Explicit approximate controllability of the Schrödinger equation with a polarizability term.

Transcription

1 Explicit approximate controllability of the Schrödinger equation with a polarizability term. Morgan MORANCEY CMLA, ENS Cachan Sept Control of dispersive equations, Benasque. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

2 Outline 1 Model studied and strategy Bilinear controlled Schrödinger equation and polarizability LaSalle invariance principle 2 Previous results Semiglobal weak stabilization in the dipolar approximation Finite dimension approximation of the polarizability system 3 Explicit approximate controllability with polarizability Study of the averaged system The averaging strategy in infinite dimension Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

3 1 Model studied and strategy Bilinear controlled Schrödinger equation and polarizability LaSalle invariance principle 2 Previous results Semiglobal weak stabilization in the dipolar approximation Finite dimension approximation of the polarizability system 3 Explicit approximate controllability with polarizability Study of the averaged system The averaging strategy in infinite dimension Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

4 Model studied Model of a quantum particle in a potential V i t ψ = ( + V (x) ) ψ + u(t)q 1 (x)ψ, x D, ψ D = 0, ψ(0, ) = ψ 0, (1.1) where ψ is the wave function, D R m is a bounded regular domain, V C (D, R) is the potential, the control u is the real amplitude of the electric field, Q 1 C (D, R) is the dipolar moment, organ MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

5 Model studied Model of a quantum particle in a potential V with a polarizability term. i t ψ = ( + V (x) ) ψ + u(t)q 1 (x)ψ+u(t) 2 Q 2 (x)ψ, x D, ψ D = 0, ψ(0, ) = ψ 0, (1.1) where ψ is the wave function, D R m is a bounded regular domain, V C (D, R) is the potential, the control u is the real amplitude of the electric field, Q 1 C (D, R) is the dipolar moment, Q 2 C (D, R) is the polarizability moment. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

6 Notations S := { ψ L 2 (D, C); ψ L 2 = 1 }. < f, g >:= f (x)g(x)dx, for f, g L 2. D Let (λ k ) k N ( ) be the non decreasing sequence of eigenvalues of the operator + V with domain H 2 H0 1. Let (ϕ k ) k N be the associated sequence of eigenvectors in S. C := {cϕ 1 ; c C, c = 1}. Goal : Find a control u such that ψ ϕ 1. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

7 LaSalle invariance principle in infinite dimensions. I 1 Lyapunov function. L : H R non negative, L(x) = 0 x = x and L(x) x +. 2 Non increasing along trajectories so t L(x(t)) non increasing, L(x(t)) α. t + 3 Invariant set. We assume x solution of the PDE and d L(x(t)) 0, t 0 = x(t) x, t 0. dt 4 Let (t n ) n N +. L(x(t n )) L(x(t 0 )) so (x(t n )) n N is bounded. x(t n ) x. n + Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

8 LaSalle invariance principle in infinite dimensions. II 5 Continuity with respect to the initial condition. Let x ( ) initiated from x. x n (t) := x(t + t n ) x (t), t 0. n 6 Conclusion. Continuity of the Lyapunov function for the weak topology. L(x n (t)) L(x (t)), t 0, n L(x(t n + t)) α, t 0. n So (invariant set) hence x (t) = x, t 0, x(t) t x. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

9 1 Model studied and strategy Bilinear controlled Schrödinger equation and polarizability LaSalle invariance principle 2 Previous results Semiglobal weak stabilization in the dipolar approximation Finite dimension approximation of the polarizability system 3 Explicit approximate controllability with polarizability Study of the averaged system The averaging strategy in infinite dimension Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

10 System under the dipole approximation System studied in [Beauchard and Nersesyan, 2010]. i t ψ = ( + V (x) ) ψ + u(t)q(x)ψ, ψ D = 0, ψ(0, ) = ψ 0. (2.1) Lyapunov function L(ψ) := γ ( + V )Pψ 2 L 2 + ( 1 ψ, ϕ 1 2), with P the orthogonal projection on Span {ϕ k, k 2} and γ > 0. Hypotheses Qϕ 1, ϕ k 0, for all k 2. λ 1 λ j λ p λ q, for all {1, j} = {p, q} and j 1. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

