# American Institute of Mathematical Sciences

December  2018, 15(6): 1495-1515. doi: 10.3934/mbe.2018069

## Review of stability and stabilization for impulsive delayed systems

 a. School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China b. Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, Shandong Normal University, Ji'nan 250014, China c. School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Ji'nan 250014, China

* Corresponding author. Email address: lxd@sdnu.edu.cn (X.Li)

Received  August 03, 2018 Revised  August 19, 2018 Published  September 2018

This paper reviews some recent works on impulsive delayed systems (IDSs). The prime focus is the fundamental results and recent progress in theory and applications. After reviewing the relative literatures, this paper provides a comprehensive and intuitive overview of IDSs. Five aspects of IDSs are surveyed including basic theory, stability analysis, impulsive control, impulsive perturbation, and delayed impulses. Then the research prospect is given, which provides a reference for further study of IDSs theory.

Citation: Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069
##### References:

show all references

##### References:
(a) State trajectory of system (3) without impulse control. (b) State trajectory of system (3) with impulsive stabilization
(a) State trajectory of system (4) without impulsive perturbation. (b) State trajectory of system (4) with impulsive perturbation
Impulsive sequence
(a) State trajectories of system (9) with $\omega$ = 2 without impulsive control. (b) Phase portraits of system (9) with $\omega$ = 2 without impulsive control. (c) State trajectories of system (9) with $\omega$ = 8 without impulsive control. (d) Phase portraits of system (9) with $\omega$ = 8 without impulsive control
(a) State trajectory of system (9) with $\omega = 2,\rho = 1.5,\mu = 0.5.$ (b) Phase portraits of system (9) with $\omega = 2,\rho = 1.5,\mu = 0.5.$ (c) State trajectory of system (9) with $\omega = 8,\rho = 1.5,\mu = 0.5.$ (d) Phase portraits of system (9) with $\omega = 8,\rho = 1.5,\mu = 0.5.$
(a) State trajectories of (10) without impulse. (b) State trajectories of system (10) with impulsive perturbation
 [1] Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 [2] Sanling Yuan, Xuehui Ji, Huaiping Zhu. Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1477-1498. doi: 10.3934/mbe.2017077 [3] Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 [4] Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1 [5] C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435 [6] Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164 [7] Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185 [8] Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087 [9] Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial & Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17 [10] Aram Arutyunov, Dmitry Karamzin, Fernando L. Pereira. On a generalization of the impulsive control concept: Controlling system jumps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 403-415. doi: 10.3934/dcds.2011.29.403 [11] George Ballinger, Xinzhi Liu. Boundedness criteria in terms of two measures for impulsive systems. Conference Publications, 1998, 1998 (Special) : 79-88. doi: 10.3934/proc.1998.1998.79 [12] John R. Graef, János Karsai. Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy. Conference Publications, 2001, 2001 (Special) : 166-173. doi: 10.3934/proc.2001.2001.166 [13] Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020138 [14] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [15] Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457 [16] Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321 [17] Canghua Jiang, Kok Lay Teo, Ryan Loxton, Guang-Ren Duan. A neighboring extremal solution for an optimal switched impulsive control problem. Journal of Industrial & Management Optimization, 2012, 8 (3) : 591-609. doi: 10.3934/jimo.2012.8.591 [18] Guirong Jiang, Qishao Lu, Linping Peng. Impulsive Ecological Control Of A Stage-Structured Pest Management System. Mathematical Biosciences & Engineering, 2005, 2 (2) : 329-344. doi: 10.3934/mbe.2005.2.329 [19] Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067 [20] Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020106

2018 Impact Factor: 1.313