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-τ))dt + h(t,x(t),x(t—τ))d(B)(t)+ σr(t,x(t),x(t—σ))dB(t),(1)where x0,＝ξ＝{ξ(θ);—τ ≤θ≤ 0)},C([-τ,0];Rn)the family of continuous functionsψ from[-τ,0]to Rn with the norm ||ψ||= sup-τ≤θ≤0 |ψ(θ)| and random variable ξ such that E||ξ||p<∞,B(·)is a G-Brownian motion,<B>(·)is the quadratic variation process of the G-Brownian motionB(·).Here f,h,σ:R+ x Rn x R n →Rn,f,h,σ∈ 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.τ(t)is the time-varying delay with 0≤ τ t)≤ τ,τ(t)≤τ＜1.The initial value ξ ={ξ(θ);-τ ≤θ ≤0 ｝ C([-τ,0];Rm)denotes the family of continuous functions ψ from[-τ,0]to Rm with the norm ||ψ|| = sup-τ≤θ≤0|ψ(θ)| and random variable ξ such that E||ξ||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)+ σ(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)+ σ(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.