Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

Abstract

<p style='text-indent:20px;'>We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [<xref ref-type="bibr" rid="b5">5</xref>], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.</p>

Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition

“…Then, a systematic approach to recover STLC when the linearized system misses a finite number of directions is described in Section 3. Before applying this method to the Schrödinger equation, we recall in Section 4 its well-posedness and the controllability result in projection of [12]. Then, the power series expansion of the Schrödinger equation is computed in Section 5.…”

“…For the bilinear Schrödinger equation (1.1), exact controllability results were first demonstrated locally around the ground state, thanks to the linear test [6,12]. These results require assumptions on the dipolar moment µ entailing that the linearized system around the ground state is exactly controllable, and they hold in arbitrarily small time, because so does the controllability of the linearized system (see [23,34] for generalizations).…”

“…Remark 2.1. For more toy examples in finite dimension illustrating the method developed in this paper, the interested reader can refer to [11,Section 3].…”

“…The core of the paper is to prove that such a linear correction does not destroy the work done in (2.9). This is done in Proposition 6.6, using sharp and simultaneous estimates on the linear control v b obtained in [12,Theorem 1.2].…”

“…There exists a unique solution of (4.1), i.e. a function ψ ∈ C 2 ([0, T ], H 7 (0) (0, 1)) with ψ(T ) ∈ H 11 (0) (0, 1) such that the following equality holds in…”