Barrier Construction of the Homicidal Chauffeur Game Considering Speed Descent of the Car

The Homicidal Chauffeur game is a classic pursuit-evasion problem for its significant application in aerospace, robot control, and military. Different from the well-known Isaacs' Homicidal Chauffeur game in which the car's speed is constant, we assume that the car's speed decreases during turns, reflecting the energy limitations in real-world applications. The paper aims to derive the optimal strategies and construct a barrier separating the capture and evasion zones. Initially, we simplify the six- dimensional problem to a three-dimensional one. Subsequently, we deduce the optimal strategy and construct the barrier using the Hamilton-Jacobi-Isaacs equation and backward induction method. A new barrier appears where the optimal strategy of the pursuer is different from the traditional one. That is, when the pursuer's speed is greater than a critical value, the optimal strategy is to apply maximum control; when the pursuer's speed is less than the critical speed, the optimal strategy is to apply zero control.