On the Existence of Periodic Solutions with Applications to Extremum-Seeking

TL;DR

This paper establishes new theoretical results on the existence and stability of periodic solutions in time-periodic dynamical systems, with applications to extremum-seeking control, avoiding traditional averaging and perturbation methods.

Contribution

It introduces two novel results on periodic solutions for scalar and general systems, and applies them to extremum-seeking problems without relying on averaging or singular perturbation techniques.

Findings

Established equivalence between periodic and bounded solutions in scalar systems

Provided sufficient conditions for existence and stability of periodic solutions

Applied results to extremum-seeking control with novel non-local insights

Abstract

This paper provides two results that are useful in the study of the existence and the stability properties of a periodic solution for a given dynamical system. The first result deals with scalar time-periodic systems and establishes the equivalence of the existence of a periodic solution and the existence of a bounded solution. The second result provides sufficient conditions for the existence and the stability of a periodic solution for a time-periodic dynamical system. Both results are applied to extremum seeking problems for a static output map with no plant dynamics and novel non-local results are provided without the use of averaging theorems and singular perturbation arguments.