Stability analysis for a class of nonlinear time-changed systems

TL;DR

This paper studies the stability of nonlinear differential systems altered by a time-change process, developing new inequalities and formulas to analyze various stability types and linking them to classical systems.

Contribution

It introduces a time-changed Gronwall's inequality and a generalized Itô formula, connecting stability of time-changed systems with their classical counterparts.

Findings

Established stability criteria for different types of time-changed systems

Developed a new analytical framework using generalized inequalities and formulas

Linked stability properties of time-changed systems to non-time-changed systems

Abstract

This paper investigates the stability of a class of differential systems time-changed by $E_{t}$ which is the inverse of a $\beta$-stable subordinator. In order to explore stability, a time-changed Gronwall's inequality and a generalized It\^o formula related to both the natural time $t$ and the time-change $E_{t}$ are developed. For different time-changed systems, corresponding stability behaviors such as exponential sample-path stability, $p$th moment asymptotic stability and $p$th moment exponential stability are investigated. Also a connection between the stability of the time-changed system and that of its corresponding non-time-changed system is revealed.