11 Semiglobal weak stabilization using feedback law We consider the feedback law Theorem u(ψ) := Im [ γ( + V )P(Qψ), ( + V )Pψ Qψ, ϕ 1 ϕ 1, ψ ]. (2.2) Under the previous hypotheses, there exists J R + finite or countable such that for any ψ 0 S H 1 0 H2 not belonging to C, there exists γ := γ ( ψ 0 L 2) > 0 such that the solution of the system (2.1) with control u defined in (2.2) with γ (0, γ )\J and initial condition ψ 0 satisfies (up to a global phase) ψ(t) t ϕ 1, in H 2 w. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

12 Beyond the dipolar approximation I i d dt ψ(t) = ( H 0 + u(t)h 1 + u(t) 2 ) H 2 ψ(t), ψ(0, ) = ψ 0. (2.3) with ψ( ) C n, H 0, H 1 and H 2 are n n Hermitian matrices. λ 1,, λ n eigenvalues of H 0 and ϕ 1,, ϕ n the associated eigenvectors. Studied in [Grigoriu et al., 2009]. Improved in [Coron et al., 2009]. Strategy : Use of a time periodic feedback ( t u(t, ψ) := α(ψ) + β(ψ) sin. ε) Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

13 Beyond the dipolar approximation II i d ( t (H dt ψ(t) = 0 + α(ψ)h 1 + β(ψ) sin ε) + β 2 (ψ) sin 2 ( t ε ( t H 1 + α 2 (ψ)h 2 + 2α(ψ)β(ψ) sin H 2 ε) ) H 2 ) ψ(t). (2.4) Use of the averaged system. Let f be T periodic and f av (x) = 1 T f (t, x)dt. This leads to T 0 i d dt ψ av (t) = ẋ(t) = f (t, x(t)) = ẋ av (t) = f av (x av (t)). (H 0 + α(ψ av )H 1 + ( α 2 (ψ av ) β2 (ψ av ) ) H 2 ) ψ av (t). (2.5) Stabilization of the averaged system. Approximation by averaging. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

14 Application of the LaSalle invariance principle I Lyapunov function. Choice of the feedbacks. L(ψ av (t)) := ψ av (t) ϕ 1 2. d dt L(ψ av (t)) = 2αI 1 (ψ av (t)) + (2α 2 + β 2 )I 2 (ψ av (t)), where I j (ψ ( av (t)) ) = Im( H j ψ av (t), ϕ 1. 1 Let k 0, H 2. The choice of feedbacks α(ψ av (t)) : = ki 1 (ψ av (t)), β(ψ av (t)) : = (I 2 (ψ av (t))), leads to d ( dt L(ψ av (t)) = 2 ki 1 (ψ av (t)) 2 (1 ki 2 (ψ av (t))) (I 2(ψ av (t)) ) 3) 0. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

15 Application of the LaSalle invariance principle II Invariant set. Assume λ j λ l for j l and for any j {2,, n}, H 1 ϕ j, ϕ 1 0 or H 2 ϕ j, ϕ 1 0. Then ψ av ( ) solution of (2.5) with L(ψ av ( )) constant implies ψ av ( ) ±ϕ 1. Under the previous hypothesis, the averaged system is globally asymptotically stable on S 2n 1 \{ ϕ 1 }. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

16 Approximation by averaging Lemma of approximation Let T > 0. There exists C and ε 0 > 0 such that, for every τ R and for every ε (0, ε 0 ), if ψ : [τ, τ + T ] S 2n 1 is a solution of (2.4), and ψ av is the solution of (2.5) such that ψ av (τ) = ψ(τ), then ψ(t) ψ av (t) < Cε, t [τ, τ + T ]. Combining this with the convergence of ψ av we obtain Main result Assume that the coupling assumption and the non degeneracy of the spectrum hold. Let V be a neighborhood of ϕ 1 and δ > 0. There exists a time T > 0 and ε 0 > 0 such that every solution of (2.4) with ε (0, ε 0 ) that satisfies ψ(τ) S 2n 1 \V for some τ > 0 also satisfies ψ(t) ϕ 1 < δ, t τ + T. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

17 1 Model studied and strategy Bilinear controlled Schrödinger equation and polarizability LaSalle invariance principle 2 Previous results Semiglobal weak stabilization in the dipolar approximation Finite dimension approximation of the polarizability system 3 Explicit approximate controllability with polarizability Study of the averaged system The averaging strategy in infinite dimension Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

