Optimal control of mean field equations with monotone coefficients and applications in neuroscience

TL;DR

This paper develops an optimal control framework for mean-field equations with monotone coefficients, applying it to neuroscience models like FitzHugh-Nagumo neuron networks, including existence, maximum principle, and numerical algorithms.

Contribution

It extends optimal control theory to mean-field equations with monotone coefficients, providing existence results, a maximum principle, and numerical methods for practical applications.

Findings

Existence of minimizers established via martingale approach

Derived a maximum principle for the control problem

Numerical gradient algorithm effectively approximates optimal control

Abstract

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^\alpha$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcal L(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $\alpha_t$ is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form $\pi(X_t)\leq0$. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control.