Value of Multi-pursuer Single-evader Pursuit-evasion Game with Terminal Cost of Evader's Position: Relaxation of Convexity Condition

TL;DR

This paper analyzes a multi-pursuer single-evader pursuit-evasion game with a terminal cost, proving the value function is a viscosity solution of the HJI PDE without requiring convexity of the terminal cost, thus broadening applicability.

Contribution

It relaxes the convexity condition on the terminal cost, proving the value function is a viscosity solution under weaker assumptions, and provides a method to compute it.

Findings

Value function is the viscosity solution of the HJI PDE.

Convexity of terminal cost is not required, only local Lipschitz continuity.

Method applicable to non-convex terminal costs or targets.

Abstract

In this study, we consider a multi-pursuer single-evader quantitative pursuit-evasion game with payoff function that includes only the terminal cost. The terminal cost is a function related only to the terminal position of the evader. This problem has been extensively studied in target defense games. Here, we prove that a candidate for the value function generated by geometric method is the viscosity solution of the corresponding Hamilton-Jacobi-Isaacs partial differential equation (HJI PDE) Dirichlet problem. Therefore, the value function of the game at each point can be computed by a mathematical program. In our work, the convexity of the terminal cost or the target is not required. The terminal cost only needs to be locally Lipschitz continuous. The cases in which the terminal costs or the targets are not convex are covered. Therefore, our result is more universal than those of previous studies, and the complexity of the proof is improved. We also discuss the optimal strategies in this game and present an intuitive explanation of this value function.