Controllability of the linear 1D wave equation with inner moving for

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1 Controllability of the linear D wave equation with inner moving forces ARNAUD MÜNCH Université Blaise Pascal - Clermont-Ferrand - France Toulouse, May 7, 4 joint work with CARLOS CASTRO (Madrid) and NICOLAE CÎNDEA (Clermont-Ferrand)

2 Problem statement Q T = (, ) (, T ), q T Q T, V := H (, ) L (, ), a, b C([, T ], ], [) 8 < : y tt y xx = v qt, (x, t) Q T y =, (x, t) Ω (, T ) (y(, ), y t (, )) = (y, y ) V, x (, ). γ(t) j ff q T = (x, t) Q T ; a(t) < x < b(t), t (, T ) t qt QT Goals of the works - For some T > and q T, prove the existence of uniform null L (q T )-controls. Approximate numerically the control of minimal L (q T )-norm..5 x Dependent domains q T included in Q T.

3 Problem statement Q T = (, ) (, T ), q T Q T, V := H (, ) L (, ), a, b C([, T ], ], [) 8 < : y tt y xx = v qt, (x, t) Q T y =, (x, t) Ω (, T ) (y(, ), y t (, )) = (y, y ) V, x (, ). γ(t) j ff q T = (x, t) Q T ; a(t) < x < b(t), t (, T ) t qt QT Goals of the works - For some T > and q T, prove the existence of uniform null L (q T )-controls. Approximate numerically the control of minimal L (q T )-norm..5 x Dependent domains q T included in Q T.

4 Combination of two works This contribution is a combination of two recent works : C. Castro : Exact controllability of the D wave equation from a moving interior point, COCV - 3 ( ytt y xx = v(x, t) x=γ(t), (x, t) Q T, γ C ([, T ], (, )), < γ (t) <, t (, T ). Existence of H ( t (,T ) γ(t) (, T )) null controls for (y, y ) L (, ) H (, ), T > N. Cîndea and AM : A mixed formulation for the direct approximations of the control of minimal L -norm for linear type wave equations, CALCOLO 4 y tt (a(x)y x ) x + b(x, t)y = v ω, (x, t) Q T Robust numerical approximation of the control of minimal L (ω (, T ))-norm using a space-time formulation, well-adapted to our non cylindrical case.

5 Combination of two works This contribution is a combination of two recent works : C. Castro : Exact controllability of the D wave equation from a moving interior point, COCV - 3 ( ytt y xx = v(x, t) x=γ(t), (x, t) Q T, γ C ([, T ], (, )), < γ (t) <, t (, T ). Existence of H ( t (,T ) γ(t) (, T )) null controls for (y, y ) L (, ) H (, ), T > N. Cîndea and AM : A mixed formulation for the direct approximations of the control of minimal L -norm for linear type wave equations, CALCOLO 4 y tt (a(x)y x ) x + b(x, t)y = v ω, (x, t) Q T Robust numerical approximation of the control of minimal L (ω (, T ))-norm using a space-time formulation, well-adapted to our non cylindrical case.

6 Literature - Controllability in the non-cylindrical situation A. Khapalov, Controllability of the wave equation with moving point control, 995 For any T >, construction of punctual L -controls for smooth initial data (y, y ) (to avoid strategic point) A. Khapalov, Interior point control and observation for the wave equation, 996 L. Cui, X. Liu, H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, 3 Controllability of the D wave with moving boundary of the form + kt, k (, ] C. Castro, Exact controllability of the D wave equation from a moving interior point, 3 Punctual controllability of the D wave, T > with H -control an C -curve P. Martin, L. Rosier, P. Rouchon, Null controllability of the structurally damped wave equation with moving control, 3. Controllability of y tt y xx εy txx = on the D-Torus, to avoid essential spectrum

7 Literature - Controllability in the non-cylindrical situation A. Khapalov, Controllability of the wave equation with moving point control, 995 For any T >, construction of punctual L -controls for smooth initial data (y, y ) (to avoid strategic point) A. Khapalov, Interior point control and observation for the wave equation, 996 L. Cui, X. Liu, H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, 3 Controllability of the D wave with moving boundary of the form + kt, k (, ] C. Castro, Exact controllability of the D wave equation from a moving interior point, 3 Punctual controllability of the D wave, T > with H -control an C -curve P. Martin, L. Rosier, P. Rouchon, Null controllability of the structurally damped wave equation with moving control, 3. Controllability of y tt y xx εy txx = on the D-Torus, to avoid essential spectrum

8 Literature - Controllability in the non-cylindrical situation A. Khapalov, Controllability of the wave equation with moving point control, 995 For any T >, construction of punctual L -controls for smooth initial data (y, y ) (to avoid strategic point) A. Khapalov, Interior point control and observation for the wave equation, 996 L. Cui, X. Liu, H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, 3 Controllability of the D wave with moving boundary of the form + kt, k (, ] C. Castro, Exact controllability of the D wave equation from a moving interior point, 3 Punctual controllability of the D wave, T > with H -control an C -curve P. Martin, L. Rosier, P. Rouchon, Null controllability of the structurally damped wave equation with moving control, 3. Controllability of y tt y xx εy txx = on the D-Torus, to avoid essential spectrum

9 Literature - Controllability in the non-cylindrical situation A. Khapalov, Controllability of the wave equation with moving point control, 995 For any T >, construction of punctual L -controls for smooth initial data (y, y ) (to avoid strategic point) A. Khapalov, Interior point control and observation for the wave equation, 996 L. Cui, X. Liu, H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, 3 Controllability of the D wave with moving boundary of the form + kt, k (, ] C. Castro, Exact controllability of the D wave equation from a moving interior point, 3 Punctual controllability of the D wave, T > with H -control an C -curve P. Martin, L. Rosier, P. Rouchon, Null controllability of the structurally damped wave equation with moving control, 3. Controllability of y tt y xx εy txx = on the D-Torus, to avoid essential spectrum

10 Literature - Controllability in the non-cylindrical situation A. Khapalov, Controllability of the wave equation with moving point control, 995 For any T >, construction of punctual L -controls for smooth initial data (y, y ) (to avoid strategic point) A. Khapalov, Interior point control and observation for the wave equation, 996 L. Cui, X. Liu, H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, 3 Controllability of the D wave with moving boundary of the form + kt, k (, ] C. Castro, Exact controllability of the D wave equation from a moving interior point, 3 Punctual controllability of the D wave, T > with H -control an C -curve P. Martin, L. Rosier, P. Rouchon, Null controllability of the structurally damped wave equation with moving control, 3. Controllability of y tt y xx εy txx = on the D-Torus, to avoid essential spectrum

