On the Relation between Discounted and Average Optimal Value Functions

We investigate the relation between discounted and average deterministic optimal control problems for nonlinear control systems. In particular we are interested in the corresponding optimal value functions. Using the concepts of Viability, Chain Controllability, and Controllability, a global convergence result for vanishing discount rate is obtained. Basic ingredients for the analysis are an Abelian type theorem, controllability properties of the system, and the Morse decomposition of the corresponding control flow