Schr\"odinger Approach to Optimal Control of Large-Size Populations

TL;DR

This paper introduces a novel Schr"odinger equation-based method for designing optimal controllers for large populations with stochastic dynamics, ensuring stability and providing explicit control constraints.

Contribution

It reformulates coupled PDEs as decoupled Schr"odinger equations, enabling explicit control design constraints and a new algorithm for finite-time optimal control.

Findings

Spectral analysis of Schr"odinger operators informs stability conditions.

Decoupling PDEs simplifies the control design process.

Proposed algorithm computes finite-time optimal controls efficiently.

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 stochastic control problem of designing optimal state-feedback controllers which guarantee the 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 coupled forward-backward PDEs as decoupled Schr\"odinger equations, by applying two variable transforms. Spectral properties of the linear Schr\"odinger operator which underlie the stability analysis are used to obtain explicit control design constraints. Our interpretation of the Schr\"odinger potential as the cost function of a closely related optimal control problem motivates a quadrature based algorithm to compute the finite-time optimal control.