由G-布朗运动驱动的随机系统稳定性研究
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摘要
本文在Peng[57]的G-期望、G-布郎运动和G-随机微分方程理论的基础上,在经典的Lyapunov隐定性理论[44],[45],[61]和比控制理论[29],[76],[78]的启发下,主要讨论了由G-布朗运动驱动的随机系统的均方稳定性和H∞控制问题.具体内容包括：G-随机微分方程解对初值的导数及性质,G-随机微分方程均方稳定性的Lyapunov准则和应用,G-随机系统的稳定化最优控制和H∞控制等问题.详细内容如下：
第一章,主要回顾了G-期望理论框架和本文将用到的基本知识,并证明了如下形式的无穷时间区域G-随机微分方程和G-倒向随机微分方程解的存在性和唯一性.
定理1.16设b,hij,σj满足Lipschitz条件(1.3.25)和(1.3.27),则G-随机微分方程(1.3.24)在MG,lp(R+,Rn)中存在唯一解.
定理1.20设.f,hij满足Lipschitz条件(1.3.25)及(1.3.27),ξ∈LG1(Ω；R),则时域无穷的G-倒向随机微分方程(1.4.32)在MG,l1(R+；Rn)中存在唯一解.
第二章,主要讨论G-随机微分方程解对初值x的导数,在G-随机微分方程生成元算子的基础上,证明了函数u(t,x)=E|xstx|2在粘性解意义下所满足的微分性质.具体内容有：
定理2.19设b,hij,σj关于t是连续的,V∈C1,2(R+×Rn;R)),V,(?)tV关于x的二阶导数有界且满足Lipschitz条件,则生成元算子L具有下列形式：其中<(?)xV(t,x),h(t,x)>+<(?)xx2Vσ(t,x),σ(t,x)>是Sd(R)中的d×d对称矩阵,具体定义为
并求出了G-随机微分方程解对初值的一、二二阶均方导数.
定理2.24设G-随机微分方程(1.3.24)的系数b,hij,σj∈Cb1,2(R+×Rn;R),相应的解{XtS,x}t≥s∈MGA[0,T];Rn)则(1.3.24)的解Xts.x于x具有连续的。阶均方导数,并且一阶均方偏导数(?)xkXts,x满足下列G-随机微分方程阶G-偏导数(?)xkxlXts,x满足如下方程其中,(?)xx2b,(?)xx2hij,(?)xx2σj农示向量值函数对x偏导数所对应的分块矩阵,如(?)xx2b=((?)xxbv(u,Xus,x))v=1n∈Rn2×n,这里(?)xxbv是函数bv的Hermite阵.
命题2.2设G-随机微分方程满足引理2.25的条件,则u(s,x)=E|Xts,r|2是方程的粘性解.
第三章,主要讨论了G-随机微分方程均方稳定的Lyapunov判断准则及其在含不确定系数的随机系统稳定性判断中的应用.具体内容有：
定理3.6如果存在V∈C1,2(R+Rn;R),满足如下两条件：
(i)对任意(t,x)∈R+×Rn有fV(t,x)≤0,
(ii)存在常数c1,c2>0,使得C1｜x｜2≤V(t,x)≤C2｜x｜.则G-随机微分方程(3.2.1)是均方稳定的.
定理3.7如果存在非负函数V∈C1,2(R+×Rn;R)满足下而两个条件：
(i)存在常数A>0,使得(?)V(t.x)≤-λV(t.x).
(ii)存在常数c1,c2>0使得,对任意(t,x)∈R+×Rn,C1｜x｜(t,x)≤C2｜x｜2.,则G-随机微分方程(3.2.1)是均方指数渐近稳定的.
对如下形式的线性随机微分方程,其稳定性有相应的代数判据.其中B,Hij,Cj∈Rn×n",而Lyapunov,函数V(x)具有形式V(x)=xTPx,p∈S+n(R).记分块矩阵H,C如下
命题3.13设尸∈S+n(R),满足下列一对线性矩阵不等式其中α取-1或1,则P满足(3.3.16).从而线性G-随机微分方程(3.3.15)是均方指数渐近稳定的.
并在一定的条件下讨论了G-随机微分方程Lyapunov准则的必要性.
定理3.16设G-随机微分方程(3.3.15)的系数b,hij,σ满足引理3.14的的条件如果G-随机微分方程(3.3.15)是均方指数渐近稳定的,则存在V∈C12(R+×Rn；R)满足不等式(3.3.2)和(3.3.5).
在G-随机微分方程Lyapunov判断准则的基础上,我们还讨论了其如下含不确定参数的随机系统的稳定性首先,构造G雨数,令G：Sd(R)→R按如下定义则存在着G-期望及G-正态随机变量η(?)N(0,∑),使得G(A)=1/2E,相应的次线性期望空间为(Ω,(?),E).
定理3.21设b,hij,σj满足引理3,14的条件,则不确定随机系统(3.4.3)一致均方指数渐近稳定的允分必要条件是G-随机微分方程(3.4.6)是均方指数渐近稳定的.
