Abstract
This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.
Highlights
During the last decades, the stability theory of impulsive delay differential systems has been undergoing fast development due to its important applications in various areas such as population management, disease control, image processing, and secure communication ([5], [9], [13], [14], [19], [20])
Most of the current research on stability analysis has been focused on the impulsive delay differential equations with time delay occurred only in the differential equations
An impulsive delay differential model with delayed impulses has been investigated in impulsive synchronization of chaotic systems in secure communication where time delays appeared in both differential and difference equations of the error dynamics due to the presence of transmission delays in the process [6, 7]
Summary
The stability theory of impulsive delay differential systems has been undergoing fast development due to its important applications in various areas such as population management, disease control, image processing, and secure communication ([5], [9], [13], [14], [19], [20]). An impulsive delay differential model with delayed impulses has been investigated in impulsive synchronization of chaotic systems in secure communication where time delays appeared in both differential and difference equations of the error dynamics due to the presence of transmission delays in the process [6, 7]. This type of equations have potential applications in other fields.
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