Impulsive Stability of Stochastic Functional Differential Systems Driven by G-Brownian Motion

Abstract

This paper is concerned with the p-th moment exponential stability and quasi sure exponential stability of impulsive stochastic functional differential systems driven by G-Brownian motion (IGSFDSs). By using G-Lyapunov method, several stability theorems of IGSFDSs are obtained. These new results are employed to impulsive stochastic delayed differential systems driven by G-motion (IGSDDEs). In addition, delay-dependent method is developed to investigate the stability of IGSDDSs by constructing the G-Lyapunov-Krasovkii functional. Finally, an example is given to demonstrate the effectiveness of the obtained results. disturbance and reduce the influence of the impulsive effect. [24,25] established the stability for impulsive stochastic differential equations driven by G-Brownian motion with the help of G-Lyapunov function technique. In the evolution of dynamical systems, it is impossible for systems to contact at the same time owing to time-delays. To get over the adverse impact of time-delays, delay-dependent scheme is a vigorous tool to verify the stability of dynamical systems (see [26][27][28]).To our best knowledge, there is no literature reported on the pth moment exponential stability and quasi sure exponential stability of the zero solution for impulsive stochastic functional differential systems driven by G-Brownian motion (IGSFDSs) or impulsive stochastic delayed differential systems driven by G-Brownian motion. Therefore, the influence of between delay and impulse for stochastic systems driven by G-Brownian motion provide a motivation of the current study. The aim of this paper is to investigate G-Lyapunov method for IGSFDSs. The contributions in this paper are concluded as follows: (i) Some theorems on pth moment exponential stability and quasi sure exponentially stability of the zero solution of IGSFDSs are established by G-Lyapunov method and impulsive analysis. (ii) These new results are employed to impulsive stochastic delayed systems driven by G-Brownian motion. (iii) If the upper bound of delay may not surpass the length of impulsive gap, delay-dependent technique is utilized to get over the influences of impulses and time delay by G-Lyapunov-Krasovkii functional. The remaining part of this paper is arranged as follows. In Section 2, some definitions and lemmas on G-expectation are proposed and the model descriptions of IGSFDSs are presented. In Section 3, some pth moment exponential stability and quasi sure exponential stability criteria are obtained via stochastic analysis and impulse technique. Section 4 extends the above theorems to impulsive stochastic delayed differential systems driven by G-Brownian motion. In addition, delay-dependent method is employed to establish the stability theorem. In Section 5, an example is provided to show our results. Section 6 gives some conclusions.

A novel adaptive control design for exponential stabilization of memristor‐based CVNNs with time‐varying delays using matrix measures

“…Consequently, the dynamical analysis problems for delayed NNs have gained considerable research attention. Up to now, a great deal of results have been reported in the literature, see for example 22–25 . For instance, in 22 the authors investigate the mean‐square stability of uncertain time‐delay stochastic systems 23 .…”

“…Up to now, a great deal of results have been reported in the literature, see for example 22–25 . For instance, in 22 the authors investigate the mean‐square stability of uncertain time‐delay stochastic systems 23 . is devoted to stability analysis of stochastic nonlinear delay systems subject to multiple periodic impulses 24 .…”

Optimal controls of impulsive fractional stochastic differential systems driven by Rosenblatt process with state‐dependent delay

“…Stochastic systems have several applications in diverse fields, for example, population dynamics, ecology, biology, engineering, and in finance. For the basic theory and applications of the stochastic systems, see [6][7][8] and the references therein. Ye and Yu [9] investigated the exact controllability of linear stochastic systems and observability inequality for backward stochastic differential equations.…”

Practical stability of stochastic functional differential equations with infinite delay

“…Random phenomena with memory appear in many real-world applications like physics, finance, and biology. In the literature, there is a various works in the stabilization of SFDEFD (for more details, see [1][2][3][4][5]). A well-known class of such dynamical systems are SFDEFD which become an essential area of research and investigation (see [6][7][8][9][10]).…”