Representation formulas for limit values of long run stochastic optimal controls

A classical problem in stochastic ergodic control consists of studying the limit behavior of the optimal value of a discounted integral in infinite horizon (the so called Abel mean of an integral cost) as the discount factor $\lambda$ tends to zero or the value defined with a Cesàro mean of an integral cost when the horizon $T$ tends to $+ \infty$. We investigate the possible limits in the norm of uniform convergence topology of values defined through Abel means or Ceàro means when $ \lambda \to 0^+ $ and $T \to + \infty $, respectively. Here we give two types of new representation formulas for the accumulation points of the values when the averaging parameter converges. We show that there is only one possible accumulation point which is the same for Abel means or Cesàro means. The first type of representation formula is based on probability measures on the product of the state space and the control state space, which are limits of occupational measures. The second type of representation formula is based on measures which are the projection of invariant measure on the space of relaxed controls. We also give a result comparing the both sets of measures involved in both classes of representation formulas. An important consequence of the representation formulas is the existence of the limit value when one has the equicontinuity property of Abel or Cesàro mean values. This is the case, for example, for nonexpansive stochastic control systems. In the end some insightful examples are given which help to better understand the results.