Hilbert Uniqueness Method and regularity

Transcription

1 Hilbert Uniqueness Method and regularity Sylvain Ervedoza 1 Joint work with Enrique Zuazua 2 1 Institut de Mathématiques de Toulouse & CNRS 2 Basque Center for Applied Mathematics Institut Henri Poincaré 29

2 Outline of the talk 1 Introduction: The Hilbert Uniqueness Method 2 An alternate HUM type method The η-weighted HUM Main result Regularity of the controlled trajectory 3 Applications Distributed controls Boundary control 4 Conclusion

3 1 Introduction: The Hilbert Uniqueness Method 2 An alternate HUM type method The η-weighted HUM Main result Regularity of the controlled trajectory 3 Applications Distributed controls Boundary control 4 Conclusion

4 An abstract control problem Let A be the generator of a group on a Hilbert space X. Consider the following model: y (t) = A y(t) + Bv(t), y() = y X, where B L(U, D(A) ) and v L 2 (, T ; U). Assumption For all v L 2 (, T ; U), solutions can be defined in the sense of transposition in C ([, T ]; X). Goal : Exact controllability Fix a time T > and y X. Can we find v L 2 (, T ; U) such that y(t ) =?

5 Hypotheses Main Assumption A : D(A) X generates a group. Consequences: one can solve the equations forward and backward the same holds for the adjoint equation.

6 Examples Wave equation in a bounded domain+ BC with distributed control u u + a(x). u + b(x)u + c(x)u = χ ω v, (t, x) R Ω, u Ω =, (u(), u()) = (u, u 1 ) H 1 (Ω) L2 (Ω), ( A = Id a(x). b c(x) ( B = χ ω ), X = H 1 (Ω) L2 (Ω), ), U = L 2 (ω). Wave equation in a bounded domain with boundary control Schrödinger equation A = i + BC, Linearized KdV A = xxx + BC, Maxwell equation,...

7 Duality Use the adjoint system to characterize the controls! For all z solution of z = A z, z(t ) = z T X, we have y(t ), z T X y, z() X = T v(t), B z(t) U dt. In particular, v is a control if and only if z T X = T v(t), B z(t) U dt + y, z X.

8 Fundamental hypotheses B : D(A) U, B L(D(A), U). Definition B is admissible if T >, K T >, T B z(t) 2 U dt K T z T 2 X, z D(A). Definition B is exactly observable at time T > if k >, T k z() 2 X B z(t) 2 U dt, z X.

9 Important remark If B is admissible and exactly observable in some time T, then the following norms are equivalent: z T X, z() X z T 2 obs,1 = T B z(t) 2 U dt T z T 2 obs,2 = η(t) B z(t) 2 U dt, for η, η α > on some interval of length T.

10 Important remark If B is admissible and exactly observable in some time T, then the following norms are equivalent: z T X, z() X z T 2 obs,1 = T B z(t) 2 U dt T z T 2 obs,2 = η(t) B z(t) 2 U dt, for η, η α > on some interval of length T.

11 Important remark If B is admissible and exactly observable in some time T, then the following norms are equivalent: z T X, z() X z T 2 obs,1 = T B z(t) 2 U dt T z T 2 obs,2 = η(t) B z(t) 2 U dt, for η, η α > on some interval of length T.

12 The Hilbert Uniqueness Method (Lions 86) Let T T. Define, for z T X, J(z T ) = 1 2 T B z(t) 2 U dt + y, z(), where z satisfies z = A z, z(t ) = z T. Observability Existence and Uniqueness of a minimizer Z T. Then V = B Z is such that the solution y of y = Ay + BV, y() = y, satisfies y(t ) =. Besides, V is the control of minimal L 2 (, T ; U)-norm.

13 On the Hilbert Uniqueness Method Recall the following facts: the Hilbert Uniqueness Method does not prove observability/controllability, but prove their equivalence. Observability/controllability properties have to be proved first: several possible methods: multipliers, see Lions, Komornik, Alabau... non-harmonic Fourier series, see Ingham, Haraux, Tucsnak... Carleman estimates, see Puel, Osses, Zhang, etc pseudo-differential calculus, see Bardos, Lebeau, Rauch, Burq, Gerard,... Resolvent estimates, see Burq, Zworski, Miller, Tucsnak...

