Optimal search for a passing target

A problem of catching a target which starts from box 1, passes through box 2 and leaves to box 3 is considered. Only box 2 is searchable and the search is partially detectable. At each period the searcher selects one of three decisions: wait, look and stop. A searching cost, a waiting cost and a reward of detection are introduced. The problem is to minimize the expected net cost until the process terminates (detection or stop). Some properties of the optimal policy are obtained, a method of obtaining the optimal policy is presented and some numerical examples are given. Furthermore a solution for an analogous model of continuous time is derived.