Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

Abstract

The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.

Stability of perturbed dynamic system on time scales with initial time difference

“…When such a change of initial time for each solution is considered, then the problem of measuring the difference between any 2 solutions starting at different times arises. The solution of this interesting problem was investigated in [10,11,12] in different ways.…”

Robust Stability Analysis of Nonlinear Fractional‐Order Time‐Variant Systems

“…The problem of stability is a very essential and crucial issue for control systems including fractional-order system. Very recently, the stability problem of fractional-order system has been investigated both from an algebraic and from an analytic point of view [8][9][10][11]. By analyzing the characteristic equation of the Jacobian matrix, an asymptotically stable critical was proposed in [12].…”

Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

“…Ever since then, many results for various differential and difference systems have been obtained. The results obtained by the former method can be seen in [3][4][5][6][7][8][9][10]; and those done by the latter can be found in [11][12][13][14][15]. However, up till now, to the best of our knowledge, there are few results for impulsive differential equations with initial time difference.…”