Sufficient Conditions for Optimality and Asymptotic Stability in Two-Player Zero-Sum Hybrid Games

Abstract

In this paper, we formulate a two-player zero-sum game under dynamic constraints given in terms of hybrid dynamical systems. We present sufficient conditions with Hamilton-Jacobi-Isaacs-like equations to guarantee attaining a solution to the game. It is shown that when the players select the optimal strategy, the value function can be evaluated without the need of computing solutions. Under additional conditions, we show that the optimal feedback laws render a set of interest asymptotically stable. Using this framework, we address an optimal control problem under the presence of an adversarial action in which the decision-making agents have dynamics that might exhibit both continuous and discrete behavior. Applications of this problem, as presented here, include disturbance rejection and security scenarios, for which the effect of the worst-case adversarial action is minimized.