Definition (Average infinite horizon optimization)

Given a policy $\gamma\in\Gamma_{A}$ with the objective of minimizing $$J_{\infty}(X,\gamma) =\limsup_{ N \to \infty } \frac{1}{N} E_{x}^{\gamma}\left[ \sum_{k=0}^{N-1}c(X_{k},U_{k}) \right]$$this is known as the Average Cost Optimal Control problem.