The Stability Of Stochastic Systems Driven By G-Brownian Motion

Based on the Peng’s theories of G-expectation, G-Brownian motion and the stochastic differential equations driven by G-Brownian motion[57], and inspired by the canonical theories of Lyapunov’s stability criteria[44],[45],[61] and the theories of H∞control for the deterministic or stochastic systems[29],[76],[78], we mainly discussed the mean square stability and the H∞problems for the stochastic systems driven by G-Brownian motion. The main contents of this paper include:the derivatives of the solutions of stochastic differential equations driven by G-Brownian motions(G-SDEs) and their properties, the stability criteria of Lyapunov’s method for G-SDEs and its applications to the systems with uncertainty coefficients, the stabilization for the systems driven by G-Brownian motions and the state feedback H∞control designing for such systems. This paper is organized as followings.In Chapter1, the theories of G-expectations and some basic knowledge are reviewed. Based on those, the existence and unique-ness of the infinite-time interval G-SDEs and G-BSDEs with follow-ing types are prove.Theoreml.16Suppose b，hij,σj satisfy the Lipschitz conditions (1.3.25) and (1.3.27), then there exists a unique solution in MG,l(R+,Rn) for G-stochastic differential equation (1.3.24) with infinite time interval.Theoreml.20Suppose f,hij satisfy the Lipschitz condition (1.3.25) and (1.3.27),(?)∈LG1(Ω;R), then the infinite-time interval G-SDEs(1.4.32) has a unique solution in MG,l(R+,Rn).In Chapter2, the derivatives of G-SDEs w.r.t. the initial value x are discussed, and the generator for such equations are also intro-duced, based on which we proved that the differential properties of function u(l,x)=E|Xst,x|2under the meaning of viscosity solution.Theorem2.19Suppose b, hij and Oj are continuous w.r.t t, V∈C1.2(R+×Rn;R)) and the second derivatives of V,(?)iV w.r.t. x are hounded and satisfy the Lipschitz conditions, then the generator operator C can be represented as the following forms where <(?),rV(t,.r),h.(t,.r)>+<(?)xx2Vσ(t,.r),σ(t,r)) is the symmetric matrix in Sd(R) with the formMoreover, the derivatives for the G-SDEs can be obtained.Theorem2.24Suppose the coefficients of the G-SDE(1.3.24) b,hij,σje Cb1,2(R+×Rn;R), and the solution of G-SDE (1.3.24){Xts,r}t≥s is in the pro cess space MG4([O,T];Rn), then Xts,r is twice differentiable w.r.t. x. Moreover,(?)xkXts,x satisfies the following G-SDEs and (?)xkx,Xts,x is the solution of following G-SDEs where (?)xx2b,(?)xx2hij,(?)xx2σj is the block matrix instructed by the partial deriva-tives w.r.t. x, i.e.,(?)xx2b=((?)xxbv(u,Xus,x))vn=1∈Rn2×n, and (?)xxbv is the Hermite matrix of bv, andProposition2.28Suppose the G-SDEs satisfy the conditions of Lemma2.25. then u(s,x)=E|Xts,x|2is the viscosity solution ofIn chapter3, the Lyapunov’s method of the stability of G-SDEs is applied, and the corresponding Lyapunov’s criteria are obtained. Moreover, the main results are applied to the analysis of the stability of the systems with uncertainty coefficients.Theorem3.6If, there exists a function V∈C1,2(R+×Rn;R), satifies the following condtions:(ⅰ) For all (t,x)∈R+xRn. there exists (?)v(t,x)≤0,(ⅱ) There exists c1, c2>0, such that c1|x|2≤V(t,x)<c2|x|2, then the G-SDE(3.2.1) is mean square stable.Theorem3.7If there exists a function V∈C1,2(R+×Rn;R) satisfies the following conditions: (ⅰ) There exists A>0, such that (?)V(t,x)≤-λV(t,x),(ⅱ) There exists cuc2>0, such that, for all (t.x)∈R+×Rn, c12≤V(t,x)≤c22, then the solution