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# Review of stability and stabilization for impulsive delayed systems

• 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.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • 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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