω) L2 (Ω); H 1 (Ω) L 2 (Ω)) and (Theorem 12) ϕ ζ Λ(ϕ 0, ϕ 1 ), (ζ 0, ζ 1 ) = n n =: ((ϕ 0, ϕ 1 ), (ζ 0, ζ 1 )) F. Γ(x 0 ) (0,T )

4.3 Boundary exact controllability. H.U.M. If we combine the direct and inverse inequalities (12) and (13) we infer [T T (x 0 )] ( ϕ 0 2 + ϕ 1 2) R(x 0 ) Γ(x 0 ) (0,T ) C(T + 1)R(x 0 )( ϕ 0 2 + ϕ 1 2 ). ϕ n If T > T (x 0 ) then the scalar product (, ) F is equivalent to the usual one and Λ Isom (H 1 0 (Ω) L2 (Ω); H 1 (Ω) L 2 (Ω)). Finally, if (ϕ 0, ϕ 1 ) = Λ 1 (y 1, y 0 ) and v = ϕ n 1 Γ(x 0 ) we have y v (0) = y 0, y v (T ) = 0, t y v (0) = y 1 and t y v (T ) = 0. 2

4.3 Boundary exact controllability. H.U.M. Remark The control v provided by the previous proof can be calculated as follows: (ϕ 0, ϕ 1 ) H 1 0 (Ω) L2 (Ω) is the solution to Minimize 1 ϕ 2 2 Γ(x 0 ) (0,T ) n (y 1, y 0 ), (ϕ 0, ϕ 1 ) Subject to (ϕ 0, ϕ 1 ) H0 1(Ω) L2 (Ω). Then, v = ϕ n 1 Γ(x 0 ). Moreover, there is C > 0 s.t. v L 2 (Σ) C ( y 0 + y 1 H 1). WE HAVE PROVED:

4.3 Boundary exact controllability. H.U.M. Theorem Let us fix x 0 R N and T > T (x 0 ) 2R(x 0 ). Then, (y 0, y 1 ) L 2 (Ω) H 1 (Ω), v L 2 (Σ) s.t. the solution y v to (2) satisfies y v (, T ) t y v (, T ) 0 in Ω. Moreover, the control v can be chosen by solving the previous minimizing problem and, there is C > 0 s.t. Remark v L 2 (Σ) C ( y 0 + y 1 H 1). If U ad (T ; (y 0, y 1 )) = {v L 2 (Σ) : y v (T ) = t y v (T ) = 0 in Ω}, then, U ad (T ; (y 0, y 1 )) is a nonempty convex closed set with an infinite number of elements. It would be very interesting to give a method that selects the best control, for instance, the control of minimal norm in L 2 (Σ).

5. Some complementary results

5. Some complementary results 1. Distributed control. From the previous section, it is possible to deduce a distributed controllability result for system (1). Let us consider ω Ω s.t. for x 0 R N and ε > 0, ω(x 0 ) = O(x 0 ) Ω, O(x 0 ) = B(x; ε). x Γ(x 0 ) Then, we can prove that the abstract space provided by H.U.M. is F = H 1 0 (Ω) L2 (Ω). Theorem Let us fix x 0 R N and T > T (x 0 ) 2R(x 0 ). Then, (y 0, y 1 ) H 1 0 (Ω) L2 (Ω), v L 2 (Q) s.t. the solution y v to (1) satisfies y v (, T ) t y v (, T ) 0 in Ω.

5. Some complementary results 2. The control with minimal L 2 -norm. It is possible to characterize the control with minimal L 2 -norm: Consider the boundary controllability problem (2) for γ = Γ(x 0 ) and assume T > T (x 0 ). Let us fix (y 0, y 1 ) L 2 (Ω) H 1 (Ω) and consider and U ad (T ; (y 0, y 1 )) = {v L 2 (Σ) : y v (T ) = t y v (T ) = 0 in Ω} 1 Minimize v 2 2 Σ Subject to v U ad (T ; (y 0, y 1 )). Then, there exists a unique minimizer v U ad (T ; (y 0, y 1 )). Moreover, v = ϕ n 1 Γ(x 0 ), with ϕ the solution to (3) associated to ( ϕ 0, ϕ 1 ) which solves Minimize 1 ϕ 2 2 Γ(x 0 ) (0,T ) n (y 1, y 0 ), (ϕ 0, ϕ 1 ) Subject to (ϕ 0, ϕ 1 ) H0 1(Ω) L2 (Ω).

5. Some complementary results 3. In this session we have proved an abstract exact controllability result for ω Ω and T > T 0 (Ω, ω) and two results of exact controllability for γ = Γ(x 0 ) and ω = ω(x 0 ) when T > T (x 0 ). But, is it possible to establish an exact controllability result for general (γ, T ) (resp. (ω, T ))? The answer is given in C. BARDOS, G. LEBEAU, J. RAUCH, SIAM J. Control Optim., 30, (1992). In this paper it is proved what system (1) is exactly controllable in H0 1 (Ω) (0, T ) at time T (ω, T ) satisfies the following geometric control condition in Ω: Every ray of Geometric Optic that propagates in Ω and is reflected on its boundary Ω enters ω at a time less than T. From this condition it is not difficult to give examples in which this condition does not hold. Then, in general, system (1) is not exactly controllable at time T. 4. We can also consider other boundary conditions.

REFERENCES J.-L. LIONS Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Masson, Paris, 1988. J.-L. LIONS Exact controllability, stabilization and perturbations for distributed systems, SIAM Review 30 (1988). E. ZUAZUA Controllability and observability of PDE: Some results and open problems Handbook of Differential Equations: Evolutionary Equations. Vol. III, Elsevier, Amsterdam, 2007.