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