DeepAI
Log In Sign Up

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

07/02/2020
by   Antoine Hocquet, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

09/16/2019

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

In this paper, we study a regularised relaxed optimal control problem an...
07/03/2018

A Mean-Field Optimal Control Formulation of Deep Learning

Recent work linking deep neural networks and dynamical systems opened up...
10/15/2020

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

This paper studies optimal control and stabilization problems for contin...
09/21/2020

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identi...
04/16/2018

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...
09/12/2022

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

Decentralized stochastic control (DSC) considers the optimal control pro...
01/10/2019

Accelerated Flow for Probability distributions

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