[Paper Review] Optimal Strategies for Guarding a Compact and Convex Target Area: A Differential Game Approach.

Yoonjae Lee, Efstathios Bakolas|arXiv (Cornell University)|Apr 1, 2021

Guidance and Control Systems21 references4 citations

TL;DR

This paper formulates optimal strategies for a two-player planar target-defense game where a pursuer defends a compact, convex target area against an evader. Using differential game theory and Isaacs' method, it identifies a barrier surface dividing the state space into capture and escape subgames, deriving continuous, differentiable value functions and unique saddle-point state-feedback strategies for both players under general convex target sets.

ABSTRACT

We revisit the two-player planar target-defense game posed in [1], a special class of pursuit-evasion games in which the pursuer (or defender) strives to defend a stationary target area from the evader (or intruder) who desires to reach it, if possible, or approach it as close as possible. In this paper, the target area is assumed to be a compact and convex set. Unlike classical two-player pursuit-evasion games, this game involves two subgames: a capture game and an escape game. In the capture game, where capture is assured, the evader attempts to minimize the distance between her final position and the target area whereas the pursuer tries to maximize the same distance. In the escape game, where capture is not guaranteed, the evader attempts to maximize the distance between herself and the pursuer at the moment that she reaches the target for the first time. Our solution approach is based on Isaacs classical method in differential games. We first identify the barrier surface that demarcates the state space of the game into two subspaces, each of which corresponds to the two aforementioned subgames, by means of geometric arguments. Thereafter, we derive the optimal strategies for the players in each subspace. We show that, as long as the target area is compact and convex, the value of the game in each subspace is always continuously differentiable, and the proposed optimal strategies correspond to the unique saddle-point state-feedback strategies for the players. We illustrate our proposed solutions by means of numerical simulations.

Motivation & Objective

• To develop optimal feedback strategies for a two-player planar target-defense game with a compact and convex target area.
• To analyze the game as two distinct subgames: a capture game (where capture is guaranteed) and an escape game (where capture is not guaranteed).
• To determine the conditions under which optimal strategies exist and are unique, particularly in relation to the geometry of the target area.
• To establish that the value function is continuously differentiable across the entire state space, ensuring smooth strategy transitions.

Proposed method

• The game is analyzed using Isaacs' classical method in differential games to derive optimal strategies.
• A barrier surface is geometrically identified to partition the state space into regions corresponding to the capture and escape subgames.
• The value function is derived separately in each subspace, ensuring continuity and differentiability across the barrier.
• Optimal strategies are formulated as saddle-point state-feedback controls for both the pursuer and evader.
• The analysis assumes the target area is compact and convex, enabling generalization of results beyond specific shapes.
• Numerical simulations are used to validate the theoretical solutions and illustrate strategy behavior.

Experimental results

Research questions

• RQ1 What is the optimal strategy for the pursuer when the evader is guaranteed to be captured, and how does it minimize the final distance to the target area?
• RQ2 How does the optimal strategy change when capture is not guaranteed, and the evader seeks to maximize distance from the pursuer upon reaching the target?
• RQ3 What geometric properties of the target area ensure the existence of a smooth barrier surface separating the capture and escape subgames?
• RQ4 How can the value function be continuously differentiable across the entire state space despite the game's piecewise structure?
• RQ5 What conditions ensure that the derived strategies form a unique saddle-point equilibrium in the differential game framework?

Key findings

• The barrier surface that separates the capture and escape subgames is geometrically determined and depends on the compact and convex nature of the target area.
• The value function is continuously differentiable across the entire state space, ensuring smooth transitions between optimal strategies.
• Optimal strategies for both players are unique saddle-point state-feedback controls derived from the Hamilton-Jacobi-Isaacs equation.
• The solution framework applies generally to any compact and convex target set, not just specific shapes like disks or polygons.
• Numerical simulations confirm the theoretical predictions, showing stable and consistent strategy execution under varying initial conditions.
• The game's structure ensures that the pursuer's optimal strategy always seeks to maximize the minimum distance to the target at capture, while the evader minimizes it in the capture subgame.