Necessary and Sufficient Conditions for Optimal Search Plans for Moving Targets

Abstract

Let {Xt, t ≥ 0} be a stochastic process representing the motion of a target in Euclidean n-space. Search effort is applied at the rate m(t) > 0 for 0 ≤ t ≤ T, and it is assumed that this effort can be distributed as finely as desired over space. We seek an optimal search plan, i.e., an allocation of effort in time and space which maximizes the probability of detecting the target by time T. When the detection function is concave, theorem 1′ of this paper gives necessary and sufficient conditions for an optimal search plan for a class of target motion processes {Xt, t ≥ 0} which includes virtually any reasonable model of target motion. In the special case of a discrete-time target motion process and an exponential detection function, the necessary and sufficient conditions have the following intuitive interpretation: For t = 0, 1, 2, …, T, let g̃t, be the probability distribution of the target's location at time t given that the effort at all times before and after t failed to detect the target. The optimal plan allocates the effort for time t so as to maximize the probability of detecting a stationary target with distribution g̃t within the effort constraint m(t). This special case is a generalization of Brown's (Brown, S. S. 1978. Optimal search for a moving target in discrete space and time. Submitted for publication.) result for discrete time and space target motion.