Exponential Stability Results on Random and Fixed Time Impulsive Differential Systems with Infinite Delay

Abstract

In this paper, we investigated the stability criteria like an exponential and weakly exponential stable for random impulsive infinite delay differential systems (RIIDDS). Furthermore, we proved some extended exponential and weakly exponential stability results for RIIDDS by using the Lyapunov function and Razumikhin technique. Unlike other studies, we show that the stability behavior of the random time impulses is faster than the fixed time impulses. Finally, two examples were studied for comparative results of fixed and random time impulses it shows by simulation.

Stability analysis for singular random impulsive control system: A novel neutral impulsive delay system approach

“…Lemma 1 (Previous works [28,29]). Assume there exists exactly m impulses arrival until the time t, t ≥ t 0 , and…”

“…The literature is elaborated as a testimony to explain a few results of random impulsive systems. In the literature [28][29][30], several stability results, like ES, weakly ES, and globally ES, were investigated for random impulsive systems with and without time delay. In previous studies [31][32][33], the stochastic time delay systems with random impulsive effects were studied.…”

Some stability results on non-linear singular differential systems with random impulsive moments

“…Random impulses are different from fixed-time impulse effects. Recently in [40], the authors studied the exponential stability based on fixed and random time effect of the impulses while they proved the robust mean square stability for random impulsive control systems in [41]. Then, by considering the impulse moments at random time points in [42], the authors proved the stability results for differential systems.…”

An exponential stabilization of random impulsive control systems and its application to chaotic systems

“…Lemma 23,31 If there is an impulse of exactly $$$ n $$$ up to time $$$ t,t\ge {t}_0 $$$ and the wait time between two consecutive impulses follows the exponential distribution of the parameter $$$ \gamma $$$, then the probability $$$$ P\left({I}_{\left[{\chi}_n^{\prime },{\chi}_{n+1}^{\prime}\right)}(t)\right)=\frac{\gamma^n{\left(t-{t}_0\right)}^n}{n! }{e}^{-\gamma \left(t-{t}_0\right)}, $$$$ where the events $$$ {I}_{\left[{\chi}_n^{\prime },{\chi}_{n+1}^{\prime}\right)}(t)=\left\{\omega \in \Omega :{\chi}_n^{\prime}\left(\omega \right)<t<{\chi}_{n+1}^{\prime}\left(\omega \right)\right\},n\in {\mathbb{Z}}_{+} $$$.…”

“…Lemma 1. 23,31 If there is an impulse of exactly n up to time t, t ≥ t 0 and the wait time between two consecutive impulses follows the exponential distribution of the parameter 𝛾, then the probability…”