Hamilton-Jacobi Equations for Two Classes of State-Constrained Zero-Sum Games

TL;DR

This paper develops Hamilton-Jacobi formulations for two classes of two-player zero-sum games with state constraints, providing analytical tools and numerical algorithms to determine optimal strategies in dynamic and time-invariant settings.

Contribution

It introduces novel HJ equations for constrained zero-sum games and demonstrates their application with a water system example.

Findings

Derived HJ equations for both time-varying and time-invariant cases.

Provided a numerical algorithm for solving the HJ equations.

Validated the approach with a water system case study.

Abstract

This paper presents Hamilton-Jacobi (HJ) formulations for two classes of two-player zero-sum games: one with a maximum cost value over time, and one with a minimum cost value over time. In the zero-sum game setting, player A minimizes the given cost while satisfying state constraints, and player B wants to prevent player A's success. For each class of problems, this paper presents two HJ equations: one for time-varying dynamics, cost, and state constraint; the other for time-invariant dynamics, cost, and state constraint. Utilizing the HJ equations, the optimal control for each player is analyzed, and a numerical algorithm is presented to compute the solution to the HJ equations. A two-dimensional water system is introduced as an example to demonstrate the proposed HJ framework.