18 Averaged system { i t ψ = ( + V (x) ) ψ + u(t)q 1 (x)ψ + u(t) 2 Q 2 (x)ψ, ψ D = 0, with feedback control u(t, ψ(t)) := α(ψ(t)) + β(ψ(t)) sin(t/ε) leads to the averaged system i t ψ av = ( + V (x) ) ψ av + α(ψ av )Q 1 ψ av + (α(ψ av ) ) β(ψ av ) 2 Q 2 ψ av, ψ av D = 0. (3.1) Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

19 Study of the averaged system and choice of the feedbacks I Lyapunov function. L(z) := γ ( + V )Pz 2 L 2 + ( 1 z, ϕ 1 2). Feedback laws with α(z) := ki 1 (z), β(z) := g(i 2 (z)), I j (z) := Im ( γ ( + V )P(Q j z), ( + V )Pz Q j z, ϕ 1 ϕ 1, z ), k > 0 small enough and g C 2 (R, R + ) satisfying g(x) = 0 if and only if x 0, g bounded. d Then, dt L(ψ av (t)) 0. Under the following assumptions (H1) Q 1ϕ 1, ϕ k = 0 = Q 2ϕ 1, ϕ k = 0, (H2) Card {k 2; Q 1ϕ 1, ϕ k = 0} <, Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

20 Study of the averaged system and choice of the feedbacks II (H3) λ 1 λ k λ p λ q for {1, k} = {p, q} and k 1, (H4) λ p λ q for p q, the invariant set is included in C. Continuity with respect to the initial condition and continuity of the feedback law for the weak H 2 topology. Assume that hypotheses (H1)-(H4) hold. If ψ 0 X 0 := { z S H 1 0 H2 ; z H 1 0 H2} with 0 < L(ψ 0 ) < 1, the solution of (3.1) satisfies (up to a global phase) ψ av (t) ϕ 1, in H 2. t + Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

21 Approximation by averaging I For an initial condition ψ 0 X 0, we consider the control u ε (t) := α(ψ av (t)) + β(ψ av (t)) sin(t/ε), with ψ av the solution of (3.1) satisfying ψ av (0, ) = ψ 0. Let L > 0, ψ 0 X 0 with 0 < L(ψ 0 ) < 1. Let ψ av be the solution of the closed loop system (3.1) with initial condition ψ 0. For any δ > 0, there exists ε 0 > 0 such that if ψ ε is the solution of (1.1) with initial condition ψ 0 and control u ε with ε (0, ε 0 ), then ψ ε (t) ψ av (t) H 2 δ, t [0, L]. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

22 Explicit approximate controllability Main result Assume that hypotheses (H1)-(H4) hold. For any s < 2, for any ψ 0 X 0 with 0 < L(ψ 0 ) < 1, there exist a strictly increasing time sequence (T n ) n N in R + tending to + and a decreasing sequence (ε n ) n N in R + such that if ψ ε is the solution of (1.1) associated to the control u ε with ε (0, ε n ) and initial condition ψ 0, dist H s (ψ ε (t), C) 1 2 n, t [T n, T n+1 ]. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

23 Open Problems Convergence in the H 2 norm. Card {j 2; Q 1 ϕ 1, ϕ j = 0} =. Approximation property on infinite time interval [s, + ). Semi global exact controllability using [Beauchard and Laurent, 2010] in the 1D case with V = 0. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

24 Open Problems Convergence in the H 2 norm. Card {j 2; Q 1 ϕ 1, ϕ j = 0} =. Approximation property on infinite time interval [s, + ). Semi global exact controllability using [Beauchard and Laurent, 2010] in the 1D case with V = 0. Thank you for you attention. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22

25 Beauchard, K. and Laurent, C. (2010). Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl. (9), 94(5): Beauchard, K. and Nersesyan, V. (2010). Semi-global weak stabilization of bilinear Schrödinger equations. C. R. Math. Acad. Sci. Paris, 348(19-20): Coron, J.-M., Grigoriu, A., Lefter, C., and Turinici, G. (2009). Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling. New. J. Phys., 11(10). Grigoriu, A., Lefter, C., and Turinici, G. (2009). Lyapunov control of Schrödinger equation:beyond the dipole approximations. In Proc of the 28th IASTED International Conference on Modelling, Identification and Control, pages , Innsbruck, Austria. Morgan MORANCEY (CMLA, ENS Cachan) Approximate controllability Sept / 22