Exponential Stability of Stochastic Differential Equations with Impulse Effects at Random Times

Abstract

In this paper, we investigate the exponential stability for a class of impulse stochastic differential equations. Different from the previous literature, the impulsive times considered in this paper are random times. By applying stochastic processes theory, stochastic analysis theory and Lyaponov method, we establish several novel exponential stability criteria of the suggested system. Finally, several simple examples are provided to show the validity and significance of the results.

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

“…Remark 1. In fact, another continuous asymptotically stabilizing controller can be found for system (3). If the assumptions of Theorem 1 hold, then the functions a and b are continuous, and since V is a SCLF for system (3), that is b(x) = 0 x ≠ 0 implies a(x) < 0, it is easy to verify that the feedback…”

“…Theorem 1 [31]: If there exists a stochastic control Lyapunov function V for the system (3), satisfying the small control property, then the feedback u = k(x), defined in (7), is continuous on R n and globally asymptotically stabilizes in probability system (3).…”

“…For system (3), suppose that there exists a SCLF V with controls in  1 , satisfying the small control property, Then, the origin solution to (3) is globally asymptotically stable in probability with the continuous feedback u = p(x) defined in (20) which takes values in  1 .…”

“…Many real systems are nonlinear and are also prone to stochastic phenomena. These features have naturally attracted attention in the control literature with studies examining stabilization problems for stochastic nonlinear systems, and some interesting results obtained during the past few decades, see [1][2][3][4][5][6][7] and references therein. These results were obtained using classical stochastic theory as found, for instance, in [8] and [9].…”

Almost sure stability of stochastic neutral Cohen–Grossberg neural networks with Lévy noise and time‐varying delays

“…Because of the influence for numerous complex factors in reality, which may lead to the stability or instability for the stochastic hybrid system (SHS), the stability problems of SHS have become a hot research theme [1][2][3][4][5][6][7][8][9], especially the neural networks (NNs) [10][11][12][13][14][15]. The Cohen-Grossberg neural networks (CGNNs) molding is a special type of NNs which was originally raised and researched by Cohen Grossberg in 1983 [16].…”