11 Generalized Observability inequality: weaker hypothesis Set H = L (, ) H (, ). Let T >. q T -non-cylindrical domain, Lϕ = ϕ tt ϕ xx. Let q T (, ) (, T ) be an open set. n o Φ = ϕ C([, T ]; L (, )) C ([, T ]; H (, )), such that Lϕ L (, T ; H (, )). Theorem (Castro, Cîndea, Münch) Assume that q T (, ) (, T ) is a finite union of connected open sets and satisfies the following hypotheses: Any characteristic line starting at a point x (, ) at time t = and following the optical geometric laws when reflecting at the boundary Σ T must meet q T. Then, there exists C > such that the following estimate holds : «ϕ(, ), ϕ t (, )) H C ϕ L (q T ) + Lϕ L (,T ;H, ϕ Φ. () (,))

12 Generalized Observability inequality: weaker hypothesis Set H = L (, ) H (, ). Let T >. q T -non-cylindrical domain, Lϕ = ϕ tt ϕ xx. Let q T (, ) (, T ) be an open set. n o Φ = ϕ C([, T ]; L (, )) C ([, T ]; H (, )), such that Lϕ L (, T ; H (, )). Theorem (Castro, Cîndea, Münch) Assume that q T (, ) (, T ) is a finite union of connected open sets and satisfies the following hypotheses: Any characteristic line starting at a point x (, ) at time t = and following the optical geometric laws when reflecting at the boundary Σ T must meet q T. Then, there exists C > such that the following estimate holds : «ϕ(, ), ϕ t (, )) H C ϕ L (q T ) + Lϕ L (,T ;H, ϕ Φ. () (,))

13 Sketch of the Proof : step Step : Let ϕ be a smooth solution of the wave eq. ϕ can be extended in a unique way to a function, still denoted by ϕ, in (x, t) (, ) R satisfying Lϕ = and the boundary conditions ϕ(, t) = ϕ(, t) = for all t R. For such extension the following holds: For each t R, x (, ) and δ > Z t+δ ϕ x (, s) ds ZZ t δ δ U (x,t+x) δ Z t+δ ϕ x (, s) ds ZZ t δ δ U (x,t x) δ ϕ x (y, s) + ϕ t (y, s) dy ds, ϕ x (y, s) + ϕ t (y, s) dy ds, where U(x,t) δ is a neighborhood of (x, t) of the form. U(x,t) δ = {(y, s) such that x y + t s < δ}

14 Step (Picture) Z t+δ ϕ x (, s) ds t δ ZZ ϕ x (y, s) + ϕ t (y, s) dy ds δ U δ (x,t+x) Z t+δ ϕ x (, s) ds t δ ZZ ϕ x (y, s) + ϕ t (y, s) dy ds δ U δ (x,t x) Dependent domains q T included Arnaud in Münch Q T.

15 Proof : step Step : Wave equation is symmetric with respect to the time and space variables. The D Alembert formulae can be used changing the time and space role, i.e. Z t+x ϕ x (, s) ds = ϕ(x, t), (x, t) (, ) R, () t x where we have taken into account the boundary condition ϕ(, t) =. Consider now x = x t in () and differentiate with respect to time. Then, ϕ x (, t x ) = ϕ x (x t, t) + ϕ t (x t, t), that written in the original variables (x, t) gives, ϕ x (, t x) = ϕ x (x, t) ϕ t (x, t), or equivalently ϕ x (, t) = ϕ x (x, t + x) ϕ t (x, t + x), t R, x (, ). (3) Integrating the square of (3) in (y, s) U(x,t+x) δ with the parametrization y = x + u v, s = t + u + v, u, v < δ, we obtain Z δ/ δ/ Z δ/ δ/ ϕ x (, t + v/ ZZ ) dudv = ϕ x (y, s) ϕ t (y, s) dyds. U x,t+x δ Therefore, with the change s = t + v/ in the first integral leads the result.

16 Proof : step Step : Wave equation is symmetric with respect to the time and space variables. The D Alembert formulae can be used changing the time and space role, i.e. Z t+x ϕ x (, s) ds = ϕ(x, t), (x, t) (, ) R, () t x where we have taken into account the boundary condition ϕ(, t) =. Consider now x = x t in () and differentiate with respect to time. Then, ϕ x (, t x ) = ϕ x (x t, t) + ϕ t (x t, t), that written in the original variables (x, t) gives, ϕ x (, t x) = ϕ x (x, t) ϕ t (x, t), or equivalently ϕ x (, t) = ϕ x (x, t + x) ϕ t (x, t + x), t R, x (, ). (3) Integrating the square of (3) in (y, s) U(x,t+x) δ with the parametrization y = x + u v, s = t + u + v, u, v < δ, we obtain Z δ/ δ/ Z δ/ δ/ ϕ x (, t + v/ ZZ ) dudv = ϕ x (y, s) ϕ t (y, s) dyds. U x,t+x δ Therefore, with the change s = t + v/ in the first integral leads the result.

17 Proof: Step Set W = {ϕ : ϕ Φ such that Lϕ = } Φ. We show that for some constant C >, «ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕx L, (4) (q T ) for any ϕ W and initial data in V. We may assume that ϕ is smooth since the general case can be obtained by a usual density argument. We also assume that ϕ is extended to (x, t) (, ) R by assuming that it satisfies the wave equation and boundary conditions in this region. This extension is unique and -periodic in time. The region q T is also extended to q T to take advantage of the time periodicity of the solution ϕ. We define, q T = [ {(x, t) such that (x, t + k) q T }. k Z

18 Proof: Step Set W = {ϕ : ϕ Φ such that Lϕ = } Φ. We show that for some constant C >, «ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕx L, (4) (q T ) for any ϕ W and initial data in V. We may assume that ϕ is smooth since the general case can be obtained by a usual density argument. We also assume that ϕ is extended to (x, t) (, ) R by assuming that it satisfies the wave equation and boundary conditions in this region. This extension is unique and -periodic in time. The region q T is also extended to q T to take advantage of the time periodicity of the solution ϕ. We define, q T = [ {(x, t) such that (x, t + k) q T }. k Z

19 Proof : Step (Picture)

20 Proof: Step q T = [ {(x, t) such that (x, t + k) q T }. k Z The key point now is to observe that the hypotheses on q T, namely the fact that "any characteristic line starting at point (x, ) and following the optical geometric laws when reflecting at the boundary must meet q T ", "for any point (, t) with t [, ] there exists one characteristic line (either (x, t + x) with x (, ) or (x, t x)) that meets q T ".

