Stability Of Impulsive Differential Equations Driven By Fractional Brownian Motion

As a Gaussian stochastic process with zero mean,fractional Brownian motion exhibits the properties of self-similarity and long-range dependence.Since the pioneering work of Mandelbrot and Van Ness,fractional Brownian motion has attracted much attention in the field of hydrology,telecommunication and mathematical finances,etc.The property of fractional Brownian motion heavily relies on the Hurst parameter H?(0,1),if H?(1/2,1),it exhibits the property of long-range dependence.Researches show that long-range dependence phenomenon was observed in water accumulation in hydrology,ATM traffic in telecommunication and stock prices in mathematical finances.On the other hand,since many systems in our daily life exhibit the phenomenon of instantaneous perturbation,it is more reasonable to use impulse differential equations to model these systems.Thus,it has both theoretical significance and application value to study impulsive differential equations driven by fractional Brownian motion.This paper investigates the existence,stability,boundedness and synchronization of several classes of impulsive differential equations.The main results are list as follows:In Chapter 1,the research background,significance and study status at home and abroad are stated.In Chapter 2,the definition and properties of fractional Brownian motion are recalled.Some lemmas and definitions needed in this paper are also recalled,which include:the basic concepts of semigroups and fractional powers of infinitesimal generators,the concept of mild solution,the definition of exponential stability in pth moment,mean square uniformly ultimately boundedness,practical synchronization.In Chapter 3,a class of impulsive neutral stochastic integrodifferential equations driven by fractional Brownian motion is studied.In terms of fractional power of operators,semigroup theory and Banach fixed point theorem,the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained.Moreover,the pth moment exponential stability conditions of the system are obtained by means of an impulsive integral inequality.Finally,an example is given to test the validity of the obtained results.In Chapter 4,a class of delayed impulsive differential equations driven by multiplicative fractional Brownian motion is studied.First of all,the sufficient conditions to ensure the existence and uniqueness of solutions are obtained.Secondly,the pth moment exponential stability conditions are obtained by means of the generalized stochastic Lyapunov method,fractional Ito formula as well as the impulsive control theory.In Chapter 5,a class of delayed impulsive differential equations driven by additive fractional Brownian motion is studied.Using stochastic analysis theory and impulsive control theory,sufficient conditions to assure mean square uniformly ultimately boundedness are obtained.As application,the obtained results are used to do synchronization with respect to a class of chaotic systems,in which the response systems are perturbed by additive fractional Brownian motion noise.The sufficient conditions to assure practical synchronization are obtained,the corresponding impulsive controller are designed.Examples are presented to test the validity and applicability of the results at length.In Chapter 6,we draw our conclusion and list the research focus in our future work.