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

Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, 33400 Talence, France

^* Corresponding author: megane.bournissou@ens-rennes.fr

Received: 2 May 2022
Accepted: 29 October 2023

Abstract

We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known [K. Beauchard and M. Morancey, Math. Control Relat. Fields 4 (2014) 125-160, M. Bournissou, J. Diff. Equ. 351 (2023) 324−360] that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's S (θ) sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.