  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.
