Log In Sign Up

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

by   Antoine Hocquet, et al.

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 ℝ^d dX_t=b(t,X_t,ℒ(X_t),α_t)dt+σ(t,X_t,ℒ(X_t),α_t)dW_t, X_0∼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)≤0. 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.


page 1

page 2

page 3

page 4


Mean-field Langevin System, Optimal Control and Deep Neural Networks

In this paper, we study a regularised relaxed optimal control problem an...

A Mean-Field Optimal Control Formulation of Deep Learning

Recent work linking deep neural networks and dynamical systems opened up...

Optimal control and stablilization for linear continuous-time mean-field systems with delay

This paper studies optimal control and stabilization problems for contin...

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identi...

Sparse solutions in optimal control of PDEs with uncertain parameters: the linear case

We study sparse solutions of optimal control problems governed by PDEs w...

Mean-Field Control Approach to Decentralized Stochastic Control with Finite-Dimensional Memories

Decentralized stochastic control (DSC) considers the optimal control pro...

Accelerated Flow for Probability distributions

This paper presents a methodology and numerical algorithms for construct...