The scope of this article is the well-known wall pursuit game, which has been used in the literature to illustrate the existence of a singular surface (dispersal line) and the associated game dilemma. We derive an analytical expression for the value function of the game, which is the viscosity solution of the Hamilton–Jacobi–Isaacs equation. Then, we introduce a hold time analysis and the rate of change for the loss of time to capture along the dispersal line, and show that the rate has a well-defined saddle point along the dispersal line, which can be used to resolve the dilemma. Moreover, we prove that the saddle point of the rate characterizes optimal game actions not only on the dispersal line, but also for all other states of the game. Finally, we analyze the same game in a version with a nonzero hold time and show that in that case, the actions from the dispersal line have to be applied both on the dispersal line and in a narrow band around it. To illustrate that, we use an example to compute the band around the line.

中文翻译：

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基于深度学习的FPGA脉冲激光测距方法研究


本文的范围是著名的追墙游戏，该游戏已在文献中用于说明奇异表面（分散线）的存在以及相关的游戏困境。我们推导出博弈价值函数的解析表达式，它是 Hamilton-Jacobi-Isaacs 方程的粘度解。然后，我们引入了保持时间分析和沿扩散线捕获时间损失的变化率，并表明该速率沿扩散线有一个明确定义的鞍点，可用于解决困境。此外，我们证明速率的鞍点不仅表征了分散线上的最佳游戏行为，而且还表征了游戏的所有其他状态。最后，我们分析了具有非零保持时间的版本中的同一游戏，并表明在这种情况下，来自分散线的动作必须应用于分散线及其周围的窄带中。为了说明这一点，我们使用一个示例来计算线周围的频带。