arXiv subjects

Théo Gherdaoui

Publications and source records attributed to Théo Gherdaoui.

3 recordsLinked to original sources

arXiv2025

Quadratic obstructions to small-time local controllability for multi-input systems

We present a necessary condition for the small-time local controllability of multi-input control-affine systems on $R^d$ . This condition is formulated on the vectors of $R^d$ resulting from the evaluation at zero of the Lie brackets of the vector fields: it involves both their direction and their amplitude. The proof is an adaptation to the multi-input case of a general method introduced by Beauchard and Marbach in the single-input case. It relies on a Magnus-type representation formula: the state is approximated by a linear combination of the evaluation at zero of the Lie brackets of the vector fields, whose coefficients are functionals of the time and the controls. Finally, obstructions to small-time local controllability result from interpolation inequalities.

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arXiv2025

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 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: they are bilinear with respect to two different controls. For ODEs, our result is a consequence of Sussman'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.

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arXiv2025

Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schrödinger equation

We investigate the small-time local controllability (STLC) near the ground state of a bilinear Schrödinger equation when the linearized system is not controllable. It is well known that, for single-input systems, quadratic terms in the state expansion can then lead to obstructions to the STLC of the nonlinear system. In this work, we extend this phenomenon to the multi-input setting, presenting the first example of multi-input quadratic obstructions for PDEs. Our results build upon our previous study of such obstructions for ODEs and provide a functional framework for analyzing them in the bilinear Schrödinger equation.

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