# Optimal control of infinite-dimensional Piecewise Deterministic Markov Processes: a BSDE approach. Application to the control of an excitable cell membrane

## TL;DR

This paper develops a novel approach using BSDEs to solve optimal control problems for infinite-dimensional PDMPs, with applications to controlling excitable cell membranes modeled by Hodgkin-Huxley equations.

## Contribution

It introduces a BSDE-based representation for the value function of infinite-dimensional PDMPs and proves uniqueness of solutions to the associated Hamilton-Jacobi-Bellman equation.

## Key findings

- Established a Feynman-Kac formula for the value function.
- Proved a comparison theorem ensuring uniqueness of viscosity solutions.
- Applied the framework to control an excitable cell membrane model.

## Abstract

In this paper we consider the optimal control of Hilbert space-valued infinite-dimensional Piecewise Deterministic Markov Processes (PDMP) and we prove that the corresponding value function can be represented via a Feynman-Kac type formula through the solution of a constrained Backward Stochastic Differential Equation. A fundamental step consists in showing that the corresponding integro-differential Hamilton-Jacobi-Bellman equation has a unique viscosity solution, by proving a suitable comparison theorem. We apply our results to the control of a PDMP Hodgkin-Huxley model with spatial component, previously studied in [22], [21] and inspired by optogenetics.