Linear Programming Estimates for Cesaro and Abel Limits of Optimal Values in Optimal Control Problems

TL;DR

This paper develops linear programming estimates for the Cesaro and Abel limits in infinite horizon optimal control problems, providing bounds and conditions for their equality and existence.

Contribution

It introduces IDLP-based bounds and duality conditions for Cesaro and Abel limits in long-term optimal control, advancing theoretical understanding.

Findings

Cesaro and Abel limits are bounded by IDLP problem values

Limits are equal if no duality gap exists

Provides IDLP-based optimality conditions and an illustrative example

Abstract

We consider infinite horizon optimal control problems with time averaging and time discounting criteria and give estimates for the Cesaro and Abel limits of their optimal values in the case when they depend on the initial conditions. We establish that these limits are bounded from above by the optimal value of a certain infinite dimensional (ID) linear programming (LP) problem and that they are bounded from below by the optimal value of the corresponding dual problem. (These estimates imply, in particular, that the Cesaro and Abel limits exist and are equal to each other if there is no duality gap). In addition, we obtain IDLP-based optimality conditions for the long run average optimal control problem, and we illustrate these conditions by an example.