A quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting

Transcription

1 A quantitative Fattorini-Hautus test: the minimal null control time problem in the parabolic setting Morgan MORANCEY I2M, Aix-Marseille Université August 2017 "Controllability of parabolic equations : new results and open problems" Morgan MORANCEY Quantitative Fattorini-Hautus test 1

2 1 Control theory and duality Abstract setting Unique continuation Observability inequalities 2 Some known criterion for controllability Hautus test Fattorini criterion A necessary inequality 3 A quantitative Fattorini-Hautus test Definition of the minimal time Examples of minimal null controllability time in the parabolic setting A necessary and sufficient condition for null controllability 4 Open problems Morgan MORANCEY Quantitative Fattorini-Hautus test 2

3 1 Control theory and duality Abstract setting Unique continuation Observability inequalities 2 Some known criterion for controllability 3 A quantitative Fattorini-Hautus test 4 Open problems Morgan MORANCEY Quantitative Fattorini-Hautus test 3

4 Abstract linear control problem { y (t) = Ay(t) + Bu(t), t (0, T ), y(0) = y 0, (S) where A generates a C 0 -semigroup on the Hilbert space (H, ), the space of controls is the separable Hilbert space (U, U ), the control operator B L(U, D(A ) ) is assumed to be admissible K T > 0, such that T 0 B e ta z 2 U dt K T z 2, z D(A ). Morgan MORANCEY Quantitative Fattorini-Hautus test 4

5 Wellposedness Let T > 0. For any y 0 H and any u L 2 (0, T ; U), a solution y C 0 ([0, T ], H) is defined by y(t), z y 0, e ta z = t 0 u(τ), B e (t τ)a z U dτ, t [0, T ], z H. Wellposedness For any y 0 H and any u L 2 (0, T ; U), there exists a unique solution y C 0 ([0, T ], H). Moreover, there exists C > 0 such that for any y 0, u, the solution satisfies y(t) C ( y 0 + u L 2 (0,T ;U)), t [0, T ]. Morgan MORANCEY Quantitative Fattorini-Hautus test 5

6 Approximate controllability/unique continuation Approximate controllability in time T : for any y 0, y 1 H, for any ε > 0, there exists u L 2 (0, T ; U) such that y(t ) y 1 ε. Let y 0 = 0 and define the input-to-state map L T : L 2 (0, T ; U) H u y(t ) Approximate controllability in time T L T ( L2 (0, T ; U) ) = H Ker L T = {0}. For any z H, L T z : t (0, T ) B e (T t)a z. Unique continuation: ( { tz(t) + A z(t) = 0 z(t ) = z T + ) B z(t) = 0, t (0, T ) = z T = 0. Morgan MORANCEY Quantitative Fattorini-Hautus test 6

7 Some necessary conditions for approximate controllability Observation of eigenvectors. ex: pointwise control. A ϕ k = λ k ϕ k = B ϕ k 0. dim U = 1: simple eigenvalues. { A ϕ k,1 = λ k ϕ k,1 A ϕ k,2 = λ k ϕ k,2 = dim Span(ϕ k,1, ϕ k,2 ) = 1. ex: a single boundary control in 1D. Morgan MORANCEY Quantitative Fattorini-Hautus test 7

8 Exact controllability Exact controllability in time T : u L 2 (0, T ; U) such that y(t ) = y 1. for any y 0, y 1 H, there exists Let y 0 = 0 and recall L T : L 2 (0, T ; U) H u y(t ) Exact controllability in time T L T ( L 2 (0, T ; U) ) = H Observability in time T : T 0 c > 0; L T z L 2 (0,T ;U) c z, z H. B z(t) 2 U dt c z T 2, z T D(A ), where { tz(t) + A z(t) = 0 z(t ) = z T Remark: observability = quantitative unique continuation. Morgan MORANCEY Quantitative Fattorini-Hautus test 8

9 Controllability to trajectories and null controllablility Regularization is an obstacle to exact controllability. Controllability to trajectories in time T : for any y 0, y 0 H, for any u L 2 (0, T ; U), there exists u L 2 (0, T ; U) such that y(t ) = y(t ) where { y (t) = Ay(t) + Bu(t), t (0, T ), y(0) = y 0. Null controllability in time T : for any y 0 H, there exists u L 2 (0, T ; U) such that y(t ) = 0. Controllability to trajectories in time T Null controllability in time T. Morgan MORANCEY Quantitative Fattorini-Hautus test 9

