Adaptive average control for piecewise deterministic Markov processes

The main goal of this paper is to study the adaptive infinite-horizon average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) for the case in which there are no boundary jumps. It is assumed that the control acts continuously on the jump intensity $\lambda$ and on the transition measure $Q$ of the process, and that jump parameters ($\lambda$ and $Q$), as well as the cost $C$, depend on an unknown parameter ${\beta }^{\ast }$. It is shown that the principle of estimation and control holds, that is, the strategy consisting of choosing, at each stage $n$, an action according to an optimal stationary policy, where the true but unknown parameter ${\beta }^{\ast }$ is replaced by its estimated value ${\beta }_n^{\ast }$, is average optimal, provided that the sequence of estimators $\left\{{\beta }_n^{\ast }\right\}$ of ${\beta }^{\ast }$ is strongly consistent, that is, ${\beta }_n^{\ast }$ converge to ${\beta }^{\ast }$ almost surely.