This paper deals with the design of time-invariant memoryless control\npolicies for robots that move in a finite two- dimensional lattice and are\ntasked with persistent surveillance of an area in which there are forbidden\nregions. We model each robot as a controlled Markov chain whose state comprises\nits position in the lattice and the direction of motion. The goal is to find\nthe minimum number of robots and an associated time-invariant memoryless\ncontrol policy that guarantees that the largest number of states are\npersistently surveilled without ever visiting a forbidden state. We propose a\ndesign method that relies on a finitely parametrized convex program inspired by\nentropy maximization principles. Numerical examples are provided.\n