Two-Player Zero-Sum Hybrid Games

TL;DR

This paper formulates a two-player zero-sum hybrid game with dynamic constraints, providing conditions for optimal strategies and stability, applicable to robust control problems like disturbance rejection and security.

Contribution

It introduces a novel hybrid game framework with Hamilton-Jacobi-Bellman-Isaacs equations and offers solution conditions without solving hybrid system trajectories.

Findings

Optimal strategies can be computed without solving hybrid system solutions.

The value function satisfies Hamilton-Jacobi-Bellman-Isaacs equations.

Optimal feedback laws ensure asymptotic stability of the hybrid system.

Abstract

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 conveniently 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 these games, presented here as robust control problems, include disturbance rejection and security problems.