Lion and Man problems are classical examples of pursuit and evasion games. However, the traditional analytic methods and indirect numerical methods only can handle the generalization of Lion and Man problems in games with small scales and simple scenarios. In this study, we first extend the original Lion and Man problems to a more complicated and time-varying game environment. Then we propose the simultaneous orthogonal collocation decomposition (SOCD) method, which is a direct method for exploring solutions of Lion and Man problems in a complicated game environment. Compared to indirect methods, SOCD method is much easier to apply. The max-minimization problem in Lion and Man problems is decomposed into two subproblems of optimal control, which are discretized by using the orthogonal collocation method. Local solutions of the resulting nonlinear programming problems lead to the optimal control problems. We also develop the receding horizon optimization method based on SOCD method to solve Lion and Man problems online in a time-varying game environment. In this method, the whole optimization time domain is divided into several short optimization cycles, and Lion and Man problems in each cycle are based on real-time observations of the game environment. The validity of these two methods is tested by conducting numerical simulations, and the results demonstrate that these methods provide a unified framework for solving extended Lion and Man problems.

中文翻译：

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扩展的Lion and Man问题的同时正交搭配分解方法

Lion and Man问题是追逐和逃避游戏的经典示例。但是，传统的分析方法和间接数值方法只能处理小规模和简单场景游戏中Lion and Man问题的泛化。在这项研究中，我们首先将原始的《狮子与人》问题扩展到一个更复杂且随时间变化的游戏环境。然后，我们提出了同时正交配置分解法（SOCD），这是一种在复杂游戏环境中探索Lion and Man问题解决方案的直接方法。与间接方法相比，SOCD方法更易于应用。《狮子与人》中的最大最小化问题问题被分解为两个最优控制子问题，通过正交配置法离散化。所产生的非线性规划问题的局部解导致最优控制问题。我们还开发了基于SOCD方法的后退视野优化方法，以在时变游戏环境中在线解决Lion and Man问题。在这种方法中，整个优化时域被分为几个短的优化周期，每个周期的Lion和Man问题都是基于对游戏环境的实时观察。通过数值模拟测试了这两种方法的有效性，结果表明这两种方法为求解扩展的Lion and Man提供了一个统一的框架。 问题。