14 A regularity problem On the regularity If y D(A), Does the function Z T computed that way belongs to D(A)? Is the controlled solution (y, v) a strong solution? Here, strong solutions means: y C 1 ([, T ]; X) C([, T ]; X 1 ), for some space X 1 smoother than X (for instance D(A)). General Answer : NO!

15 Consider the wave equation y tt y xx =, < x < 1, < t < T, y(, t) =, y(1, t) = v(t), < t < T, (y(x, ), y t (x, )) = (y (x), y 1 (x)) L 2 (, 1) H 1 (, 1). The adjoint problem is z tt z xx =, z(, t) = z(1, t) =, (z, z 1 ) H 1 (, 1) L2 (, 1), and the solutions write ( z(x, t) = 2 k 1 ẑ k cos(kπt) + ẑk 1 kπ sin(kπt) Controllability in time T = 4 : If ( y (x), y 1 (x) ) = 2 k 1 (ŷ k, ŷ k 1 ) sin(kπx), Ẑ k = ŷ k 1 4k 2 π 2, Ẑ k 1 = ŷk 4. ) sin(kπx),

16 In particular, the HUM control can be computed explicitly v(t) = Z x (1, t) ( = 1 ŷ ( 1) k k kπ 1 4 k 2 π 2 cos(kπt) ŷ ) k kπ sin(kπt). k 1 = v() = 1 ( 1) k ŷ 1 k 4 kπ! k 1 = If y H 1 (, 1), the controlled solution is not a strong solution in general because of the failure of the compatibility conditions y (1) = v() =.

17 Main question Goal To design a control method, and only one, which respects the regularity of the solutions. If y D(A), we want Z T D(A ), the controlled equation y = Ay + BV is satisfied in a strong sense. Related result - Dehman Lebeau 9 and Lebeau Nodet 9: The wave equation with distributed control B = χ ω where χ ω is smooth, and where the HUM operator is modified by a weight function η(t) vanishing at t {, T }.

18 The η-weighted HUM Main result Trajectory 1 Introduction: The Hilbert Uniqueness Method 2 An alternate HUM type method The η-weighted HUM Main result Regularity of the controlled trajectory 3 Applications Distributed controls Boundary control 4 Conclusion

19 The η-weighted HUM Main result Trajectory The modified HUM method: The η-weighted HUM Let y X, and δ > such that T 2δ T, where T is the time of observability. Define, for z T X, J(z T ) = 1 2 T η(t) B z(t) 2 U dt + y, z(), where z satisfies z = A z, z(t ) = z T and { on (, ] [T, ) η = 1 on [δ, T δ] η. Observability Existence and Uniqueness of a minimizer Z T. Then V = ηb Z is such that the solution y of y = Ay + BV, y() = y, satisfies y(t ) =. Besides, V is the control of minimal L 2 ((, T ), dt/η; U)-norm.

20 The η-weighted HUM Main result Trajectory Main result Theorem (SE Zuazua 9) Assume that admissibility and observability property hold. Also assume that η C 1 (R). If y D(A), then the minimizer Z T and the control function V = ηb Z computed by the η-weighted HUM are more regular: Z T D(A ), V H 1 (, T ; U). Moreover, there exists a constant C such that Z T D(A ) + V H 1 (,T ;U) C y D(A).

21 The η-weighted HUM Main result Trajectory Before the proof First remark that, due to the classical observability property, Z T X + v L 2 (,T ;U) C y X. Also remark that admissibility and observability properties yield T k z T D(A ) k z() D(A ) η(t) B z (t) 2 U dt K z T D(A ) It is sufficient to prove that T η(t) B Z (t) 2 U dt <. Indeed, this implies Z T D(A ) and v H 1 (, T ; U).

22 The η-weighted HUM Main result Trajectory Idea of the proof-i Write the characterization of the control V = ηb Z : = T η(t) B Z (t), B z(t) U dt + y, z() X, for all z solution of z = A z, z(t ) = z T. Then take formally z = Z = (A ) 2 Z : But T and η(t) B Z (t) 2 U dt = Ay, A Z () X T η (t) B Z (t), B Z (t) U dt. Ay, A Z () X C Ay X A Z T X T A Z T X C η(t) B Z (t) 2 U dt.

