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

TL;DR

The paper tackles measure functional differential equations with state-dependent delays, introducing a regulated-phase-space framework and leveraging the Perron–Stieltjes integral to define solutions. It establishes existence and uniqueness and proves a novel periodic averaging principle, enabling simplification of complex delay dynamics. The theory is then applied to a predictor-feedback extremum-seeking scheme for static maps with state-dependent delays, including a PDE-based stability analysis and an illustrative simulation. Collectively, the results extend the mathematical foundations of measure-type delay equations and offer practical tools for robust, real-time control design in systems with adaptive delays.

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.

Figures (5)

• Figure 1: Block diagram of the extremum seeking control system with nonconstant delays.
• Figure 2: The state-dependent-delay $D(\theta(t))=\frac{1}{2}\sin(5\theta(t))^2$, the delayed time $\phi(t)=t-\frac{1}{2}\sin(5\theta(t))^2$, and the prediction time $\sigma(t)=\phi^{-1}(t)$.
• Figure 3: Convergence of the state $\alpha(x,t)$ employed to represent the delayed system with state-dependent delays BK:2013 in a three-dimensional space. In blue, we can check $\Theta(t)=\alpha(0,t)$, while in red, we have $\theta(t)=\alpha(1,t)$, both reaching a neighborhood of $\theta^\ast$.
• Figure 4: Convergence of the output $y(t)$ to a neighborhood of $y^{*}$.
• Figure 5: Convergence of the control signal $U(t)$ to a neighborhood of $0$.

Theorems & Definitions (18)

• Theorem 1.1: 14
• Theorem 1.2: 43, Gronwall Inequality for Perron-Stieltjes
• Theorem 1.3: Schauder Fixed--Point Theorem
• Example 2.1: 15
• Lemma 2.2: 15
• Lemma 2.3: 15
• Remark 3.1
• Remark 3.2
• Theorem 3.3: Existence of Solutions
• proof