21 Proof: Step Thus, given t [, ] we can apply STEP with (x, t + x) q T or with (x, t x) q T with δ sufficiently small so that U δ (x,t+x) q T or U δ (x,t x) q T. In particular we see that for any t [, ] there exists δ t > and C t > such that Z t+δt ϕ x (, s) ds C t t δ t ZZ qt ϕ t + ϕ x dxdt. By compacity, there exists a finite number of times t,..., t n such that i=,..,n (t i δ ti, t i + δ ti ) covers the whole interval [, ] and therefore, by adding the corresponding inequalities, there exists C > such that Z ϕ x (, s) ds C ϕ t ZZ qt + ϕ x dxdt. (5) The fact that we can replace q T by q T is due to the -periodicity of ϕ in time. Finally, the result is a consequence of the well-known boundary observability inequality Z ϕ(, ), ϕ t (, )) V C ϕ x (, s) ds.

22 Proof: Step Thus, given t [, ] we can apply STEP with (x, t + x) q T or with (x, t x) q T with δ sufficiently small so that U δ (x,t+x) q T or U δ (x,t x) q T. In particular we see that for any t [, ] there exists δ t > and C t > such that Z t+δt ϕ x (, s) ds C t t δ t ZZ qt ϕ t + ϕ x dxdt. By compacity, there exists a finite number of times t,..., t n such that i=,..,n (t i δ ti, t i + δ ti ) covers the whole interval [, ] and therefore, by adding the corresponding inequalities, there exists C > such that Z ϕ x (, s) ds C ϕ t ZZ qt + ϕ x dxdt. (5) The fact that we can replace q T by q T is due to the -periodicity of ϕ in time. Finally, the result is a consequence of the well-known boundary observability inequality Z ϕ(, ), ϕ t (, )) V C ϕ x (, s) ds.

23 Proof: Step Thus, given t [, ] we can apply STEP with (x, t + x) q T or with (x, t x) q T with δ sufficiently small so that U δ (x,t+x) q T or U δ (x,t x) q T. In particular we see that for any t [, ] there exists δ t > and C t > such that Z t+δt ϕ x (, s) ds C t t δ t ZZ qt ϕ t + ϕ x dxdt. By compacity, there exists a finite number of times t,..., t n such that i=,..,n (t i δ ti, t i + δ ti ) covers the whole interval [, ] and therefore, by adding the corresponding inequalities, there exists C > such that Z ϕ x (, s) ds C ϕ t ZZ qt + ϕ x dxdt. (5) The fact that we can replace q T by q T is due to the -periodicity of ϕ in time. Finally, the result is a consequence of the well-known boundary observability inequality Z ϕ(, ), ϕ t (, )) V C ϕ x (, s) ds.

24 Proof: Step 3 Step 3. We show that we can substitute ϕ x by ϕ in the right hand side of (4), i.e. «ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕ L, (6) (q T ) for any ϕ W and initial data in V. In fact, this requires to extend slightly the observation zone q T. Instead, we observe that if q T satisfies the hypotheses in Proposition then there exists a smaller open subset q T q T that still satisfies the same hypotheses and such that the closure of q T is included in q T. Thus, (4) must hold as well for q T. Let us introduce now a function η which satisfies the following hypotheses: η C ((, ) (, T )), supp(η) q T, η t L + η x /η L C in q T η > η > in q T, with η > constant. As q T is a finite union of connected open sets, the function η can be easily obtained by convolution of the characteristic function of q T with a positive mollifier.

25 Proof: Step 3 Step 3. We show that we can substitute ϕ x by ϕ in the right hand side of (4), i.e. «ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕ L, (6) (q T ) for any ϕ W and initial data in V. In fact, this requires to extend slightly the observation zone q T. Instead, we observe that if q T satisfies the hypotheses in Proposition then there exists a smaller open subset q T q T that still satisfies the same hypotheses and such that the closure of q T is included in q T. Thus, (4) must hold as well for q T. Let us introduce now a function η which satisfies the following hypotheses: η C ((, ) (, T )), supp(η) q T, η t L + η x /η L C in q T η > η > in q T, with η > constant. As q T is a finite union of connected open sets, the function η can be easily obtained by convolution of the characteristic function of q T with a positive mollifier.

26 Proof: Step 3 Multiplying the equation of ϕ by ηϕ and integrating by parts we easily obtain ZZ q T η ϕ x dx dt = ZZ ZZ η ϕ t dx dt + (η t ϕϕ t η x ϕϕ x ) dx dt q T q T ZZ η ϕ t dxdt + η ZZ t L (q T ) ( ϕ + ϕ t ) dx dt q T q T + ZZ ( η x q T η ϕ + ηϕ x ) dx dt. Therefore, ZZ ZZ η ϕ x dx dt C ( ϕ t + ϕ ) dx dt, q T q T for some constant C >, and we obtain ϕ x L ( q T ) C This combined with (4) for q T provides (6). ZZ ZZ η ϕ x dx dt C C ( ϕ t + ϕ ) dx dt. q T q T

27 Proof : Step 4 Step 4. Here we prove that we can remove the second term in the right hand side of (6), i.e. ϕ(, ), ϕ t (, )) V C ϕ t L (q T ), (7) for any ϕ W and initial data in V. We substitute the inequality (energy estimate) ϕ L (q T ) T ϕ(, ), ϕ t (, )) H. in (6) ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕ(, ), ϕ t (, )) H «. Inequality (7) is finally obtained by contradiction. Assume that it is not true. Then, there exists a sequence (ϕ k (, ), ϕ k t (, ))) k> V such that ϕ k (, ), ϕ k t (, )) V =, k >, ϕk t L, as k. (q T ) There exists a subsequence such that (ϕ k (, ), ϕ k t (, )) (ϕ (, ), ϕ t (, )) weakly in V and strongly in H. Passing to the limit in the equation we see that the solution associated to (ϕ (, ), ϕ t (, )), ϕ must vanish at q T and therefore, by (6), ϕ =.

