Optimal Strategies for Guarding a Compact and Convex Target Set: A Differential Game Approach

TL;DR

This paper extends the analytical solutions for a two-player pursuit-evasion game to more complex convex target sets, deriving optimal strategies and barrier surfaces using geometric and Isaacs' methods.

Contribution

It develops a generalized solution for guarding convex target sets in pursuit-evasion games, including the derivation of barrier surfaces and optimal feedback strategies.

Findings

Derived the barrier surface for convex target sets.

Established saddle point strategies for both players.

Validated solutions through numerical simulations.

Abstract

We revisit the two-player planar target-defense game initially posed by Isaacs where a pursuer (or defender) attempts to guard a target set from an attack by an evader (or attacker). This paper builds on existing analytical solutions to games of defending a simple shape of target area to develop a generalized and extended solution to the same game with a compact convex target set with smooth boundary. Isaacs' method is applied to address the game of kind and games of degree. A geometric solution approach is used to find the barrier surface that demarcates the winning sets of the players. A value function coupled with a set of optimal state feedback strategies in each winning set is derived and proven to correspond to the saddle point solution of the game. The proposed solutions are illustrated by means of numerical simulations.