L-catch and L-escape Differential Game

In this work, we investigate a simple motion differential game involving one pursuer and one evader, where the dynamics of the pursuer and the evader are governed by first-order and second-order differential equations, respectively. The control functions for both players are assumed to satisfy generalized integral constraints, which limit their maneuverability over time. Our main focus is on the analysis of pursuit under the l-capture condition. Specifically, we demonstrate that pursuit is completed when the l-distance between the positions of the pursuer x(t) and the evader y(t) becomes less than or equal to a given threshold l, that is, , at some finite time t. We derive an explicit formula for the guaranteed time of pursuit, which holds for all admissible strategies of the evader. Additionally, we show that an optimal escape strategy for the evader, known as l-escape, can also be realized at this same optimal time. This simultaneous realization of pursuit and escape conditions indicates a balance in the strategies of both players, thus leading to an optimal pursuit time. Our results contribute to the broader understanding of differential games and optimal control theory.