标题： 二次障碍对多输入系统小时间局部能控性的阻碍 显示英文标题

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

摘要： 我们提出了一种多输入控制仿射系统在$R^d$上的小时间局部能控性的必要条件。 该条件是基于在零点处计算向量场的李括号所得的$R^d$向量：它涉及它们的方向和幅度。 证明是对 Beauchard 和 Marbach 在单输入情况下引入的一般方法的适应。 它依赖于一种类似 Magnus 的表示公式：状态通过向量场在零点处的李括号的线性组合来近似，其系数是时间和控制的功能。 最后，小时间局部能控性的障碍来自于插值不等式。 显示英文摘要

摘要： 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.