On Minimum-Dispersion Control of Nonlinear Diffusion Processes

TL;DR

This paper introduces a novel method for controlling nonlinear stochastic diffusion processes by leveraging $ abla$-order variational analysis, enabling efficient learning of time-dependent controls through Monte Carlo simulations.

Contribution

It develops an $ abla$-order variational analysis framework to analytically represent cost increments, facilitating the design of law-feedback controls for nonlinear diffusions.

Findings

The method effectively learns time-dependent control coefficients.

Numerical experiments demonstrate the approach's viability.

The framework simplifies the control problem via duality and linearization.

Abstract

This work collects some methodological insights for numerical solution of a "minimum-dispersion" control problem for nonlinear stochastic differential equations, a particular relaxation of the covariance steering task. The main ingredient of our approach is the theoretical foundation called $\infty$-order variational analysis. This framework consists in establishing an exact representation of the increment ($\infty$-order variation) of the objective functional using the duality, implied by the transformation of the nonlinear stochastic control problem to a linear deterministic control of the Fokker-Planck equation. The resulting formula for the cost increment analytically represents a "law-feedback" control for the diffusion process. This control mechanism enables us to learn time-dependent coefficients for a predefined Markovian control structure using Monte Carlo simulations with a modest population of samples. Numerical experiments prove the vitality of our approach.