28 Proof : Step 4 Step 4. Here we prove that we can remove the second term in the right hand side of (6), i.e. ϕ(, ), ϕ t (, )) V C ϕ t L (q T ), (7) for any ϕ W and initial data in V. We substitute the inequality (energy estimate) ϕ L (q T ) T ϕ(, ), ϕ t (, )) H. in (6) ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕ(, ), ϕ t (, )) H «. Inequality (7) is finally obtained by contradiction. Assume that it is not true. Then, there exists a sequence (ϕ k (, ), ϕ k t (, ))) k> V such that ϕ k (, ), ϕ k t (, )) V =, k >, ϕk t L, as k. (q T ) There exists a subsequence such that (ϕ k (, ), ϕ k t (, )) (ϕ (, ), ϕ t (, )) weakly in V and strongly in H. Passing to the limit in the equation we see that the solution associated to (ϕ (, ), ϕ t (, )), ϕ must vanish at q T and therefore, by (6), ϕ =.

29 Proof : Step 4 Step 4. Here we prove that we can remove the second term in the right hand side of (6), i.e. ϕ(, ), ϕ t (, )) V C ϕ t L (q T ), (7) for any ϕ W and initial data in V. We substitute the inequality (energy estimate) ϕ L (q T ) T ϕ(, ), ϕ t (, )) H. in (6) ϕ(, ), ϕ t (, )) V C ϕ t L (q T ) + ϕ(, ), ϕ t (, )) H «. Inequality (7) is finally obtained by contradiction. Assume that it is not true. Then, there exists a sequence (ϕ k (, ), ϕ k t (, ))) k> V such that ϕ k (, ), ϕ k t (, )) V =, k >, ϕk t L, as k. (q T ) There exists a subsequence such that (ϕ k (, ), ϕ k t (, )) (ϕ (, ), ϕ t (, )) weakly in V and strongly in H. Passing to the limit in the equation we see that the solution associated to (ϕ (, ), ϕ t (, )), ϕ must vanish at q T and therefore, by (6), ϕ =.

30 Proof Step 5. We now write (7) with respect to the weaker norm. In particular, we obtain ϕ(, ), ϕ t (, )) H C ϕ L (q T ), (8) for any ϕ Φ with Lϕ =. Let η Φ be defined by η(x, t) = η(x, ) + R t ϕ(x, s) ds, for all (x, t) Q T such that (η(, ), η t (, )) = ( ϕ t (, ), ϕ(, )) V where designates the Dirichlet Laplacian in (, ). Then Lη = in Q T. Then, inequality (7) on η and the fact that is an isomorphism from H (, ) to L (, ), provide (ϕ(, ), ϕ t (, ), ) H = ( ϕ t (, ), ϕ(, )) V = (η(, ), η t (, )) V C η t L (q T ) = C ϕ L (q T ).

31 Proof Step 5. We now write (7) with respect to the weaker norm. In particular, we obtain ϕ(, ), ϕ t (, )) H C ϕ L (q T ), (8) for any ϕ Φ with Lϕ =. Let η Φ be defined by η(x, t) = η(x, ) + R t ϕ(x, s) ds, for all (x, t) Q T such that (η(, ), η t (, )) = ( ϕ t (, ), ϕ(, )) V where designates the Dirichlet Laplacian in (, ). Then Lη = in Q T. Then, inequality (7) on η and the fact that is an isomorphism from H (, ) to L (, ), provide (ϕ(, ), ϕ t (, ), ) H = ( ϕ t (, ), ϕ(, )) V = (η(, ), η t (, )) V C η t L (q T ) = C ϕ L (q T ).

32 Proof Step 6. Here we finally obtain (). Given ϕ Φ we can decompose it as ϕ = ϕ + ϕ where ϕ, ϕ Φ solve j Lϕ = Lϕ, ϕ (, ) = (ϕ ) t (, ) = j Lϕ =, ϕ (, ) = ϕ(, ), (ϕ ) t (, ) = ϕ t (, ). From Duhamel s principle, we can write Z t ϕ (, t) = ψ(, t s, s)ds where ψ(x, t, s) solves, for each value of the parameter s (, t), j Lψ(,, s) =, ψ(,, s) =, ψ t (,, s) = Lϕ(, s). Therefore, Z T Z T ϕ L (q T ) ψ(,, s) L (q T ) ds C ψ(,, s), ψ t (,, s)) H ds C Lϕ L (,T ;H (,)) (9) Combining (9) and estimate (8) for ϕ we obtain ϕ(, ), ϕ t (, )) H = ϕ (, ), (ϕ ) t (, )) H C ϕ L (q T ) C ϕ L (q T ) + ϕ L C ϕ (q T ) L (q T ) + Lϕ L (,T ;H. )

33 Proof Step 6. Here we finally obtain (). Given ϕ Φ we can decompose it as ϕ = ϕ + ϕ where ϕ, ϕ Φ solve j Lϕ = Lϕ, ϕ (, ) = (ϕ ) t (, ) = j Lϕ =, ϕ (, ) = ϕ(, ), (ϕ ) t (, ) = ϕ t (, ). From Duhamel s principle, we can write Z t ϕ (, t) = ψ(, t s, s)ds where ψ(x, t, s) solves, for each value of the parameter s (, t), j Lψ(,, s) =, ψ(,, s) =, ψ t (,, s) = Lϕ(, s). Therefore, Z T Z T ϕ L (q T ) ψ(,, s) L (q T ) ds C ψ(,, s), ψ t (,, s)) H ds C Lϕ L (,T ;H (,)) (9) Combining (9) and estimate (8) for ϕ we obtain ϕ(, ), ϕ t (, )) H = ϕ (, ), (ϕ ) t (, )) H C ϕ L (q T ) C ϕ L (q T ) + ϕ L C ϕ (q T ) L (q T ) + Lϕ L (,T ;H. )

34 Proof Step 6. Here we finally obtain (). Given ϕ Φ we can decompose it as ϕ = ϕ + ϕ where ϕ, ϕ Φ solve j Lϕ = Lϕ, ϕ (, ) = (ϕ ) t (, ) = j Lϕ =, ϕ (, ) = ϕ(, ), (ϕ ) t (, ) = ϕ t (, ). From Duhamel s principle, we can write Z t ϕ (, t) = ψ(, t s, s)ds where ψ(x, t, s) solves, for each value of the parameter s (, t), j Lψ(,, s) =, ψ(,, s) =, ψ t (,, s) = Lϕ(, s). Therefore, Z T Z T ϕ L (q T ) ψ(,, s) L (q T ) ds C ψ(,, s), ψ t (,, s)) H ds C Lϕ L (,T ;H (,)) (9) Combining (9) and estimate (8) for ϕ we obtain ϕ(, ), ϕ t (, )) H = ϕ (, ), (ϕ ) t (, )) H C ϕ L (q T ) C ϕ L (q T ) + ϕ L C ϕ (q T ) L (q T ) + Lϕ L (,T ;H. )

