Guarding a convex target set in a particular case of the game of two cars

This paper presents a geometric solution framework for a target defense problem, formulated as a variant of the classical Game of Two Cars. The setting considers a Dubins defender that is faster and more maneuverable and aims to intercept a Dubins attacker attempting to reach a convex target set. To address the computational complexity of solving the associated Hamilton-Jacobi-Isaacs (HJI) equations, a geometric approach based on the concept of the Attacker Dominance Region (ADR) is developed. The ADR is constructed piecewise from the boundaries of the players’ reachable sets. The complete solution consists of two components: a Game of Kind, which determines the outcome based on the spatial relationship between the ADR and the target set, and a Game of Degree, which derives optimal strategies that achieve equilibrium. Simulation results demonstrate the effectiveness of the proposed method under realistic motion constraints and indicate its potential applicability to practical target defense scenarios.