Abstract We prove an inverse mapping theorem on a metric space of controls that allows to “control” final points of trajectories of a nonlinear system. More precisely, our result provides local distance estimates of arbitrary controls from feasible ones. As an application we derive second-order necessary optimality conditions for L 1 -local minima for the Mayer optimal control problem with a general control constraint U ⊂ R m , state constraints described by inequalities and final point constraints, possibly having empty interior. Thanks to this inverse mapping theorem we first get a second-order variational inequality as a necessary optimality condition. Then the separation theorem leads in a straightforward way to second-order necessary conditions.

中文翻译：

---

具有最终点约束和二阶必要最优性条件的系统的可行控制的距离估计

摘要 我们在控制的度量空间上证明了逆映射定理，该定理允许“控制”非线性系统的轨迹的最终点。更准确地说，我们的结果提供了对可行控制的任意控制的局部距离估计。作为一个应用，我们推导出了 Mayer 最优控制问题的 L 1 局部最小值的二阶必要最优性条件，一般控制约束 U ⊂ R m ，状态约束由不等式和最终点约束描述，可能有空的内部。由于这个逆映射定理，我们首先得到一个二阶变分不等式作为必要的最优性条件。然后分离定理以一种直接的方式导出二阶必要条件。