Minimax solution of the multiple-target problem

The convexization procedure developed for a class of minimax problems is applied to the determination of the minimax solution of the multiple-target problem. The method is parallel to that used in earlier works, but the results are completely independent. It is shown that the state space may be partitioned into subregions in which the minimax strategy is a pure strategy and into subregions in which it is a mixed strategy in the terminology developed in the theory of games, which aptly characterizes the nature of the minimax solution in this problem. It is also shown that the minimax strategy in open-loop form is a piecewise linear function of the initial state and a linear function of the state along the resulting trajectory. In feedback form, it is a piecewise linear function of the state and the cost incurred in the elapsed interval of play.