Here is the setup. A prisoner is being held in the center a circular yard with radius $r$ and can run in any direction at some velocity $v$, there is a spotlight which illuminates a line on the circle and can rotate with angular velocity $\frac{d\theta}{dt}$, the spotlight can change direction at will.
The prisoner is free if they get out of the yard without being spotted, and loses if the spotlight catches them at any point. The prisoner must run in a straight line, and cannot run back after starting but must complete the journey (they are rather dull.. other than knowing a bunch of probability and game theory).
Is there an optimal strategy for the prisoner and how does it depend on the parameters? We assume that the spotlight is following its optimal strategy, so the question of what the spotlight will do comes first.
For example, if the spotlight can make an entire sweep in less time than it takes for the prisoner to run to the edge, then the optimal strategy for the spotlight is to sweep consistently in one direction, knowing that if the prisoner runs it will catch them guaranteed. This is a bad case because there is no optimal strategy for the prisoner, they will be caught no matter what.