Jump Control of Probability Densities With Applications to Autonomous Vehicle Motion

We investigate the problem of controlling the probability density of the state of a process that is observed by the controller via a fixed but unknown scalar non-negative function of the state. The goal is to control the process so that its probability density at a point in the state-space becomes proportional to the value of the function observed at that point. Our solution, inspired by bacterial chemotaxis, involves a randomized controller that switches among different deterministic modes. We show that under appropriate existence conditions, this controller guarantees convergence of the probability density to the desired function. The results can be applied to the problem of in loco optimization of a measurable signal using a team of autonomous vehicles that measure the signal but do not have access to position measurements. Alternative applications in the area of mobile robotics include deployment and environmental monitoring.