35 Generalized Observability inequality: weaker hypothesis Set H = L (, ) H (, ). Let T >. Theorem (Castro, Cîndea, Münch) Assume q T (, ) (, T ) is a finite union of connected open sets and satisfies the following hypotheses: "Any characteristic line starting at the point x (, ) at time t = and following the optical geometric laws when reflecting at the boundaries x =, must meet q T ". Then, there exists C > such that the following estimate holds : «ϕ(, ), ϕ t (, )) H C ϕ L (q T ) + Lϕ L (,T ;H, ϕ Φ. () (,))

36 Remarks. The hypotheses on q T stated in the Theorem are optimal in the following sense: If there exists a subinterval ω (, ) for which all characteristics starting in ω and following the geometrical optics conditions when getting to the boundary x =,, do not meet q T, then the inequality fails to hold. This is easily seen by considering particular solutions of the wave equation which initial data supported in ω.. The proof of inequality () above does not provide an estimate on the dependence of the constant with respect to q T. 3. In the cylindrical situation, i.e. q T = (a, b) (, T ), a generalized Carleman inequality, valid for the wave equation with variable coefficients, have been obtained in Cindea, Fernandez-Cara and Munch (3) (see also Yao ). The extension of Proposition to the wave equation with variable coefficients is still open and a priori can not be obtained by the method used in this section.

37 Norm on Φ Corollary Under the hypotheses on q T, the space Φ is a Hilbert space with the scalar product, ZZ Z T (ϕ, ϕ) Φ = ϕ(x, t)ϕ(x, t) dx dt + η < Lϕ, Lϕ > H (,),H (,) dt, () q T for any fixed η >. PROOF: The seminorm associated to this inner product Φ is a norm from (). We check that Φ is closed with respect to this norm. Let us consider a convergence sequence {ϕ k } k Φ such that ϕ k ϕ in the norm Φ. From (), there exist (ϕ, ϕ ) H and f L (, T ; H (, )) such that (ϕ k (, ), ϕ k,t (, )) (ϕ, ϕ ) in H and Lϕ k f in L (, T ; H (, )). Therefore, ϕ k can be considered as a sequence of solutions of the wave equation with convergent initial data and second hand term Lϕ k f. By the continuous dependence of the solutions of the wave equation on the data, ϕ k ϕ in C([, T ]; L (, )) C ([, T ]; H (, )), where ϕ is the solution of the wave equation with initial data (ϕ, ϕ ) H and second hand term Lϕ = f L (, T ; H (, )). Therefore ϕ Φ.

38 Strong convergent (numerical approximation)" [CINDEA, FERNANDEZCARA, MUNCH, COCV3], Minimize J(y, v) := ZZ y dx dt + ZZ v dx dt. Q T q T The optimal pair (y, v) is y = L ϕ in Q T, v = ϕ ω in Q T where ϕ Φ solves the variational problem ZZ ZZ Lϕ Lϕ dx dt + ϕ ϕ dx dt, = (y, ϕ t (, )) H,H (y, ϕ(, )) L ϕ Φ. Q T q T Let ϕ h Φ h Φ solves the variational problem ZZ ZZ Lϕ h Lϕ h dx dt+ ϕ h ϕ h dx dt, = (y, ϕ ht (, )) H,H (y, ϕ h (, )) L ϕ h Φ h. Q T q T ϕ h ϕ in Φ as h = y h := Lϕ h y in L (Q T ) and v h := ϕ h qt v in L (q T )

39 Strong convergent (numerical approximation)" [CINDEA, FERNANDEZCARA, MUNCH, COCV3], Minimize J(y, v) := ZZ y dx dt + ZZ v dx dt. Q T q T The optimal pair (y, v) is y = L ϕ in Q T, v = ϕ ω in Q T where ϕ Φ solves the variational problem ZZ ZZ Lϕ Lϕ dx dt + ϕ ϕ dx dt, = (y, ϕ t (, )) H,H (y, ϕ(, )) L ϕ Φ. Q T q T Let ϕ h Φ h Φ solves the variational problem ZZ ZZ Lϕ h Lϕ h dx dt+ ϕ h ϕ h dx dt, = (y, ϕ ht (, )) H,H (y, ϕ h (, )) L ϕ h Φ h. Q T q T ϕ h ϕ in Φ as h = y h := Lϕ h y in L (Q T ) and v h := ϕ h qt v in L (q T )

40 Control of minimal L -norm: a mixed formulation [CINDEA, MUNCH, CALCOLO4], min (ϕ,ϕ ) H J (ϕ, ϕ ) = ZZ Z ϕ dx dt+ < ϕ, y > H (,),H q (,) ϕ y dx. T where Lϕ = in Q T ; ϕ = on Σ T, (ϕ, ϕ t )(, ) = (ϕ, ϕ ) and Z < ϕ, y > H (,),H (,)= x (( ) ϕ )(x) x y (x) dx where is the Dirichlet Laplacian in (, ). Since the variable ϕ is completely and uniquely determined by (ϕ, ϕ ), the idea of the reformulation is to keep ϕ as variable and consider the following extremal problem: min Ĵ (ϕ) = ZZ Z ϕ dx dt+ < ϕ t (, ), y > ϕ W H (,),H q (,) ϕ(, ) y dx, T n o W = ϕ : ϕ L (q T ), ϕ = on Σ T, Lϕ = L (, T ; H (, )). () From (), the property ϕ W implies that (ϕ(, ), ϕ t (, )) H, so that the functional Ĵ is well-defined over W.

41 Control of minimal L -norm: a mixed formulation [CINDEA, MUNCH, CALCOLO4], min (ϕ,ϕ ) H J (ϕ, ϕ ) = ZZ Z ϕ dx dt+ < ϕ, y > H (,),H q (,) ϕ y dx. T where Lϕ = in Q T ; ϕ = on Σ T, (ϕ, ϕ t )(, ) = (ϕ, ϕ ) and Z < ϕ, y > H (,),H (,)= x (( ) ϕ )(x) x y (x) dx where is the Dirichlet Laplacian in (, ). Since the variable ϕ is completely and uniquely determined by (ϕ, ϕ ), the idea of the reformulation is to keep ϕ as variable and consider the following extremal problem: min Ĵ (ϕ) = ZZ Z ϕ dx dt+ < ϕ t (, ), y > ϕ W H (,),H q (,) ϕ(, ) y dx, T n o W = ϕ : ϕ L (q T ), ϕ = on Σ T, Lϕ = L (, T ; H (, )). () From (), the property ϕ W implies that (ϕ(, ), ϕ t (, )) H, so that the functional Ĵ is well-defined over W.

