On the Optimal Cost and Asymptotic Stability in Two-Player Zero-Sum Set-Valued Hybrid Games

In this paper, we formulate a two-player zero-sum game under dynamic constraints formulated in terms of a hybrid inclusion. The game consists of a min-max problem involving a cost functional associated to the actions and corresponding (potentially nonunique) solutions to the system. We present sufficient conditions given in terms of Hamilton-Jacobi-Isaacs-like equations to establish a bound on the worst-case cost under the optimal strategy and to exactly evaluate it. Under additional conditions, we show that the proposed optimal state-feedback laws render a set of interest pre-asymptotically stable for the resulting hybrid closed-loop system. The results are illustrated in a numerical example.