On average control generating families for singularly perturbed optimal control problems with long run average optimality criteria

TL;DR

This paper develops a method to approximate and construct near optimal controls for singularly perturbed systems with long-term average criteria by using averaged control problems and linear programming techniques.

Contribution

It introduces a novel approach combining asymptotic approximation, infinite-dimensional LP reformulation, and semi-infinite LP problems to generate near optimal controls for SP systems.

Findings

Method effectively constructs near optimal controls.

Numerical example demonstrates practical applicability.

Approach bridges asymptotic analysis and LP techniques.

Abstract

The paper aims at the development of tools for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems with long run average optimality criteria. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional (ID) linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with a numerical example.