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arXiv · 2412.17384

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

Théo Gherdaoui
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Abstract

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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Théo Gherdaoui. 2024-12-23. Quadratic obstructions to small-time local controllability for multi-input systems. https://doi.org/10.1007/s10883-025-09747-y

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