This paper analyzes the stability of optimal policies in the long-run stochastic control framework with an averaged risk-sensitive criterion for discrete-time MDPs on finite state-action space. In particular, we study the robustness of optimal controls when perturbations to the risk-aversion parameter are applied, and investigate the Blackwell property, together with its link to the risk-sensitive vanishing discount approximation framework. Finally, we present examples that help to better understand the intricacies of the risk-sensitive control framework.
