Multi-cumulant Control for Zero-Sum Differential Games: Performance-Measure Statistics and State-Feedback Paradigm

Abstract

This chapter presents an extension of cost-cumulant control theory over a finite horizon for a class of stochastic zero-sum differential games wherein the evolution of the states of the game in response to decision strategies selected by two players from sets of admissible controls is described by a stochastic linear differential equation and a standard integral-quadratic cost. A direct dynamic programming approach for the Mayer optimization problem is used to solve for a multi-cumulant based solution when both players measure the states and minimize the first finite number of cumulants of the standard integral-quadratic cost associated with this special class of differential games. This innovative decision-making paradigm is proposed herein to provide not only a mechanism in which the conflicting interests of noncooperative players can be optimized, but also an analytical tool which is used to provide a complete statistical description of the global performance of the stochastic differential game.