The Equivalence Of Two Kinds Of Variational Problem Analysis

In this paper, we consider the equivalence of a minimax disparity problem and the corresponding minimum variance problem by optimal control theory at first, and secondly a general minimax problem and the corresponding Lagrange problem are considered. We mainly transform minimax problem into equivalent nonsmooth Lagrange optimal control problem by taking its inverse problem into consideration, and then approximate the nonsmooth system by a smooth one to which the classical Pontryagin maximum principle is available. We get conclu-sion that the nonsmooth system accepts Bang-Bang property, which guarantees the equivalence of two problems. Besides, we demonstrate that avoiding point-wise state constraint makes it easier to get optimal control. Furthermore, our equivalence result can be extended to other restraints.