23 The η-weighted HUM Main result Trajectory Idea of the proof-ii But T η (t) B Z (t), B Z (t) U dt ( T C B Z (t) ) 1/2 ( ) 1/2 T 2 U dt B Z (t) 2 U dt Using observability, T C A Z T X Z T X C A Z T X y X η(t) B Z (t) 2 U dt C y 2 D(A).

24 The η-weighted HUM Main result Trajectory More general results Theorem (SE-Zuazua 9) Let s N and assume that η C s (R). If the initial datum y D(A s ), then the minimizer Z T and the control function V given by the η-weighted HUM satisfy Z T D((A ) s ) V H s (, T ; U). Besides, there exists a positive constant C s = C s (η, k, K T ) independent of y D(A s ) T Z T 2 D((A ) s ) + V (s) (t) 2 dt C s y 2 D(A U s ).

25 The η-weighted HUM Main result Trajectory Comments If η C (R), the η-weighted HUM map defined by { X X L Θ : 2 (, T ; dt/η, U) y (Z T, V ) defines by restriction a map on D(A s ): Θ : D(A s ) D((A ) s ) H s (, T ; U). No smoothness assumption on B, or on [A, BB ]. generalizes Dehman Lebeau 9 to the cases B = χ ω not smooth and boundary control cases, but less precise: pseudo-differential estimates on Θ for the wave equation with a distributed controlled. We have not make precise yet the regularity of the control trajectory!!!!

26 The η-weighted HUM Main result Trajectory Comments If η C (R), the η-weighted HUM map defined by { X X L Θ : 2 (, T ; dt/η, U) y (Z T, V ) defines by restriction a map on D(A s ): Θ : D(A s ) D((A ) s ) H s (, T ; U). No smoothness assumption on B, or on [A, BB ]. generalizes Dehman Lebeau 9 to the cases B = χ ω not smooth and boundary control cases, but less precise: pseudo-differential estimates on Θ for the wave equation with a distributed controlled. We have not make precise yet the regularity of the control trajectory!!!!

27 The η-weighted HUM Main result Trajectory And the regularity of the controlled trajectory? Corollary if the initial datum y D(A s ), then the controlled trajectory y with control function V given by the η-weighted HUM satisfies y(t) s k= C k ([, T ]; Z s k ), where the spaces Z j are defined by induction by Z = X, Z j = A 1 (Z j 1 + BB D((A ) j )). Important Remark In several situations, these spaces can be computed.

28 The η-weighted HUM Main result Trajectory Idea of the proof Z T D((A ) s ) and Z solves Z = A Z = Z s k= C k ([, T ]; D((A ) s k ) = V = ηb Z s k= C k ([, T ]; B D((A ) s k ) = BV = ηbb Z s k= C k ([, T ]; BB D((A ) s k ) Besides, V H s (, T ; U) since (A ) s Z T X. Hence the result by standard regularity results.

29 Distributed controls Boundary control 1 Introduction: The Hilbert Uniqueness Method 2 An alternate HUM type method The η-weighted HUM Main result Regularity of the controlled trajectory 3 Applications Distributed controls Boundary control 4 Conclusion

30 Distributed controls Boundary control Distributed Controls Let Ω R N and ω Ω. Wave equation: y y = vχ ω, y =, (y(), y ()) = (y, y 1 ) in Ω (, ), on Ω (, ), H 1(Ω) L2 (Ω), v L 2 ((, T ) ω) is the control function χ ω (x) supported in ω, strictly positive on some ω ω (no smoothness assumption)

31 Distributed controls Boundary control Geometric Control Condition Assume GCC in time T : (Bardos Lebeau Rauch 92 & Burq Gerard 96) All the rays of Geometric Optics enter in the control region ω before the time T >. { on (, ] [T, ) Take T 2δ > T, and η = 1 on [δ, T δ] η.

32 Distributed controls Boundary control The η-weighted HUM Our functional reads: T J(z, z 1 ) = 1 η(t)χ 2 2 ω(x) z(x, t) 2 dxdt ω + y, z (, ) H 1 (Ω) H 1 (Ω) y 1 (x) z(x, ) dx Ω where z is the solution of z z =, z =, (z(t ), z (T )) = (z, z 1 ) in Ω (, ), on Ω (, ), L 2 (Ω) H 1 (Ω).

