Optimal Control of a Large Population of Randomly Interrogated Interacting Agents

This article investigates a stochastic optimal control problem with linear Gaussian dynamics, quadratic performance measure, but non-Gaussian observations. The linear Gaussian dynamics characterizes a large number of interacting agents evolving under a centralized control and external disturbances. The aggregate state of the agents is only partially known to the centralized controller by means of the samples taken randomly in time and from anonymous randomly selected agents. Due to the removal of the agent identity from the samples, the observation set has a non-Gaussian structure, and as a consequence, the optimal control law that minimizes a quadratic cost is essentially nonlinear and infinite-dimensional, for any finite number of agents. For infinitely many agents, however, this article shows that the optimal control law is the solution to a reduced order, finite-dimensional linear quadratic Gaussian problem with Gaussian observations sampled only in time. For this problem, the separation principle holds and is used to develop an explicit optimal control law by combining a linear quadratic regulator with a separately designed finite-dimensional minimum mean square error state estimator. Conditions are presented under which this simple optimal control law can be adopted as a suboptimal control law for finitely many agents.