Two-Player Zero-Sum Hybrid Games

In this paper, we formulate a two-player zero-sum game under dynamic constraints defined by hybrid dynamical equations. The game consists of a min-max problem involving a cost functional that depends on the actions and resulting solutions to the hybrid system, defined as functions of hybrid time and, hence, can flow or jump. A terminal set conviniently defined allows to recast both finite and infinite horizon problems. We present sufficient conditions given in terms of Hamilton-Jacobi-Bellman-Isaacs-like equations to guarantee to attain a solution to the game. It is shown that when the players select the optimal strategy, the value function can be evaluated without computing solutions to the hybrid system. Under additional conditions, we show that the optimal state-feedback laws render a set of interest asymptotically stable for the resulting hybrid closed-loop system. Applications of this problem, as presented here, include a disturbance rejection scenario for which the effect of the perturbation is upper bounded, and a security scenario in which we formulate an optimal control problem under the presence of a maximizing attack.
Comment: 21 pages