42 Control of minimal L -norm: a mixed formulation The main variable is now ϕ submitted to the constraint equality Lϕ = as an L (, T ; H (, )) function. This constraint is addressed introducing a Lagrangian multiplier λ L (, T ; H (Ω)): We consider the following problem : find (ϕ, λ) Φ L (, T ; H (, )) solution of ( a r (ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ (3) b(ϕ, λ) =, λ L (, T ; H (, )), where (r - augmentation parameter) ZZ Z T a r : Φ Φ R, a r (ϕ, ϕ) = ϕ ϕ dx dt + r < Lϕ, Lϕ > H,H dt q T b : Φ L (, T ; H (, )) R, b(ϕ, Z T λ) = < Lϕ, λ > H (,),H (,) dt ZZ = x ( (Lϕ)) x λ dx dt Q T Z l : Φ R, l(ϕ) = < ϕ t (, ), y > H (,),H (,) + ϕ(, ) y dx.

43 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

44 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

45 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

46 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

47 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

48 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

49 Well-posedness of the mixed formulation Theorem The mixed formulation (3) is well-posed. The unique solution (ϕ, λ) Φ L (, T ; H (, )) is the unique saddle-point of the Lagrangian L : Φ L (, T ; H (, )) R defined by L(ϕ, λ) = ar (ϕ, ϕ) + b(ϕ, λ) l(ϕ). 3 The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (, T ; H (, )) is the state of the controlled wave equation in the weak sense (associated to the control ϕ qt ). The well-posedness of the mixed formulation is a consequence of two properties [FORTIN-BREZZI 9] : a is coercive on Ker(b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (, T ; H (, ))}. b satisfies the usual "inf-sup" condition over Φ L (, T ; H (, )): there exists δ > such that b(ϕ, λ) inf sup δ. (4) λ L (,T ;H (,)) ϕ Φ ϕ Φ λ L (,T,H (,))

50 Inf-Sup condition For any λ L (H ), we define the (unique) element ϕ such that Lϕ = λ Q T, ϕ (, ) = ϕ,t (, ) = Ω, ϕ = Σ T From the direct inequality, ϕ L (Q T ) C Ω,T λ L (,T ;H (,)) C Ω,T λ L (,T ;H (,)) we get that ϕ Φ. In particular, b(ϕ, λ ) = λ and L (,T ;H (,)) sup ϕ Φ b(ϕ, λ ) ϕ Φ λ L (Q T ) b(ϕ, λ ) ϕ Φ λ L (Q T ) = λ L (,T ;H (,)) ϕ L(qT ) + η λ L(,T ;H (,)) «λ L(,T ;H (,)). Combining the above two inequalities, we obtain sup ϕ Φ b(ϕ, λ ) ϕ Φ λ L (,T ;H (,)) and, hence, (4) holds with δ = CΩ,T + η. q CΩ,T + η

51 Inf-Sup condition For any λ L (H ), we define the (unique) element ϕ such that Lϕ = λ Q T, ϕ (, ) = ϕ,t (, ) = Ω, ϕ = Σ T From the direct inequality, ϕ L (Q T ) C Ω,T λ L (,T ;H (,)) C Ω,T λ L (,T ;H (,)) we get that ϕ Φ. In particular, b(ϕ, λ ) = λ and L (,T ;H (,)) sup ϕ Φ b(ϕ, λ ) ϕ Φ λ L (Q T ) b(ϕ, λ ) ϕ Φ λ L (Q T ) = λ L (,T ;H (,)) ϕ L(qT ) + η λ L(,T ;H (,)) «λ L(,T ;H (,)). Combining the above two inequalities, we obtain sup ϕ Φ b(ϕ, λ ) ϕ Φ λ L (,T ;H (,)) and, hence, (4) holds with δ = CΩ,T + η. q CΩ,T + η

52 Dual... of the dual problem ("UZAWA" type algorithm) Lemma Let A r be the linear operator from L (H ) into L (H ) defined by A r λ := (Lϕ), λ L (H ) where ϕ Φ solves ar (ϕ, ϕ) = b(ϕ, λ), ϕ Φ. For any r >, the operator A r is a strongly elliptic, symmetric isomorphism from L (H ) into L (H ). Theorem sup inf Lr (ϕ, λ) = inf J (λ) + L r (ϕ, ) λ L (H ) ϕ Φ λ L (,T,H (,)) where ϕ Φ solves a r (ϕ, ϕ) = l(ϕ), ϕ Φ and J : L (H ) R defined by J (λ) = ZZ A r λ(x, t)λ(x, t) dx dt b(ϕ, λ) Q T

53 Dual... of the dual problem ("UZAWA" type algorithm) Lemma Let A r be the linear operator from L (H ) into L (H ) defined by A r λ := (Lϕ), λ L (H ) where ϕ Φ solves ar (ϕ, ϕ) = b(ϕ, λ), ϕ Φ. For any r >, the operator A r is a strongly elliptic, symmetric isomorphism from L (H ) into L (H ). Theorem sup inf Lr (ϕ, λ) = inf J (λ) + L r (ϕ, ) λ L (H ) ϕ Φ λ L (,T,H (,)) where ϕ Φ solves a r (ϕ, ϕ) = l(ϕ), ϕ Φ and J : L (H ) R defined by J (λ) = ZZ A r λ(x, t)λ(x, t) dx dt b(ϕ, λ) Q T

54 Conformal approximation Let then Φ h and M h be two finite dimensional spaces parametrized by the variable h such that Φ h Φ, M h L (, T ; H (, )), h >. Then, we can introduce the following approximated problems : find (ϕ h, λ h ) Φ h M h solution of ( a r (ϕ h, ϕ h ) + b(ϕ h, λ h ) = l(ϕ h ), ϕ h Φ h (5) b(ϕ h, λ h ) =, λ h M h. The well-posedness is again a consequence of two properties : the coercivity of the bilinear form a r on the subset N h (b) = {ϕ h Φ h ; b(ϕ h, λ h ) = λ h M h }. From the relation a r (ϕ, ϕ) r η ϕ Φ, ϕ Φ the form a r is coercive on the full space Φ, and so a fortiori on N h (b) Φ h Φ. The second property is a discrete inf-sup condition : there exists δ h > such that inf λ h M h b(ϕ h, λ h ) sup δ h. (6) ϕ h Φ h ϕ h Φh λ h Mh For any fixed h, the spaces M h and Φ h are of finite dimension so that the infimum and supremum in (6) are reached: moreover, from the property of the bilinear form a r, δ h is strictly positive. Consequently, for any fixed h >, there exists a unique couple (ϕ h, λ h ) solution of (5).