10 (Final time) Observability Let y 0 H. Then y(t ) = e T A y 0 + L T u. Null controllability in time T y 0 H, u L 2 (0, T ; U); L T u = e T A y 0 e T A (H) L T ( L 2 (0, T ; U) ) (Final time) Observability in time T : T 0 c > 0; L T z L 2 (0,T ;U) c e T A z, z H. B z(t) 2 U dt c z(0) 2, z T D(A ), where { tz(t) + A z(t) = 0 z(t ) = z T Morgan MORANCEY Quantitative Fattorini-Hautus test 10

11 1 Control theory and duality 2 Some known criterion for controllability Hautus test Fattorini criterion A necessary inequality 3 A quantitative Fattorini-Hautus test 4 Open problems Morgan MORANCEY Quantitative Fattorini-Hautus test 11

12 Hautus test: characterization of controllability in finite dimension Finite dimensional setting: H = R n, U = R m, A R n n and B R n m. Equivalence between exact, null and approximate controllability in time T. Hautus test: M.L.J. Hautus (1969). controllability in time T Ker(A λ) Ker B = {0}, λ C. Necessary condition : dim Ker(A λ) m for all λ C. Morgan MORANCEY Quantitative Fattorini-Hautus test 12

13 Fattorini criterion: a sufficient condition for approximate controllability Setting: assume moreover that A has compact resolvent and the root vectors of A form a complete sequence in H. Assume that C : D(C) H U is A -bounded i.e. Cz α z + β A z, z D(A ). Fattorini criterion: H.O. Fattorini (1966). ( ) z D(A ) and Ce ta z = 0, for a.e. t (0, + ) Ker(A λ) Ker C = {0}, λ C. = z = 0, Thus, if B is A -bounded and moreover the semigroup generated by A is analytic, Approximate controllability Ker(A λ) Ker B = {0}, λ C. Question: quantitative Fattorini-Hautus test for null controllability? Morgan MORANCEY Quantitative Fattorini-Hautus test 13

14 A necessary inequality T. Duyckaerts & L. Miller (2012). Assume that A generates a C 0 -semigroup on H and that B is admissible. Null controllability in time T = C T > 0; y D(A ), λ C with Re(λ) > 0, ( ) (A y 2 C T e 2 Re(λ)T + λ)y 2 + B y 2 U. Re(λ) 2 Re(λ) Necessary conditions for null controllability in time T : Sufficient observation of eigenvectors (depending on time!) A ϕ k = λ k ϕ k = B ϕ k U C T Re(λk )e Re(λ k)t. dim U = 1: distance between eigenvalues { A ϕ k = λ k ϕ k A ϕ j = λ jϕ j + ϕ k ϕ j = λ k λ j C T λ k e Re(λ k)t. Morgan MORANCEY Quantitative Fattorini-Hautus test 14

15 1 Control theory and duality 2 Some known criterion for controllability 3 A quantitative Fattorini-Hautus test Definition of the minimal time Examples of minimal null controllability time in the parabolic setting A necessary and sufficient condition for null controllability 4 Open problems Morgan MORANCEY Quantitative Fattorini-Hautus test 15

16 Classical controllability behaviour Parabolic setting Hyperbolic setting Infinite speed of propagation; Null controllability in arbitrary time; No restriction on the control domain (internal control, boundary control). H.O. Fattorini & D.L. Russell (1971), A.V. Fursikov & O.Y. Imanuvilov (1996) and G. Lebeau & L. Robbiano (1995). Finite speed of propagation; Exact controllability only in large time; Geometric condition on the control domain (internal control, boundary control). C. Bardos, G. Lebeau & J. Rauch (1992). Morgan MORANCEY Quantitative Fattorini-Hautus test 16

17 The minimal time However, there are examples of parabolic control problems exhibiting a minimal time for null controllability and/or a geometric condtion... Let { T 0 = inf T > 0 ; C T > 0; y D(A ), λ C with Re(λ) > 0, y 2 C T e 2T Re(λ) ( (A + λ)y 2 Re(λ) 2 ) } ( ) + B y 2 U Re(λ) and T 0 = + when the previous set is empty. Morgan MORANCEY Quantitative Fattorini-Hautus test 17

