On approximations of stochastic optimal control problems with an application to climate equations

TL;DR

This paper investigates controlling slow variables in multi-scale stochastic systems, especially in climate models, showing that controlling fast scales can effectively influence slow dynamics through advanced stochastic analysis techniques.

Contribution

It introduces a novel approach to approximate stochastic control problems with multi-scale dynamics, demonstrating control of slow variables via fast-scale actions without averaging.

Findings

Convergence of uncontrolled problems implies convergence of optimal controls and costs.

Control can be achieved by acting solely on fast scales in multi-scale stochastic systems.

The framework applies to systems with unbounded control actions, relevant for climate modeling.

Abstract

The paper is devoted to the optimal control of a system with two time-scales, in a regime when the limit equation is not of averaging type but, in the spirit of Wong-Zakai principle, it is a stochastic differential equation for the slow variable, with noise emerging from the fast one. It proves that it is possible to control the slow variable by acting only on the fast scales. The concrete problem, of interest for climate research, is embedded into an abstract framework in Hilbert spaces, with a stochastic process driven by an approximation of a given noise. The principle established here is that convergence of the uncontrolled problem is sufficient for convergence of both the optimal costs and the optimal controls. This target is reached using Girsanov transform and the representation of the optimal cost and the optimal controls using a Forward Backward System. A challenge in this program is represented by the generality considered here of unbounded control actions.