Pursuit-Evasion in Minimal Time with Varying Observation Constraints

Optimal control problems under incomplete information, particularly in pursuit-evasion scenarios, present significant mathematical challenges. This study extends a basic time-optimal pursuit-evasion game by introducing a time-dependent observer parameter, $\lambda(t)$, which enhances the model's realism without altering the fundamental control strategy. We derive an optimal control law for the pursuer, based on current observations, and explicitly calculate the minimum capture time for a piecewise constant $\lambda(t)$. This work provides an analytical framework for managing uncertainty in dynamic environments, with direct applications in robotics, autonomous navigation, and search-and-rescue operations.