The Pontryagin maximum principle and $Q$-functions in rough environments

TL;DR

This paper develops a Pontryagin maximum principle and $Q$-functions for optimal control in noisy rough differential equations, introducing a new differentiation method and a policy improvement algorithm for constrained environments.

Contribution

It introduces a novel differentiation procedure along spike variations and extends $Q$-functions to rough environments, enabling policy improvement with entropic cost constraints.

Findings

Derived Pontryagin maximum principle for rough differential equations

Developed a new differentiation method along spike variations

Proposed a policy improvement algorithm for constrained control

Abstract

We derive the Pontryagin maximum principle and $Q$-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal $Q$-function (also known as the $q$-function) to derive a policy improvement algorithm for settings with entropic cost constraints.