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

Abstract

Long term average or ‘ergodic’ optimal control problems on a compact manifold are considered. The problems exhibit a special structure which is typical of control problems related to large deviations theory: 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 ρ and the long term current density J. As such, control problems may be reformulated as variational problems over ρ and J. The density ρ is paired in the cost functional with a state dependent cost function V, and the current density J is paired with a vector potential or gauge field A. We discuss several optimization problems: the problem in which both ρ and J are varied freely, the problem in which ρ 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 a variational principle using infinite dimensional Lagrange multipliers. In the case where the initial dynamics are reversible the optimally controlled diffusion is also reversible. 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.