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In this paper we consider the optimal control of Hilbert space-valuedinfinite-dimensional Piecewise Deterministic Markov Processes (PDMP) and weprove that the corresponding value function can be represented via aFeynman-Kac type formula through the solution of a constrained BackwardStochastic Differential Equation. A fundamental step consists in showing thatthe corresponding integro-differential Hamilton-Jacobi-Bellman equation has aunique viscosity solution, by proving a suitable comparison theorem. We applyour results to the control of a PDMP Hodgkin-Huxley model with spatialcomponent, previously studied in [22], [21] and inspired by optogenetics.

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