On Minimum-Dispersion Control of Nonlinear Diffusion Processes

TL;DR

The paper addresses how to control nonlinear diffusion processes to minimize dispersion in a stochastic population, offering an alternative to covariance steering. It introduces $\infty$-order variational analysis to obtain an exact cost increment via duality, yielding a law-feedback control within a Fokker-Planck framework that turns the problem into a linear measure dynamics. A practical descent approach combines this duality with the Krasovskii-Subbotin algorithm and Monte Carlo adjoint computations to learn time-dependent control coefficients under a predefined Markovian structure. Numerical experiments on a Theta-model neuron population demonstrate effective dispersion reduction and robustness, with potential scalability through Monte Carlo and parallelization despite computational challenges.

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.

Figures (1)

• Figure 1: Sample paths produced by the initial (upper) and learned (bottom) controls.