Small-time local controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term

The goal of this article is to contribute to a better understanding of the relations between the exact controllability of nonlinear PDEs and the control theory for ODEs based on Lie brackets, through a study of the Schr¨odinger PDE with bilinear control. We focus on the small-time local controllability (STLC) around an equilibrium, when the linearized system is not controllable. We study the second-order term in the Taylor expansion of the state, with respect to the control. For scalar-input ODEs, quadratic terms never recover controllability: they induce signed drifts in the dynamics (see [Beauchard and Marbach, J. Differ. Equ. 264 (2018) 3704–3774]). Thus proving STLC requires to go at least to the third order. Similar results were proved for the bilinear Schrödinger PDE with scalarinput controls in [Bournissou, Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term (2022)]. In this article, we study the case of multi-input systems. We clarify among the quadratic Lie brackets, those that allow to recover STLC: they are bilinear with respect to two different controls. For ODEs, our result is a consequence of Sussmann’s sufficient condition S (θ) (when focused on quadratic terms), but we propose a new proof, designed to prepare an easier transfer to PDEs. This proof relies on a representation formula of the state inspired by the Magnus formula. By adapting it, we prove a new STLC result for the bilinear Schrödinger PDE.