Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Abstract

We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.In [10], Beauchard and Laurent prove that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. In [18], Coron proves that a positive minimal time is required for this controllability result, on a particular degenerate example.In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.Theorem 2 There exists ǫ > 0 such that, for every d = 0 and u ∈ L 2 ((0, ǫ), R) satisfying |u(t)| < ǫ, ∀t ∈ (0, ǫ), the solution (ψ, s, d) ∈ C 0 ([0, ǫ], H 1 0 ((0, 1), C)) × C 0 ([0, ǫ], R) × C 1 ([0, ǫ], R) of (1.9) such that (ψ, s, d)(0) = (ψ 1 (0), 0, 0) satisfies (ψ, s, d)(ǫ) = (ψ 1 (ǫ), 0, d).The goal of this article is to go further in this analysis:1. we propose a general context for the minimal time to be positive (in particular, the variables s and d are not required anymore in the state), 2. we propose a sufficient condition for the local controllability to hold in large time; this assumption is compatible with the previous context and weaker than (1.7), 3. we work in an optimal functional frame, for instance, our non controllability result requires u small in L 2 -norm, not in L ∞ -norm as in Theorem 2, 4. we perform a first step toward the characterization of the minimal time.

Quadratic obstructions to small-time local controllability for scalar-input systems

“…For systems governed by partial differential equations, we expect that the behaviors proved in finite dimension can also be observed. For example, the first author and Morancey obtain in [7] a drift quantified by the H −1 -norm of the control, which prevents small-time local controllability, under an assumption corresponding to [f 1 , [f 0 , f 1 ]](0) / ∈ S 1 . In [14], Coron and Crépeau observe that the behavior of the second-order expansion of a Korteweg-de-Vries system is fully determined by the position of the linear approximation (thus recovering a kind of invariant manifold up to the second order).…”

Rapid stabilization of a linearized bilinear 1-D Schrödinger equation

“…Throughout this article, we assume that µ satisfies the following assumption. Remark 1.2 A direct computation shows that, for every function µ ∈ H 3 ((0, 1); R), we have 3 1 0 (µϕ 1 ) ′′′ (x) cos(kπx)dx (1.4) and therefore there exists C > 0 such that As proved in [8], Hypothesis 1.1 is not necessary to get local exact controllability of (1.1). However (see [5]) it is a necessary and sufficient condition to get exact controllability of the following linearized equation around the ground state      i∂ t Ψ = −∆Ψ − v(t)µ(x)Φ 1 (t, x), (t, x) ∈ (0, T ) × (0, 1), Ψ(t, 0) = Ψ(t, 1) = 0, t ∈ (0, T ), Ψ(0, x) = Ψ 0 (x),…”

“…The first local controllability results on the bilinear Schrödinger equation appear in [2,3,5]. These local controllability results have been extended with weaker assumptions in [8], in a more general setting in infinite time [42] and also in the case of simultaneous controllability of a finite number of particles in [38,39]. Note that, despite the infinite speed of propagation, it was proved in [17,4,8,38] that a minimal amount of time is required for the controllability of some bilinear Schrödinger equations.…”

Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

“…The particular case of the first obstruction was encountered by Coron in [17] and Morancey and the first author in [8], in the context of a bilinear Schrödinger equation. The following theorem, proved in Section 3, shows that any integer-order quadratic obstruction is possible and has consequences for the controllability of the full nonlinear system.…”

“…• controllability is restored by means of a quadratic expansion but only in large time (see e.g. [13,14] where the authors obtain controllability in large time for Korteweg-de-Vries systems with critical lengths and [5,6,8] where the authors obtain controllability of bilinear Schrödinger equation),…”