18 With a pointwise control S. Dolecki (1973) { ty xxy = δ x=x0 u(t), t (0, T ), x (0, 1), y(t, 0) = y(t, 1) = 0. Notations: ϕ k (x) = 2 sin(kπx), λ k = k 2 π 2. Minimal time: ln ϕ k (x 0) T min = lim sup = T 0 k + λ k Morgan MORANCEY Quantitative Fattorini-Hautus test 18

19 With a zero-order coupling term supported outside the control domain F. Ammar Khodja, A. Benabdallah, M. González Burgos & L. de Teresa (2016). ( xx 0 ty + 0 xx y(t, 0) = y(t, 1) = ( 0 0 ) y + ). ( ) ( 0 q(x) y = ωu(t, x) ), t (0, T ), x (0, 1), Notations: ϕ k (x) = 2 sin(kπx), λ k = k 2 π 2. ω = (a, b), Supp(q) ω = I k (q) = 1 q(x)ϕ k (x) 2 dx, I 1,k (q) = a 0 0 where I k (q) + I 1,k (q) = 0 for any k N. q(x)ϕ k (x) 2 dx. Minimal time: min ( ln I k (q), ln I 1,k (q) ) T min = lim sup = T 0. k + λ k Morgan MORANCEY Quantitative Fattorini-Hautus test 19

20 Degenerate Grushin equation in dimension two K. Beauchard, P. Cannarsa & R. Guglielmi (2014) and K. Beauchard, L. Miller & M. M. (2015). { ty x1 x 1 y x 2 1 x2 x 2 y = 1 ωu(t, x 1, x 2), t (0, T ), (x 1, x 2) Ω y(t, x 1, x 2) = 0, (x 1, x 2) Ω. Notations: Ω = ( 1, 1) (0, 1), [ ] ω = ( b, a) (a, b) (0, 1). 1 x2 ω ω 1 0 a b 1 x 1 Minimal time: T min = a2 2 = T0. Morgan MORANCEY Quantitative Fattorini-Hautus test 20

21 A system with different diffusion coefficients F. Ammar Khodja, A. Benabdallah, M. González Burgos & L. de Teresa (2014). ( xx 0 ty + 0 d xx y(t, 0) = ( 0 u(t) ) ( ) 0 1 y + y = 0, t (0, T ), x (0, 1), 0 0 ) ( ) 0, y(t, 1) =. 0 Notations : Λ = { k 2 π 2, dk 2 π 2 ; k N }. Minimal time: T min = c(λ) = T 0. where c(λ) is the condensation index defined by ln E (λ k ) c(λ) = lim sup with E(z) = k + λ k + j=1 (1 z2 z 2 k ). Morgan MORANCEY Quantitative Fattorini-Hautus test 21

22 Setting Assume that the operator A admits a sequence of eigenvalues Λ = (λ k ) k N such that δ > 0, Re(λ k ) δ λ k, k N and + k=1 1 < +. (Spectrum) λ k For any k N, we denote by r k = dim(ker(a + λ k )) the geometric multiplicity of the eigenvalue λ k and assume that sup k N r k < +. We denote by (ϕ k,j ) k N,1 j r k the associated sequence of normalised eigenvectors and we assume that it forms a complete sequence in H i.e. ( ) Φ, ϕ k,j = 0, k N, j {1,..., r k } = Φ = 0. Morgan MORANCEY Quantitative Fattorini-Hautus test 22

23 Equivalence between quantitative Fattorini-Hautus test and null controllability F. Ammar Khodja, A. Benabdallah, M. González Burgos & M. M. (submitted) Theorem 1 Assume that the condensation index of the sequence Λ = (λ k ) k N satisfies c(λ) = 0. Let T 0 be defined by ( ). Then: If T 0 > 0 and T < T 0, system (S) is not null controllable in time T ; If T 0 < + and T > T 0, system (S) is null controllable in time T. Condition (Spectrum) in a parabolic setting: "restriction to space dimension 1". Generalization to "suitable growth" of the geometric multiplicity. Qualitative result (T min [T 0, 2T 0]) for algebraic multiplicity 2 (i.e. Jordan chain of length 2) Assumption c(λ) = 0 is not so strong. Morgan MORANCEY Quantitative Fattorini-Hautus test 23

