Randomized look strategy for a moving target when a search path is given

This paper investigates the search problem for a moving target when a search path is given in advance. The searcher's strategy is represented by a randomized look strategy, that is, the probability with which he looks in his current position at each discrete time. The searcher knows the probabilistic law about which one of some options the target selects as his path. If the detection of the target occurs by looking, the searcher gains some value but must expend some cost for looking. The searcher wants to determine his optimal look strategy in order to maximize the expected reward which is defined as the expected value minus the expected cost. He can randomize his look strategy by the probability that he looks in his current position. We prove the NP-completeness of this problem and propose a dynamic programming method to give an optimal solution which becomes the bang-bang control in the result. We derive some characteristics of the optimal look strategy and analyze them by some numerical examples.