Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics

TL;DR

This paper extends the theory of controlled piecewise deterministic Markov processes to infinite dimensions, applying it to optimize neuron control in Optogenetics using a PDE-coupled Markov process model.

Contribution

It introduces an infinite-dimensional controlled PDMP framework and demonstrates its application to neuron models like Hodgkin-Huxley for optimal control.

Findings

Existence of optimal relaxed controls established.

Conditions for optimal ordinary controls derived.

Application to neuron control in Optogenetics demonstrated.

Abstract

In this paper we define an infinite-dimensional controlled piecewise deterministic Markov process (PDMP) and we study an optimal control problem with finite time horizon and unbounded cost. This process is a coupling between a continuous time Markov Chain and a set of semilinear parabolic partial differential equations, both processes depending on the control. We apply dynamic programming to the embedded Markov decision process to obtain existence of optimal relaxed controls and we give some sufficient conditions ensuring the existence of an optimal ordinary control. This study, which constitutes an extension of controlled PDMPs to infinite dimension, is motivated by the control that provides Optogenetics on neuron models such as the Hodgkin-Huxley model. We define an infinite-dimensional controlled Hodgkin-Huxley model as an infinite-dimensional controlled piecewise deterministic Markov process and apply the previous results to prove the existence of optimal ordinary controls for a tracking problem.