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
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Figure 3.1.  (a) State trajectory of system (3) without impulse control. (b) State trajectory of system (3) with impulsive stabilization
Figure 3.2.  (a) State trajectory of system (4) without impulsive perturbation. (b) State trajectory of system (4) with impulsive perturbation
Figure 3.3.  Impulsive sequence
Figure 4.2.  (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
Figure 4.3.  (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 $\omega = 8,\rho = 1.5,\mu = 0.5.$
Figure 5.3.  (a) State trajectories of (10) without impulse. (b) State trajectories of system (10) with impulsive perturbation
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