Guarding a Convex Target Set From an Attacker in Euclidean Spaces

This letter addresses a two-player target defense game in an $n$ -dimensional Euclidean space where an attacker attempts to enter a closed convex target set whereas a defender strives to capture the attacker beforehand. We propose a generalized differential game-based solution model that can not only encompass 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 convex target sets. The game has two value functions, each of which is derived in a semi-analytical form including an optimization problem that admits a unique solution. If the latter solutions have closed-form expressions, the barrier surface that divides the state space of the game into the winning sets of players can be analytically constructed. For the case where there exists no closed-form solution but the target set has a smooth boundary, the map between the target boundary and the projection of barrier on the game space is derived. By using the Hamilton-Jacobi-Isaacs equation, we show that our semi-analytical state feedback strategies always constitute the unique saddle point of the game. We verify our solutions with related work and provide numerical simulation results.