Slow and fast dynamics in measure functional differential equations with state-dependent delays through averaging principles and applications to extremum seeking

TL;DR

This paper introduces a novel class of measure functional differential equations with state-dependent delays, establishing foundational results and applying them to extremum seeking for real-time optimization in control systems.

Contribution

It proves existence, uniqueness, and a periodic averaging principle for these equations, expanding their theoretical understanding and practical applications.

Findings

Established existence and uniqueness of solutions.

Proved a periodic averaging principle for the equations.

Validated stability of an extremum seeking algorithm.

Abstract

This paper investigates a new class of equations called measure functional differential equations with state-dependent delays. We establish the existence and uniqueness of solutions and present a discussion concerning the appropriate phase space to define these equations. Also, we prove a version of periodic averaging principle to these equations. This type of result was completely open in the literature. These equations involving measure bring the advantage to encompass others such as impulsive, dynamic equations on time scales and difference equations, expanding their application potential. Additionally, we apply our theoretical insights to a real-time optimization strategy, using extremum seeking to validate the stability of an innovative algorithm under state-dependent delays. This application confirm the relevance of our findings in practical scenarios, offering valuable tools for advanced control system design. Our research provides significant contributions to the mathematical field and suggests new directions for future technological developments.