Three problems in searching for a moving target between two sites

Suppose that a moving target moves randomly between two sites and its movement is modeled by a homogeneous Markov chain. We consider three classical problems: (1) what kind of strategies are valid? (2) what strategy is the optimal? (3) what is the infimum of expected numbers of looks needed to detect the target? Problem (3) is thoroughly solved, and some partial solutions to problems (1) and (2) are achieved.