55 Conformal approximation Let then Φ h and M h be two finite dimensional spaces parametrized by the variable h such that Φ h Φ, M h L (, T ; H (, )), h >. Then, we can introduce the following approximated problems : find (ϕ h, λ h ) Φ h M h solution of ( a r (ϕ h, ϕ h ) + b(ϕ h, λ h ) = l(ϕ h ), ϕ h Φ h (5) b(ϕ h, λ h ) =, λ h M h. The well-posedness is again a consequence of two properties : the coercivity of the bilinear form a r on the subset N h (b) = {ϕ h Φ h ; b(ϕ h, λ h ) = λ h M h }. From the relation a r (ϕ, ϕ) r η ϕ Φ, ϕ Φ the form a r is coercive on the full space Φ, and so a fortiori on N h (b) Φ h Φ. The second property is a discrete inf-sup condition : there exists δ h > such that inf λ h M h b(ϕ h, λ h ) sup δ h. (6) ϕ h Φ h ϕ h Φh λ h Mh For any fixed h, the spaces M h and Φ h are of finite dimension so that the infimum and supremum in (6) are reached: moreover, from the property of the bilinear form a r, δ h is strictly positive. Consequently, for any fixed h >, there exists a unique couple (ϕ h, λ h ) solution of (5).

56 Discretization The space Φ h must be chosen such that Lϕ h L (, T, H (, )) for any ϕ h Φ h. This is guaranteed for instance as soon as ϕ h possesses second-order derivatives in L loc (Q T ). A conformal approximation based on standard triangulation of Q T is obtained with spaces of functions continuously differentiable with respect to both x and t. We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. We note h := max{diam(k ), K T h }. We introduce the space Φ h as follows: Φ h = {ϕ h Φ h C (Q T ) : ϕ h K P(K ) K T h, ϕ h = on Σ T } where P(K ) denotes an appropriate space of polynomial functions in x and t. We consider for P(K ) the reduced Hsieh-Clough-Tocher C -element ( Composite finite element and involves as degrees of freedom the values of ϕ h, ϕ h,x, ϕ h,t on the vertices of each triangle K ). We also define the finite dimensional space M h = {λ h C (Q T ), λ h K P (K ) K T h, λ h = on Σ T } For any h >, we have Φ h Φ and M h L (, T ; H (, )).

57 Discretization The space Φ h must be chosen such that Lϕ h L (, T, H (, )) for any ϕ h Φ h. This is guaranteed for instance as soon as ϕ h possesses second-order derivatives in L loc (Q T ). A conformal approximation based on standard triangulation of Q T is obtained with spaces of functions continuously differentiable with respect to both x and t. We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. We note h := max{diam(k ), K T h }. We introduce the space Φ h as follows: Φ h = {ϕ h Φ h C (Q T ) : ϕ h K P(K ) K T h, ϕ h = on Σ T } where P(K ) denotes an appropriate space of polynomial functions in x and t. We consider for P(K ) the reduced Hsieh-Clough-Tocher C -element ( Composite finite element and involves as degrees of freedom the values of ϕ h, ϕ h,x, ϕ h,t on the vertices of each triangle K ). We also define the finite dimensional space M h = {λ h C (Q T ), λ h K P (K ) K T h, λ h = on Σ T } For any h >, we have Φ h Φ and M h L (, T ; H (, )).

58 Discretization The space Φ h must be chosen such that Lϕ h L (, T, H (, )) for any ϕ h Φ h. This is guaranteed for instance as soon as ϕ h possesses second-order derivatives in L loc (Q T ). A conformal approximation based on standard triangulation of Q T is obtained with spaces of functions continuously differentiable with respect to both x and t. We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. We note h := max{diam(k ), K T h }. We introduce the space Φ h as follows: Φ h = {ϕ h Φ h C (Q T ) : ϕ h K P(K ) K T h, ϕ h = on Σ T } where P(K ) denotes an appropriate space of polynomial functions in x and t. We consider for P(K ) the reduced Hsieh-Clough-Tocher C -element ( Composite finite element and involves as degrees of freedom the values of ϕ h, ϕ h,x, ϕ h,t on the vertices of each triangle K ). We also define the finite dimensional space M h = {λ h C (Q T ), λ h K P (K ) K T h, λ h = on Σ T } For any h >, we have Φ h Φ and M h L (, T ; H (, )).

59 Change of the norm L (H ) over the discrete space Φ h [Bramble, Gunzburger] Remark that if there exist two constants C > and α > such that ψ h L (Q T ) C h α ψ h L (,T ;H (,)), ψ h Φ h (7) then a similar inequality it holds for weaker norms. More precisely, we have ϕ h L (,T ;H (,)) C h α ϕ h L (Q T ), ϕ h Φ h. (8) Indeed, to obtain (8) it suffices to take ψ h (, t) = ( ) ϕ h (, t) in (7). That gives Z T Z ( ) ϕ h (, t) T dt C h α ( ) L ϕ h,x (, t) dt. (,) L (,) Since is a self-adjoint positive operator and ϕ h Φ h H (Q T ) we can integrate by parts in both hand-sides of the above inequality and hence we deduce estimate (8). C and α does not depend on T.

60 Change of the norm L (H ) over the discrete space Φ h [Bramble, Gunzburger] Remark that if there exist two constants C > and α > such that ψ h L (Q T ) C h α ψ h L (,T ;H (,)), ψ h Φ h (7) then a similar inequality it holds for weaker norms. More precisely, we have ϕ h L (,T ;H (,)) C h α ϕ h L (Q T ), ϕ h Φ h. (8) Indeed, to obtain (8) it suffices to take ψ h (, t) = ( ) ϕ h (, t) in (7). That gives Z T Z ( ) ϕ h (, t) T dt C h α ( ) L ϕ h,x (, t) dt. (,) L (,) Since is a self-adjoint positive operator and ϕ h Φ h H (Q T ) we can integrate by parts in both hand-sides of the above inequality and hence we deduce estimate (8). C and α does not depend on T.

