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On the Mittag-Leffler stability of mixed-order fractional homogeneous cooperative delay systems

L. V. Thinh, H. T. Tuan

Abstract

In this paper, we study a class of multi-order fractional nonlinear delay systems. Our main contribution is to show the (local or global) Mittag-Leffler stability of systems when some structural assumptions are imposed on the "vector fields": cooperativeness, homogeneity, and order-preserving on the positive orthant of the phase space. In particular, our method is applicable to the case where the degrees of homogeneity of the non-lag and lag components of the vector field are different. In addition, we also investigate in detail the convergence rate of the solutions to the equilibrium point. Two specific examples are also provided to illustrate the validity of the proposed theoretical result.

On the Mittag-Leffler stability of mixed-order fractional homogeneous cooperative delay systems

Abstract

In this paper, we study a class of multi-order fractional nonlinear delay systems. Our main contribution is to show the (local or global) Mittag-Leffler stability of systems when some structural assumptions are imposed on the "vector fields": cooperativeness, homogeneity, and order-preserving on the positive orthant of the phase space. In particular, our method is applicable to the case where the degrees of homogeneity of the non-lag and lag components of the vector field are different. In addition, we also investigate in detail the convergence rate of the solutions to the equilibrium point. Two specific examples are also provided to illustrate the validity of the proposed theoretical result.

Paper Structure

This paper contains 8 sections, 8 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations
  4. Preliminaries
  5. Mittag-Leffler stability of homogeneous cooperative delay systems
  6. Boundedness and positivity of solutions
  7. Mittag-Leffler stability of the systems
  8. Numerical examples and discussion

Key Result

Lemma 2.5

(see, e.g., NNThang_23, Shuailei_Zhang) If $\eta>0$ and $\alpha\in(0,1]$, then for all $t\ge0,\ s\ge0$, we have

Figures (4)

  • Figure 1: The solution to system \ref{['Eqmain-2']} with $\varphi(s)=\left(0.20.15\right)$ on the interval $[-1,0]$.
  • Figure 2: The solution to the system \ref{['Eqmain-2']} with $\varphi(s)=\left(1.20.4\right)$ on the interval $[-1,0]$.
  • Figure 3: The solution to system \ref{['Eqmain-1']} with $\varphi(s)=\left(0.20.4\right)$ on the interval $[-1,0]$.
  • Figure 4: The solution to system \ref{['Eqmain-1']} with $\varphi(s)=\left(2.30.2\right)$ on the interval $[-1,0]$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • proof
  • ...and 16 more