On the Distribution of the First Component $\eta_{t}$ of a Controlled Poisson Process $\{\eta_{t},\xi_{t}\}$, $t\ge 0$, without Boundary

An ergodicity condition for the first component $\eta_{t}$ of a controlled Poisson process without boundary is found. The Laplace transform of the same component $\eta_{t}$, $t\ge 0$, is obtained from the given transition probabilities of the process $\{\eta_{t},\xi_{t}\}$, $t\ge 0$. It is essential that the given process $\{\eta_{t},\xi_{t}\}$, $t\ge 0$, is a Markov process homogeneous in the second component.