Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term

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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 Schrödinger 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. Thus proving STLC requires to go at least to the third order. Similar results were proved for the bilinear Schrödinger PDE with scalar-input controls. In this article, we study the case of multi-input systems. We clarify among the quadratic Lie brackets, those that allow to recover STLC. For ODEs, our result is a consequence of Sussmann’s sufficient condition S(θ), 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.