Singularly Perturbed Control Systems with Noncompact Fast Variable

TL;DR

This paper investigates the asymptotic behavior of singularly perturbed optimal control problems with noncompact fast variables, using Weak KAM theory to analyze the convergence of value functions without compactness assumptions.

Contribution

It introduces a novel analysis of control systems with noncompact fast variables, employing Weak KAM theory and Aubry sets for asymptotic analysis.

Findings

Convergence of value functions to sub and supersolutions of a limit equation.

Extension of asymptotic analysis to noncompact fast variables.

Application of Weak KAM theory in control problems.

Abstract

We deal with a singularly perturbed optimal control problem with slow and fast variable depending on a parameter {\epsilon}. We study the asymptotic, as {\epsilon} goes to 0, of the corresponding value functions, and show convergence, in the sense of weak semilimits, to sub and supersolution of a suitable limit equation containing the effective Hamiltonian. The novelty of our contribution is that no compactness condition are assumed on the fast variable. This generalization requires, in order to perform the asymptotic proce- dure, an accurate qualitative analysis of some auxiliary equations posed on the space of fast variable. The task is accomplished using some tools of Weak KAM theory, and in particular the notion of Aubry set.