Schrödinger Approach to Optimal Control of Large-Size Populations

Abstract

Large-size populations consisting of a continuum of identical and non-cooperative agents with stochastic dynamics are useful in modeling various biological and engineered systems. This paper addresses the problem of designing mean-field type optimal state-feedback controllers which guarantee closed-loop stability of the stationary density of such agents with nonlinear Langevin dynamics, under the action of their individual steady state controls.We represent the corresponding optimality system which consists of coupled forward-backward PDEs as decoupled Schrödinger equations, by introducing a novel variable transform. Spectral properties of the Schrödinger operator which underlie the stability analysis are used to obtain explicit control design constraints, in the case that the agents do not interact explicitly via density dependent dynamics or cost function. Our interpretation of the Schrödinger potential as the cost function of a closely related optimal control problem motivates a quadrature based algorithm to compute the finite time optimal control 1 . We show the deep connection between the nonlinear Schrödinger equation and Mean Field Games for agents with nonlinear Langevin dynamics and explicit interactions.