Optimal Search Among False Contacts
D. V. Kalbaugh · SIAM Journal on Applied Mathematics · 1992
This paper considers rational methods for the design of search patterns when the intended target is among Poisson-distributed false contacts. Attention is restricted to situations in which the searcher must immediately decide, based on sensed information, whether a contact is the intended target and, if the answer is yes, end the search and commit to the contact. Discrimination of false contact from target is modeled as reliable only with given probabilities. This research seeks to maximize the probability of finding the intended target in a fixed time. Most of the paper addresses an optimal search trajectory when intended target and false contacts move according to Markov processes. The searcher’s velocity is taken as the control, and it is assumed that his speed is limited and his initial position is given. It is proved that an optimal control exists in the space $L_\infty $. A set of necessary conditions for the optimal control is derived: a maximum principle in the form of a two-point boundary value problem in both ordinary and partial differential equations. A computer algorithm is outlined that uses the necessary conditions as a guide to iteratively improve a supplied initial trajectory. The paper also derives closed-form expressions for the probability of finding an intended target in $R^2 $ when target and false contacts are stationary and detection is modeled as definite, i.e., detection probability either 0 or 1 and under complete control of the searcher out to a fixed distance. Various search patterns are evaluated.