Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds

TL;DR

This paper studies ergodic stochastic control problems on compact manifolds, deriving linear and nonlinear PDEs through variational principles to characterize optimal control strategies and their properties.

Contribution

It introduces a variational framework for ergodic control problems on manifolds, leading to novel linear and nonlinear PDE characterizations of optimal controls.

Findings

Reversible dynamics imply symmetrizable controlled diffusions.

Optimal control problems reduce to linear or nonlinear PDEs depending on constraints.

Eigenvalue problems characterize reversible controlled processes.

Abstract

We consider long term average or `ergodic' optimal control poblems with a special structure: Control is exerted in all directions and the control costs are proportional to the square of the norm of the control field with respect to the metric induced by the noise. The long term stochastic dynamics on the manifold will be completely characterized by the long term density $\rho$ and the long term current density $J$. As such, control problems may be reformulated as variational problems over $\rho$ and $J$. We discuss several optimization problems: the problem in which both $\rho$ and $J$ are varied freely, the problem in which $\rho$ is fixed and the one in which $J$ is fixed. These problems lead to different kinds of operator problems: linear PDEs in the first two cases and a nonlinear PDE in the latter case. These results are obtained through through variational principle using infinite dimensional Lagrange multipliers. In the case where the initial dynamics are reversible we obtain the result that the optimally controlled diffusion is also symmetrizable. The particular case of constraining the dynamics to be reversible of the optimally controlled process leads to a linear eigenvalue problem for the square root of the density process.