Minimum Transmission Energy Trajectories for a Linear Pursuit Problem

Abstract

In this paper we study a pursuit problem in the context of a wireless sensor network, where the pursuer (i.e., mobile sink) trying to capture a pursuee (i.e., tracked object), moving with constant velocity, is always directly communicating with a sensor node in the very near proximity of the pursuee. Assuming that the sensor nodes can adjust their transmission power depending on the distance ρ between the pursuer and pursuee according to the usual power law ρ ^− α, the task is to find the optimal trajectory of the pursuer minimizing the total transmission energy. We approach this classical control theoretic problem by the method of dynamic programming. The cost function, describing the transmission cost with an optimal policy, factorizes into radial and angular functions. The partial differential equation governing the cost function can then be reduced to an ordinary differential equation for the angular function. This equation as well as the related optimal trajectories can be solved numerically. The qualitative behavior of the trajectories is also discussed. The trajectories are self-similar in the sense that any magnification of an optimal trajectory is also an optimal trajectory for different initial conditions.