In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.

在对具有端点不等式约束的最优控制问题的强变异算法的早期分析中，Mayne和Polak提供了条件，在这些条件下，累积点满足要求一定最优函数的条件，该条件在算法中用于生成搜索方向，对于所有控制都是非负的。本文的目的是弄清此最优条件的性质，我们称其为一阶极小极大值条件，以及该条件的相关集成形式，该形式也隐含在过去的算法收敛性分析中。当问题制定中包含或不包含路径状态约束时，我们分别考虑这些条件。如果没有路径状态约束，我们表明，积分一阶极小极大条件等于最小原理，并且最小原理（和等效积分一阶极小极大条件）严格地强于一阶极小极大条件。对于具有状态约束的问题，我们确定积分的一阶极小极大条件和极小原理再次相等。但是，在状态受约束的上下文中，不再不再需要最小原理比一阶最小极大条件强，反之亦然。一个例子证实了一个令人惊讶的事实，即一阶极小极大条件是一个独特的最优性条件，可以为状态约束的问题提供信息，而在某些情况下，最小原理无法做到这一点。