of (3.2.1) is mean-square exponentially asymptotically sta-ble.For the linear systems with following forms: where D. Hij,Cj∈R×n We can choose the Lyapunov’s function V(x) with the form V(x)=xTPx,P∈S+n(R). Let We have the following results.Proposition3.13Suppose; P∈S+n(R) satisfies the following couple linear matrix inequalities: where a takes values-1or1, then the linear G-SDEs(3.3.15) mean-sqmare exponentially asymptotically stableFurthermore, under some proper conditions, the existence of the Lya-punov’s function is also discussed, which can be considered as the inverse of the Lyapunov criteria for G-SDEs.Theorem3.16Suppose the coefficients of G-SDE(3.3.15) b, hij,σj satisfy the conditions of the Lemma3.14. and (3.3.15) is mean-square exponentially asymptotically stable, then then; exists V∈C1,2(R+×Rn;R) satisfying (3.3.2) and (3.3.5). As the applications of the Lyapunov’s criteria for G-SDEs, the stability for the following systems with coefficient uncertainty is also discussed. We can construct the G-function as followings, let G:Sd(R)�'R defined by then there exist G-expectation and a random variable η(?)N(0,Σ) with G-normal distribution, such that G(A)=1/2E<Aη,η> and the corresponding non-linear space is (Ω,(?),E).Theorem3.21Suppose b,hij,σj satisfy the conditions of the Lemma3.14, then the system (3.4.3) with coefficient uncertainty is uniformly mean-square exponentially asymptotically stable, if and only if the corresponding G-SDEs(3.4.6)is uniformly mean-square exponentially asymptotically stable.In chapter4, we discussed the robustness for the G-stochastic systems, in which the H∞norm as the main objective. This part contents the optimal stabilization and state feedback H∞control designing for such systems.Consider the following systems driven by G-Brownian motion. where v∈MG2(R+;Rnv) is the exogenous disturbance, z∈Rnz is the ob-servation, and the operator (?):MG2(R+;Rnv)�'MG2(R+Rnv) is defined by:(?)v＝z(·0,v), the norm of (?) is given by the H∞norm Theorem4.4Suppose there exists7>0,V∈C1,2(R+×Rn;R), such that, for all (t,x,v)∈R+×Rn×Rne, we have and (3.3.2), then (4.2.1) is externally stable on R+For the systems driven by G-Brownian motion with the following forms we also have the following results.Theorem4.5Suppose the coefficients b, hij, σj of (4.2.11) satisfy the con-ditions of the Lemma3.14, then the following results is equivalent:(i) The system (4.4.1) is internally stable;(ii) There exists V G C1,2(R+×Rn;R). A, c1, c2>0, such that(iii) There exists an M>0, such that and there exists V with bounded (?)x,x2V satisfying (4.2.12) and (4.2.13).Theorem4.6For a given7>0, if there exists a function V∈C1.2(R+×Rn;R) such that, for every (t,x)∈R+×Rn, the following equality is true. where and (3.3.2), then (4.2.11) is externally stable on R+, and‖(?)‖≤γ.We also consider the optimal stabilization for the following system driven by G-Brownian motion.Theorem4.14Suppose V∈C1,2(R+Rn;R) and the control uo(t,x)∈U satisfies the following conditions: Then u=uo(t,x) is the optimal stabilization control for (4.3.1), such that (4.3.1) is mean-square exopential asymptotically stable and uo(t,x) is the op-timal control for the optimal problem(4.3.2). Moreover, we also have Js,xo(uo)=V(s,xo).We also discuss the state feedback H∞, control designing for the systems with following forms. Theorem4.18For a given7>0, if there exists V∈C1,2(R+×Rn;R), such that where Thru the state feedback H∞control for system (4.4.5) on R+can be given by Here. λij(t,x) takes values λij which satisfies the following equality.