On vehicle placement to intercept moving targets

We address optimal placement of vehicles with simple motion to intercept a mobile target that is generated stochastically on a line segment. The optimality of vehicle placement is measured through a cost function associated with intercepting the target. With a single vehicle, we assume that the target moves (i) with fixed speed and in a fixed direction perpendicular to the line segment, or (ii) to maximize the distance from the line segment, or (iii) to maximize intercept time. In each case, we show that the cost function is strictly convex, its gradient is smooth, and the optimal vehicle placement is obtained by a standard gradient-based optimization technique. With multiple vehicles, we assume that the target moves with fixed speed and in a fixed direction perpendicular to the line segment. We present a discrete-time partitioning and gradient-based algorithm, and characterize conditions under which the algorithm asymptotically leads the vehicles to a set of critical configurations of the cost function.