A brief survey on stability and stabilization of impulsive systems with delayed impulses

Xinyi He; Jianlong Qiu; Xiaodi Li; Jinde Cao

doi:10.3934/dcdss.2022080

Article Contents

2022, Volume 15, Issue 7: 1797-1821. Doi: 10.3934/dcdss.2022080

This issue Previous Article Distributionally robust front distribution center inventory optimization with uncertain multi-item orders Next Article Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3)

A brief survey on stability and stabilization of impulsive systems with delayed impulses

1.
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
2.
School of Automation and Electrical Engineering & Key Laboratory of Complex systems, and Intelligent Computing in Universities of Shandong, Linyi University, Linyi 276005, China
3.
School of Mathematics, Southeast University, Nanjing 211189, China
4.
Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea

^* Corresponding author: Xiaodi Li
^* Corresponding author: Xiaodi Li

Received: December 2021

Revised: February 2022

Published: July 2022

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Abstract

This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.

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Mathematics Subject Classification: 34D20, 93C27, 93D30.

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Figure 1. Outline of this paper

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Figure 2. Trajectories of system (5) with $ a = 1 $ and $ b = 0.5 $

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Figure 3. Trajectories of system (5) with $ a = -1 $ and $ b = 1.5 $

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Table 1. Summary of stability results of impulsive systems with DIs

Type of DI	Stability	Constrains	Main results
Type of DI	Stability	Constrains	Imp control	Imp disturbance	Hybrid
TDDI	UAS	Condi (6)	Th 3.2	Th 3.3	n/a
	Uniform ES	Condi (8)	Cor 1	Cor 1	Th 3.4
	Uniform ISS	Condi (11)	Th 3.6	Cor 2	Th 3.5
	FTS/FTCS	Condi (13)	Th 3.9/3.10	Th 3.7/3.8	Th 3.11
SDDI	US/UAS	Condi (6)	Th 3.12	Th 3.13	n/a
	Local ES	Condi (14)	Th 3.14	Th 3.14	n/a
	FTS/FTCS	Condi (15)	Th 3.16	Th 3.15	n/a

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