Optimal search model with compulsory stop for a randomly arriving target

One target appears in one of regions at a random time and will thereafter remain there. The searcher has the available total search time T. Exponential-type detection functions are assumed. The search is terminated compulsorily at a certain random time. The problem is to maximize a probability of detecting the target. Necessary and sufficient conditions for an optimal policy are derived and a sequential allocation method of search time is proposed. The case of one region is solved explicitly and an example of this case is given.