Generalized Abelian Theorems for Long Run Optimal Control

Abstract

We study optimal control problems involving an average defined as the mean of a Lagrangian with respect to a probability measure on R_+ . A particular example is the Cesaro average over a fixed time horizon. A fundamental and widely studied question is whether the corresponding value functions converge uniformly as the horizon tends to infinity. Classically, an Abelian theorem states that the uniform convergence of Cesaro means implies the convergence of Abel means to the same limit. We extend this result to value functions associated with more general families of probability measures on R + . This framework captures long-term optimization problems and the asymptotic behavior of value functions as the time horizon tends to infinity. We introduce suitable conditions on sequences of probability measures that ensure the convergence of the associated value functions in the topology of uniform convergence. Finally, we establish an Abelian-type theorem showing that, whenever the Cesaro values converge uniformly, the values associated with various other sequences of probability measures converge to the same limit.