24 Sketch of proof: simple eigenvalues I Definition of solutions y(t ), ϕ j y 0, e λ j T ϕ j = Complete family of eigenvectors T 0 e λ j (T t) u(t), B ϕ j U dt. y(t ) = 0 Biorthogonal family T 0 e λ j (T t) u(t), B ϕ j U dt = e λ j T y 0, ϕ j, j N. Let T > 0 and let σ = (σ k ) k N be a normally ordered complex sequence satisfying (Spectrum). Then, there exists a biorthogonal family (q k ) k N to the exponentials associated with σ i.e. T 0 e σ j t q k (t)dt = δ k,j, k, j N, such that for any ε > 0 there exists C ε > 0 such that q k L 2 (0,T ;C) C εe Re(σ k)(c(σ)+ε). Morgan MORANCEY Quantitative Fattorini-Hautus test 24

25 Sketch of proof: simple eigenvalues II "New" form of controls for the moment problem u(t) = k N α k q k (T t) B ϕ k, where the scalar coefficient α k has to be determined. T 0 e λ j (T t) u(t), B ϕ j U dt = e λ j T y 0, ϕ j, j N k N α k B ϕ k, B ϕ j T α k = e λ kt y 0, ϕ k, k N. B ϕ k 2 U 0 e λ j (T t) q k (T t)dt = e λ j T y 0, ϕ j, j N Convergence of the series thanks to quantitative Fattorini-Hautus test u(t) = e λkt B ϕ k y 0, ϕ k q k (T t). B ϕ k N k 2 U Multiple eigenvalues: construct and estimate a biorthogonal family to (B ϕ k,1,..., B ϕ k,rk ). Morgan MORANCEY Quantitative Fattorini-Hautus test 25

26 Minimal time coming from condensation of the spectrum Theorem 2 Assume that the eigenvectors of A form a Riesz basis of H. Assume that for any k j N, Ker(B ) Span(ϕ k, ϕ j) {0} and for any ε > 0 B ϕ k U e ε Re(λ k) +. k + Finally, assume that the condensation index of the sequence Λ = (λ k ) k N satisfies c(λ) = Bohr(Λ). Let T 0 be defined by ( ). Then: If T 0 > 0 and T < T 0, system (S) is not null controllable in time T ; If T 0 < + and T > T 0, system (S) is null controllable in time T. Structural assumption on B generalizes dim U = 1 = simple eigenvalues. Technical (?) assumption on the condensation index: "allow eigenvalues to get exponentially close but only two by two". Remove the sufficient observation of eigenvectors hypothesis: qualitative result T min [T 0, 2T 0]. Morgan MORANCEY Quantitative Fattorini-Hautus test 26

27 The general setting Theorem 3 Let T 0 be defined by ( ). Then, there exists T [T 0, T 0 + c(λ)] such that if T > 0 and T < T, system (S) is not null controllable in time T ; if T < + and c(λ) < +, for T > T, system (S) is null controllable in time T. Quite general setting for (systems) of one dimensional parabolic equations. Morgan MORANCEY Quantitative Fattorini-Hautus test 27

28 1 Control theory and duality 2 Some known criterion for controllability 3 A quantitative Fattorini-Hautus test 4 Open problems Morgan MORANCEY Quantitative Fattorini-Hautus test 28

29 On the spectral assumption + k=1 1 λ k < + Used for the moment method: needed for the biorthogonal family with respect to the time exponentials biorthogonal family to e λ k B ϕ k in L 2 (0, T ; U)? Crucial for the characterization of controllability through the quantitative Fattorini-Hautus test. Quantum harmonic oscillator, T. Duyckaerts & L. Miller (2012). Let H = L 2 (R), U = L 2 (R), B = 1 (,x0 ) with x 0 R and A = xx + x 2, D(A) = {y H 2 (R); x x 2 y(x) L 2 (R)}. A is self-adjoint and generates a C 0 -semigroup. Its eigenvalues are {λ k = 2k 1; k N } and its eigenvectors are the Hermite polynomials (Hilbert basis). For this operator T min = + and M, m R such that y 2 M (A + λ)y 2 + m B y 2 U, y D(A ), λ R = T 0 = 0. Notice that this example satisfies every assumption of Theorem 3 except that = +. k N 1 λ k Morgan MORANCEY Quantitative Fattorini-Hautus test 29

30 Open problems A "better" quantitative Fattorini-Hautus test: capture the minimal time coming from both the condensation of eigenvalues and the "localization" of eigenvectors. Minimal time for null controllability of parabolic systems in higher dimensions? Lack of methods... Morgan MORANCEY Quantitative Fattorini-Hautus test 30