Guarding a Target Set from a Single Attacker in the Euclidean Space

TL;DR

This paper develops a comprehensive differential game framework for target defense in high-dimensional Euclidean spaces, providing analytical solutions and barrier surfaces to determine winning strategies for attacker and defender.

Contribution

It introduces a universal differential game solution that handles complex high-dimensional target sets and derives semi-analytical value functions with barrier surfaces for optimal strategies.

Findings

Explicit value functions for the game are derived.

Barrier surfaces are constructed to separate winning regions.

Optimal strategies form a unique saddle point verified by Hamilton-Jacobi-Isaacs equation.

Abstract

This paper addresses a two-player target defense game in the $n$-dimensional Euclidean space where an attacker attempts to enter a closed convex target set while a defender strives to capture the attacker beforehand. We provide a complete and universal differential game-based solution which not only encompasses recent work associated with similar problems whose target sets have simple, low-dimensional geometric shapes, but can also address problems that involve nontrivial geometric shapes of high-dimensional target sets. The value functions of the game are derived in a semi-analytical form that includes a convex optimization problem. When the latter problem has a closed-form solution, one of the value functions is used to analytically construct the barrier surface that divides the state space of the game into the winning sets of players. For the case where the barrier surface has no analytical expression but the target set has a smooth boundary, the bijective map between the target boundary and the projection of the barrier surface is obtained. By using Hamilton-Jacobi-Isaacs equation, we verify that the proposed optimal state feedback strategies always constitute the game's unique saddle point whether or not the optimization problem has a closed-form solution. We illustrate our solutions via numerical simulations.