Ensemble Control of Stochastic Linear Systems

Abstract: In this paper, we consider the problem of steering a family of independent, structurally identical, finite-dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by using an open-loop control function. Our exploration of this class of control problems, which falls under the rising subject of ensemble control, is motivated by pulse design problems in quantum control. Here we extend the concept of ensemble control to stochastic systems with additive diffusion and jump processes, which we model using Brownian motion and Poisson counters, respectively, and consider optimal steering problems. We derive a Fredholm integral equation that is used to solve for the optimal control, which minimizes both the mean square error (MSE) and the error in the mean of the terminal state. In addition, we present several example control problems for which optimal solutions are computed by numerically approximating the singular system of the associated Fredholm operator. We use Monte Carlo simulations to illustrate the performance of the resulting controls. Our work has immediate practical applications to the control of dynamical systems with additive noise and parameter dispersion, and also makes an important contribution to stochastic control theory.