Solutions for Multiple-Defender Single-Attacker Convex Target Guarding Games: Verification via Parametric Optimization

A zero-sum differential game is considered in which a team of defenders is tasked with cooperatively capturing a single attacker who seeks to breach a convex region (or target). We show that the equilibrium strategies of the agents can be computed via a convex program, which remains tractable even in high-dimensional cases as opposed to the direct solution of the Hamilton-Jacobi-Isaacs (HJI) equation. To that end, we mainly attempt to prove the validity of a candidate for the Value of the game constructed based on Isaacs' geometric approach using classical/viscosity solution concepts of the HJI equation. A major challenge is that the candidate Value function generally lacks an analytical expression, making it difficult to derive its gradient/subdifferential. To address this challenge, we propose a parametric optimization approach to associate the candidate Value function with the optimal value function of a parametric program and investigate its continuity and differentiability properties through duality and optimality conditions. Lastly, we present a numerical example to showcase the applicability of our solution in a nontrivial high-dimensional game scenario.