The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a first-order necessary condition for a strong local minimum) using the local maximum principle (a first-order necessary condition for a weak local minimum). In the present work, we use this method to derive second-order necessary conditions for a strong local minimum on the base of such conditions for a weak local minimum. For simplicity, we confine ourselves to the consideration of the Mayer problem with endpoint equality and inequality constraints and control inequality constraints given by a finite number of twice smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide a rather short proof of second-order necessary conditions for a strong local minimum.

中文翻译：

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具有端点和控制约束的问题中强局部最小值的必要二阶条件

滑模（松弛）方法最初是在最优控制中发明的，目的是使用局部最大值原理（一阶必要条件）给出最大值原理（强局部最小值的一阶必要条件）的透明证明弱局部最小值）。在目前的工作中，我们使用这种方法在弱局部最小值的这种条件的基础上推导出强局部最小值的二阶必要条件。为简单起见，我们仅限于考虑 Mayer 问题，该问题具有端点相等和不等式约束以及由有限数量的两倍平滑函数给出的控制不等式约束。假设主动控制约束的梯度线性无关，