Ergodic switching control for diffusion-type processes

We consider the control of the discrete component nt of a switching Markov process xt = (zt, nt) when there is a running cost and an immediate cost c(i, j) for switching nt from i to j. We study the minimization of the ergodic (or long-term average) total cost. Essentially, this paper treats the case where, for nt = n fixed, zt is a reflected diffusion or a reflected diffusion with jumps, nt being, for fixed z, a continuous-time Markov chain. Using the vanishing discount approach, we extend existing results dealing with the situation where nt evolves only by the switching control action and the diffusion is non-degenerate. Moreover, we solve the ergodic problem for a class of diffusions which can be degenerate and for an example with absorbing state.