Input-to-state Stability of Impulsive Stochastic Nonlinear Systems Driven by G-Brownian Motion
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Input-to-state Stability of Impulsive Stochastic Nonlinear Systems Driven by G-Brownian Motion Lijun Pan* and Jinde Cao* Abstract: This paper studies the input-to-state stability (ISS), stochastic input-to-state stability (SISS) and eλt weighted input-to-state stability (eλt -ISS) of impulsive stochastic nonlinear systems driven by G-Brownian motion (IGSNSs). If the continuous stochastic systems are not ISS, the impulsive effects can stabilize the system for the fixed dwell-time sequences. However, if the continuous stochastic systems are ISS, the hybrid system can achieve ISS for destabilizing impulses with upper bound of the fixed dwell-time. Moreover, the average dwell-time condition is generalized to guarantee the ISS for IGSNSs based on G-Lyapunov method. Finally, an example is provided to illustrate the effectiveness of theoretical results. Keywords: G-Brownian motion, impulse, input-to-state stability, stochastic systems.
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INTRODUCTION
As is well known, the signal transmissions of systems are usually subjected to random noises, which lead to stochastic perturbation and uncertainties of the process for dynamic behavior. Correspondingly, a lot of the dynamics results of stochastic networks and stochastic systems have been obtained [1–8]. G-expectation is an emerging field drawing attention from researchers due to their wide applications in risk measures, volatility uncertainty, superpricing in finance and so on. The initial research of Gexpectation can be traced back to Peng [9]. Hereafter, its related stochastic calculus, strong laws of large numbers, central limit theorem under sublinear expectation have been obtained [10–14]. Especially, by sample path properties of G-Brownian motion, some results have been established to study stochastic systems driven by G-Brownian motion [15–19]. In particular, in [15], G-Lyapunov differential operator was introduced to deal with G-martingale problems. Discrete time feedback control was developed to quasi-sure exponential stabilization of stochastic systems induced by G-Brownian motion in [18]. Impulsive systems, as special hybrid systems, are dynamical systems simulating both continuous and instantaneous jump behavior [20, 21]. Since impulsive systems have been used practically in the fields such as synchronization of complex networks [22, 23], neural networks
[24] and chaotic systems [25], the dynamics of impulsive systems have attracted much attention in recent years. In the evolution of the impulsive dynamics, Lyapunov function and Lyapunov-Razumikhin function are often utilized to stabilize the systems, see [26, 27]. Because impulsive gain and impulsive moments are also factors that can not be ignored, different types of dwell time must be considered. In the past decades, Lyapunov function together with dwell time condition has become the main method to study the dynamics of impulsive systems [28, 29]. For example, in [29], an average dwell time and Lyapunov function are generalized to the synchroni
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