保护边界安全一直以来是一个热点问题, 边界线形状的复杂多变增加了边界防御问题研究的复杂性。本文通过对边界防御场景的刻画, 将问题转化为追逃微分博弈问题, 进而求解出此博弈中局中人的最优策略, 并给出切实可行的算法。本文运用简单运动刻画局中人的运动, 运用二次曲线的一般方程形式刻画边界形状, 定义支付函数为捕获点到边界的距离。通过几何方法对问题进行研究, 得到了圆形边界情形下更加简洁的值函数形式, 并构造出局中人的最优策略。在圆形边界的基础上, 进一步得到针对二次曲线边界情形下的通用算法。最后对模型进行从定量到定性、从二维到三维、从多追一到多追多的完善, 并通过数值仿真验证了结论与算法的正确性。

Border security has always been a topic sparking intense debate. Moreover, border morphology realities separating countries are quite complex, exacerbating the complexity of border defence issues. As per relevant literature reviews, there is no perfect universal solution to address border defence problems, which makes it a matter of utmost concern. This paper aims to transform the problem into a pursuit-evasion differential game problem by delineating the border defence scenario. The method of differential game is subsequently used to solve the optimal strategy of the players, which ultimately yields a feasible algorithm. Specifically, this paper uses simple motion to describe player movement in the game, utilises the general equation form of the quadratic curve to demarcate the shape of the boundary, and defines the payoff function as the distance from the capture point to the boundary. The problem is studied by the geometric method, and a more concise value function form is yielded in the case of a circular boundary. The optimal strategy of the player in the game is ultimately constructed. A general algorithm for the quadratic curve boundary is further obtained based on the circular boundary. The model is then stretched from the game of degree to the game of kind, from two dimensions to three dimensions, from M-pursuers against a single evader to M-pursuers against N-evaders, and the result and algorithm are validated by using a numerical simulation.