Dynamic Behavior Analysis Of Stochastic Differential Equations Driven By G-Brownian Motion

This dissertation focuses on some topics under the G-Brownian motion.This work is divided into three Sections.In the first Section,we discuss the following stochastic differential equations with delay driven by G-Brownian motion(G-SDDEs,in short):dx(t)f(t,x(t),x(t-?))dt + h(t,x(t),x(t—?))d(B)(t)+ ?r(t,x(t),x(t—?))dB(t),(1)where x0,???{?(?);—? ??? 0)},C([-?,0];Rn)the family of continuous functions? from[-?,0]to Rn with the norm ||?||= sup-????0 |?(?)| and random variable ? such that E||?||p<?,B(·)is a G-Brownian motion,<B>(·)is the quadratic variation process of the G-Brownian motionB(·).Here f,h,?:R+ x Rn x R n ?Rn,f,h,?? MpG[0,T].The asymptotical boundedness and exponential stability of the equations(1)are obtained by means of G-Lyapunov function.Motivated by the results in Section 1,we investigate the following stochastic cou-pled systems on networks with time-varying delay driven by G-Brownian motion(G-SCSNTVD,in short)in Section 2:(?)where xk(t)=(xk1(t),…,xkmk(t))T,fhh and gh:Rmh? Rmk are continuously activation functions,bk(·):Rmk? Rmk is an appropriate behaved function,akh and bkh denote the strength of the coupling.?(t)is the time-varying delay with 0? ? t)? ?,?(t)???1.The initial value ? ={?(?);-? ?? ?0 ? C([-?,0];Rm)denotes the family of continuous functions ? from[-?,0]to Rm with the norm ||?|| = sup-????0|?(?)| and random variable ? such that E||?||p<? By means of inequality technique,k-th vertex-Lyapunov functions and graph-theory,we obtain asymptotical boundedness for G-SCSNTVD.As an application,stochastic coupled oscillators networks with time-varying delay driven by G-Brownian motion are discussed.In Section 3,we study stabilization of SDEs and applications to synchronization of stochastic neural network driven by G-Brownian motion with state feedback control.In details,for an unstable stochastic system,dx(t)= f h(t,x(t))d(B)(t)+ ?(t,X(t))dB(t),t ? 0,(3)we aim to design a feedback controller embedded into the drift with the following form,dx(t)=[f(t,x(t))u(t,x(t))]dt + h(t,x(t))d(B)(t)+ ?(t,x(t))dB(t),t?0.(4)Consequently,the corresponding controlled system becomes stable.In the sequel,we consider unstable stochastic Hopfield neural networks driven by G-Brownian motion(G-SHNNs,in short)with the following form dx(t)—[-Cx(t)+ AF(x(t))]dt + H(t,x(t)d(B)(t)+J(t,d(B)dB(t),(5)wherex(t)=(x1(t),...,xn(t))T,C = diag(c1….,Cn),A =(aij)n×n,F(x(t))=(f1(x1(t)),…,fn(xn(t)))T,H(t,x(t))=(h1(t,x1(t)),…,hn(t,xn(t)))T,J(t,x(t))=(J1(t,x1,(t)),…,J.(t.xn(t)))T.Here,n corresponds to the number of units in a neural network,ci>0 denotes the rate with which the ith unite will reset its potential to the resting state in isolation when being disconnected from the network and external stochastic per-turbation,xi(t)denotes the potential of cell i at time t,fi(xi(t)denotes the activation function,aij denotes the strengths of connectivity between cell i and j.