Stochastic Optimal Control as Non-equilibrium Statistical Mechanics: Calculus of Variations over Density and Current

TL;DR

This paper reformulates stochastic optimal control as a non-equilibrium statistical mechanics problem, deriving a gauge-invariant Hamilton-Jacobi-Bellman equation through a variational approach over density and current, with applications to ergodic control.

Contribution

It introduces a novel variational framework over density and current for SOC, incorporating a gauge-invariant Hamilton-Jacobi-Bellman equation and hydrodynamic interpretation.

Findings

Derivation of a generalized gauge-invariant HJB equation.

Hydrodynamic interpretation of the control problem.

Application to ergodic control of a particle in a circle.

Abstract

In Stochastic Optimal Control (SOC) one minimizes the average cost-to-go, that consists of the cost-of-control (amount of efforts), cost-of-space (where one wants the system to be) and the target cost (where one wants the system to arrive), for a system participating in forced and controlled Langevin dynamics. We extend the SOC problem by introducing an additional cost-of-dynamics, characterized by a vector potential. We propose derivation of the generalized gauge-invariant Hamilton-Jacobi-Bellman equation as a variation over density and current, suggest hydrodynamic interpretation and discuss examples, e.g., ergodic control of a particle-within-a-circle, illustrating non-equilibrium space-time complexity.