61 Change of the norm L (H ) over the discrete space Φ h We consider, for any fixed h >, the following equivalent definitions of the form a r,h and b h over the finite dimensional spaces Φ h Φ h and Φ h M h respectively : a r,h : Φ h Φ h R, a r,h (ϕ h, ϕ h ) = a(ϕ h, ϕ h ) + rc h α ZZQ T Lϕ h Lϕ h dxdt b h : Φ h M h R, b h (ϕ h, λ h ) = C h α ZZQ T Lϕ h λ h dxdt. Let n h = dim Φ h, m h = dim M h and let the real matrices A r,h R n h,n h defined by a r,h (ϕ h, ϕ h ) = A r,h {ϕ h }, {ϕ h } R n h,r n h, ϕ h, ϕ h Φ h, where {ϕ h } R nh, denotes the vector associated to ϕ h and, R n h,r n h the usual scalar product over R n h. The problem reads: find {ϕ h } R nh, and {λ h } R mh, such that Ar,h Bh T «««{ϕh } Lh B h {λ h } R n h +m h,n h +m h R n h +m h, = R n h +m h,. The matrix of order m h + n h is symmetric but not positive definite. We use exact integration methods and the LU decomposition method. From ϕ h, an approximation v h of the control v is given by v h = ϕ h qt L (Q T ).

62 Change of the norm L (H ) over the discrete space Φ h We consider, for any fixed h >, the following equivalent definitions of the form a r,h and b h over the finite dimensional spaces Φ h Φ h and Φ h M h respectively : a r,h : Φ h Φ h R, a r,h (ϕ h, ϕ h ) = a(ϕ h, ϕ h ) + rc h α ZZQ T Lϕ h Lϕ h dxdt b h : Φ h M h R, b h (ϕ h, λ h ) = C h α ZZQ T Lϕ h λ h dxdt. Let n h = dim Φ h, m h = dim M h and let the real matrices A r,h R n h,n h defined by a r,h (ϕ h, ϕ h ) = A r,h {ϕ h }, {ϕ h } R n h,r n h, ϕ h, ϕ h Φ h, where {ϕ h } R nh, denotes the vector associated to ϕ h and, R n h,r n h the usual scalar product over R n h. The problem reads: find {ϕ h } R nh, and {λ h } R mh, such that Ar,h Bh T «««{ϕh } Lh B h {λ h } R n h +m h,n h +m h R n h +m h, = R n h +m h,. The matrix of order m h + n h is symmetric but not positive definite. We use exact integration methods and the LU decomposition method. From ϕ h, an approximation v h of the control v is given by v h = ϕ h qt L (Q T ).

63 Change of the norm : computation of C and α In order to approximate the values of the constants C, α appearing in (7)-(8) we consider the following problem : find α > and C > such that ϕ h L sup (,T ;H (,)) ϕ h Φ h ϕ h L (Q T ), h >. C hα Since dim Φ h <, the supremum is, for any fixed h >, the solution of the following eigenvalue problem : h >, j ff γ h = sup γ : K h {ψ h } = γj h {ψ h }, {ψ h } R m h \ {} We determine C and α such that C h α = γ h. We obtain C.48, α.993. We check that the constant γ h (and so C and α) does not depend on T nor on the controllability domain.

64 Change of the norm : computation of C and α In order to approximate the values of the constants C, α appearing in (7)-(8) we consider the following problem : find α > and C > such that ϕ h L sup (,T ;H (,)) ϕ h Φ h ϕ h L (Q T ), h >. C hα Since dim Φ h <, the supremum is, for any fixed h >, the solution of the following eigenvalue problem : h >, j ff γ h = sup γ : K h {ψ h } = γj h {ψ h }, {ψ h } R m h \ {} We determine C and α such that C h α = γ h. We obtain C.48, α.993. We check that the constant γ h (and so C and α) does not depend on T nor on the controllability domain.

65 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

66 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

67 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

68 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

69 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

70 The discrete inf-sup test In order to solve the mixed formulation (5), we first test numerically the discrete inf-sup condition (6). Taking η = r > so that a r,h (ϕ, ϕ) = (ϕ, ϕ) Φ for all ϕ, ϕ Φ, it is readily seen that the discrete inf-sup constant satisfies j ff δ δ h := inf : Bh A r,h BT h {λ h} = δ J h {λ h }, {λ h } R m h \ {}. The matrix B h A r,h BT h is symmetric, positive definite so that δ h > for any h >. 4 δ h h qt with T =.; + qt with T = ; qt with T = ; qt with T =.; qt with T =. qt 3 with T =.. = (Φ h, M h ) passes the discrete inf-sup test! Figure: δ h vs. h for various control domains q T, T > and r =.

71 Geometries γ T (t) γ T (t) γ T (t) γ 3 T (t) q T q T q T t t t t q 3 T QT QT QT QT.5 x.5 x.5 x.5 x Figure: Time dependent domains qt i, i {,,, 3}.

72 Triangular meshes Figure: Meshes associated with the domains qt i =. : i =,,, 3.

73 Numerical illustration T =.; y (x) = sin(πx); y = ; q T = q Mesh h v h L (qt ) Lϕ h L (,T ;H (,)) v v h L (qt ) y λ h L (QT ) κ Table: Norms vs. h for r =. r = : v v h L (qt ) O(h.3 ), Lϕ h L (,T ;H (,)) O(h.3 ), y λ h L (QT ) O(h.94 ) r = 3 : v v h L (qt ) O(h.9 ), Lϕ h L (QT ) O(h.4 ), y λ h L (QT ) O(h. ).

74 Convergence as h T =.; y (x) = sin(πx); y = ; q T = q 4 Figure: r = ; q T = q. ; Norms v v h L (q T ) ( ) and y λ h L (Q T ) ( ) vs. h. h

75 Numerical illustration T =.; y (x) = e 5(x.8) ; y = ; q T = q. 5 5 ϕ h 5 λ h.5 t.5 x.5 t.5 x Figure: r = ; q T = q. : Functions ϕ h (Left) and λ h (Right) over Q T. v v h L (q T ) e5.85 h.4, Lϕ h L (Q T ) e7.96 h.3, y λ h L (Q T ) e.58 h.6

76 Numerical illustration T =.; y (x) = x θ (,θ)(x)+ x θ (θ,)(x), y (x) =, θ (, ) q T = q. 5 ϕ h 5 λ h.5 t.5 x.5 t.5 x Figure: Example EX3 with θ = /3; r = ; q T = q. : Functions ϕ h (Left) and λ h (Right). v v h L (q T ) e.54 h.47, Lϕ h L (Q T ) e.9 h.54, y λ h L (Q T ) e.5 h.9.

77 Numerical illustration T =.; y (x) = e 5(x.8) ; y = ; q T = q ϕ h 5 λ h.5 t.5 x.5 t.5 x Figure: Example EX: q T = q 3. - Function ϕ h (Left) and λ h (Right) over Q T.