33 Distributed controls Boundary control Results Theorem-I Given any (y, y 1 ) H 1 (Ω) L2 (Ω), there exists a unique minimizer (Z, Z 1 ) of J over L 2 (Ω) H 1 (Ω). The function V (x, t) = η(t)χ ω (x)z (x, t) is the control function of minimal L 2 (, T ; dt/η; L 2 (ω))-norm, defined by T v 2 L 2 (,T ;dt/η;l 2 (ω)) = ω v(x, t) 2 dx dt η(t). If (y, y 1 ) D(A s ) for some s N, (Z, Z 1 ) D((A ) s ) = D(A s 1 ) and V H s (, T ; L2 (ω)).

34 Distributed controls Boundary control Results Theorem-II If (y, y 1 ) D(A s ) for some s N and χ ω is smooth, the η-weighted HUM yields V H s (, T ; L2 (ω)) C k ([, T ]; H s k k= (ω)) and the controlled trajectory satisfies y s s k= C k ([, T ]; H s+1 k (Ω) H s k (Ω)). Yields another proof of the one in Dehman Lebeau 9.

35 Distributed controls Boundary control Boundary control Let Ω R N, Γ = Ω, Γ Γ satisifying GCC in time T. Let ξ Γ be defined on Ω, non-vanishing on Γ. We now consider the following wave equation: y y =, in Ω (, ), y = ξ Γ v, on Γ (, ), (y(), y ()) = (y, y 1 ) L 2 (Ω) H 1 (Ω). η(t) as before.

36 Distributed controls Boundary control Our method The functional J reads: J(z, z 1 ) = T 1 Γ η(t)ξ Γ (x) 2 n z(x, t) 2 dγdt y (x) z (x, ) dx y 1, z(, ) H 1 (Ω) H 1 (Ω), where z is the solution of z z =, z =, (z(t ), z (T )) = (z, z 1 ) in Ω (, ), on Ω (, ), H 1(Ω) L2 (Ω).

37 Distributed controls Boundary control Results Theorem Assume that Γ, T satisfies GCC. Given any (y, y 1 ) L 2 (Ω) H 1 (Ω), there exists a unique minimizer (Z, Z 1 ) of J over H 1 (Ω) L2 (Ω). The function V (x, t) = η(t)ξ Γ (x) n Y (x, t) Γ1 is the control method of minimal L 2 (, T ; dt/η; L 2 (Γ))-norm, defined by T v 2 L 2 (,T ;dt/η;l 2 (Γ)) = Γ v(x, t) 2 dγ dt η(t). If (y, y 1 ) D(A s ) for some s N, then (Z, Z 1 ) D((A ) s ) = D(A s+1 ) and V H s (, T ; L2 (Γ)).

38 Distributed controls Boundary control Results Theorem Assume that the function ξ Γ is smooth. If (y, y 1 ) D(A s ) for some s N, the η- weighted HUM yields: s V H s (, T ; L2 (Γ 1 )) C k ([, T ]; H s k 1/2 (Γ 1 )) k= and (Z, Z 1 ) D((A ) s ) = D(A s+1 ). In particular, the controlled trajectory y satisfies y s k= C k ([, T ]; H s k (Ω) H s 1 k (Ω)).

39 1 Introduction: The Hilbert Uniqueness Method 2 An alternate HUM type method The η-weighted HUM Main result Regularity of the controlled trajectory 3 Applications Distributed controls Boundary control 4 Conclusion

40 Further applications Orders of convergence of discrete controls Joint work with Enrique Zuazua The exact controls computed with the η-weighted HUM converge at a rate h 2/3 to the continuous η-hum control. Controllability of semi-linear wave equations See Dehman-Lebeau 9 for such examples.

41 Open problem-i Exact controllability What happens for the exact controllability when A does not generate a group? Typical example: Linearized KdV equation Can we design a HUM type method which yields smooth controlled trajectory for smooth initial data? Do we have such regularity properties for the non linear KdV equation? Work in progress...

42 Open problem-ii Exact controllability to trajectories What happens when A does not generate a group? Typical example: The heat equation. Can we design a HUM type method which yields smooth controlled trajectory for smooth initial data? Widely open.

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