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

Abstract

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^α$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),α_t)dt+σ(t,X_t,\mathcal L(X_t),α_t)dW_t,$ $X_0\sim μ$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $α_t$ 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 $π(X_t)\leq0$. 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.