Optimal periodic patrolling trajectories of UUVs guarding a channel

Given a number of patrollers, the channel patrol problem consists of determining the periodic trajectories that the patrollers must trace out so as to maximize the probability of detection of the intruder. We formulate this problem as an optimal control problem. We assume that the patrollers' sensors are imperfect and that their motions are subject to turn rate constraints, and that the intruder travels straight down a channel with constant speed. Using discretization of time and space, we approximate the optimal control problem with a large-scale nonlinear programming problem which we solve to obtain an approximately stationary solution and a corresponding optimized trajectory for each patroller. In numerical tests, we obtain new insight - not easily obtained using geometric calculations - into efficient patrol trajectory design for up to two patrollers in a narrow channel where interaction between the patrollers is unavoidable due to their limited turn rate.