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Solutions for Games of Guarding an Arbitrary Convex Target Set with Multiple Defenders in R^n
  • Yoonjae Lee ,
  • Efstathios Bakolas
Yoonjae Lee
University of Texas at Austin

Corresponding Author:[email protected]

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Efstathios Bakolas
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Abstract

This paper addresses a target-guarding differential game involving a single attacker, a group of faster defenders, and a convex target set. The goal of the defenders is to guard the target set, which is assumed to be a convex subset of R^n, from the attacker who attempts to reach the same set. In contrast to standard geometric formulations of this type of problems in the literature, we propose a new solution approach based on parametric optimization. In the proposed approach, the high- dimensional game is reduced into a parametric convex program, whose parameter corresponds to the state of the game and its objective is to find the capture point that determines the optimal policies of the players. The solution of this program is proven to be the unique fixed point of the composite projection operator associated with the feasible set of the program and the target set. An algorithm to find the latter solution is presented. We further show that, under certain conditions, the value function of the program is continuously differentiable and satisfies the Hamilton- Jacobi-Isaacs equation, thereby verifying its equivalence with the Value function of the game. Lastly, we highlight the effectiveness of our approach by means of numerical simulations.
2024Published in IEEE Transactions on Automatic Control on pages 1-8. 10.1109/TAC.2024.3379948