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 attacker attempts to enter closed convex set whereas defender strives capture the beforehand. We propose generalized differential game-based solution model that can not only encompass recent work associated with similar problems whose sets have simple low-dimensional geometric shapes, but also address involve nontrivial shapes of high-dimensional sets. The has two value functions, each which is derived semi-analytical form including optimization problem admits unique solution. If latter solutions closed-form expressions, barrier surface divides state into winning players be analytically constructed. For case there exists no smooth boundary, map between boundary and projection on derived. By using Hamilton-Jacobi-Isaacs equation, we show our feedback strategies always constitute saddle point game. verify related provide numerical simulation results.