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

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 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 $\S(\theta)$ 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.