A generalized fractional Halanay inequality and its applications

Abstract

This paper is concerned with a generalized Halanay inequality and its applications to fractional-order delay linear systems. First, based on a sub-semigroup property of Mittag-Leffler functions, a generalized Halanay inequality is established. Then, applying this result to fractional-order delay systems with an order-preserving structure, an optimal estimate for the solutions is given. Next, inspired by the obtained Halanay inequality, a linear matrix inequality is designed to derive the Mittag-Leffler stability of general fractional-order delay linear systems. Finally, numerical examples are provided to illustrate the proposed theoretical results.

Figures (3)

• Figure 1: Orbits of the solution of the system \ref{['ex1']} with the initial condition $\varphi(s)=( 0.2-0.4\cos s,0.1+0.1s,\log({s+3})-0.5)^{\rm T}$ on $[-2,0]$.
• Figure 2: Orbits of the solution of the system \ref{['ex2']} with the initial condition $\varphi(s)=( 0.3+0.4\sin s,0.1+0.5s)^{\rm T}$ on $[-1,0]$.
• Figure 3: Orbits of the solution of the system \ref{['ex3']} with the initial condition $\varphi(s)=0.3-0.5\cos (2s)$ on $[-2,0]$.

Theorems & Definitions (27)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Remark 2.5
• Remark 2.6
• Corollary 2.7
• Lemma 3.1