  We address optimal placement of vehicles with simple motion to intercept a
mobile target that arrives 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.
