Optimal Strategies for Pursuit-evasion Differential Games of Players with Damped Double Integrator Dynamics

In this article, we investigate and derive optimal strategies for the pursuit-evasion problem of two players with damped double integrator dynamics. The pursuer is assumed to have a greater acceleration and attempt to capture the evader as quickly as possible, while the evader wants to avoid or delay the capture. Different from the problems with single integrator dynamics, optimal strategies for such a problem are not intuitive. To solve this problem, we exploit both geometrical methods and differential games. Specifically, we first illustrate some properties of the optimal strategies, using which isochrones of the players are constructed. Then, by exploring the geometrical characteristics of the isochrones of the two players, we derive the target point directly without using the explicit formula of the intersections of isochrones. We propose an algorithm to determine the capture time, which is exactly the value function of the game, and finally the optimal strategies are derived subsequently. To prove the optimality of the proposed strategies, differential games and Hamilton-Jacobi-Isaacs (HJI) equation are used, where we show that the proposed value function is the solution to the HJI equation under our proposed strategies. Moreover, we apply our results to the game with one pursuer and multiple evaders, and some useful results are given. Finally, we carry out various simulations to show the optimality and effectiveness of our proposed strategies.