Stabilizing Optimal Density Control of Nonlinear Agents with Multiplicative Noise

Control of continuous time dynamics with multiplicative noise is a classic topic in stochastic optimal control. This work addresses the problem of designing infinite horizon optimal controls with stability guarantees for large populations of identical, non-cooperative and non-networked agents with multi-dimensional and nonlinear stochastic dynamics excited by multiplicative noise. We provide constraints on the state and control cost functions which guarantee stability of the closed-loop system under the action of the individual optimal controls, for agent dynamics belonging to the the class of reversible diffusion processes. A condition relating the state-dependent control cost and volatility is introduced to prove the stability of the equilibrium density. This condition is a special case of the constraint required to use the path integral Feynman-Kac formula for computing the control. We investigate the connection between the stabilizing optimal control and the path integral formalism, leading us to a control law formulation expressed exclusively in terms of the desired equilibrium density.