第四章,讨论了以H∞范数作为主要指标的G-随机系统鲁棒性问题,具体包括G-随机微分方程的稳定化和基于状态反馈的H∞控制设计.具体内容有：
对如下含外界干扰的G-随机系统其中v∈MG2(R+；Rnv)为外界干扰项,z∈,Rnz为观测项,定义算子(?)MG2(R+；Rnv)→MG2(R+；Rnz)如下(?)v=z(·.0,v).算子的范数(?)为如正的H∞范数
定理4.4如果存和γ>0,∈C1,2(r+×Rn;R),使得,对任意(t,x.v)∈R+×Rn×Rnv,有HvoV(t,x):=(?)vV(t,x)+mT(t,x,v)m(t,x,v)-γ2｜v｜2≤0及(3.3.2),则(4.2.1)在R+上是外部稳定的.
对如下形式的含外界干扰的G-随机系统还有如下结论：
定理4.5设G-随机系统(4.2.11)的系数b.hij,σj满足引理3.14的条件,则下列三个条件是等价的：
(i)系统(4.4.1)是内部稳定的；
(ii)存在V∈C1,2(R+×Rn;R),Λ,c1,c2>0,使得(?)0V(t,x)≤-λV(t,x), c1｜x｜2≤V(t,x)≤c2｜x｜2,
(iii)存在M>0,使E｜x(x,x0,0)｜2≤M｜x0｜2.且存在二阶导数(?)xx2xV有界的V满足(4.2.12)和(4.2.13).
定理4.6如果存在V∈C1,2(R+×Rn;R)及γ>0,使得,对任意(t,x)∈R+×Rn,有其中及(3.3.2),则系统(4.2.11)在R+上是外部稳定的且‖(?)‖≤γ.
对如下的由G-随机微分方程来描述的控制系统,考虑其稳定化最优控制.有下列结论.
定理4.14设V∈C1,2(R+×Rn;R)及u0(t,x)∈u满足正列条件则,在控制u=u,0(t,x)下,系统(4.3.1)是均方指数渐近稳定的且u0(t,x)为满足最优问题(4.3.2)的最优化稳定控制,同时还有Js,x0(u0)=V(s,x0).
我们还讨论了如正形式G-随机系统的状态反馈H∞控制设计问题
定理4.18设γ>0,如果存在V∈C1,2(R+×Rn；R),满足其中则为系统(4.4.5)在R+上的状态反馈H∞控制,这里函数λij(t,x)定义为,对任意固定(t,x)∈R+×Rn,使得下式成立的λij为其值：
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 form
Moreover, 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, and
Proposition2.28Suppose the G-SDEs satisfy the conditions of Lemma2.25. then u(s,x)=E|Xts,x|2is the viscosity solution of
In 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) 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 stable
Furthermore, 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 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.
第一章,主要回顾了G-期望理论框架和本文将用到的基本知识,并证明了如下形式的无穷时间区域G-随机微分方程和G-倒向随机微分方程解的存在性和唯一性.
定理1.16设b,hij,σj满足Lipschitz条件(1.3.25)和(1.3.27),则G-随机微分方程(1.3.24)在MG,lp(R+,Rn)中存在唯一解.
定理1.20设.f,hij满足Lipschitz条件(1.3.25)及(1.3.27),ξ∈LG1(Ω；R),则时域无穷的G-倒向随机微分方程(1.4.32)在MG,l1(R+；Rn)中存在唯一解.
第二章,主要讨论G-随机微分方程解对初值x的导数,在G-随机微分方程生成元算子的基础上,证明了函数u(t,x)=E|xstx|2在粘性解意义下所满足的微分性质.具体内容有：
定理2.19设b,hij,σj关于t是连续的,V∈C1,2(R+×Rn;R)),V,(?)tV关于x的二阶导数有界且满足Lipschitz条件,则生成元算子L具有下列形式：其中<(?)xV(t,x),h(t,x)>+<(?)xx2Vσ(t,x),σ(t,x)>是Sd(R)中的d×d对称矩阵,具体定义为
并求出了G-随机微分方程解对初值的一、二二阶均方导数.
定理2.24设G-随机微分方程(1.3.24)的系数b,hij,σj∈Cb1,2(R+×Rn;R),相应的解{XtS,x}t≥s∈MGA[0,T];Rn)则(1.3.24)的解Xts.x于x具有连续的。阶均方导数,并且一阶均方偏导数(?)xkXts,x满足下列G-随机微分方程阶G-偏导数(?)xkxlXts,x满足如下方程其中,(?)xx2b,(?)xx2hij,(?)xx2σj农示向量值函数对x偏导数所对应的分块矩阵,如(?)xx2b=((?)xxbv(u,Xus,x))v=1n∈Rn2×n,这里(?)xxbv是函数bv的Hermite阵.
命题2.2设G-随机微分方程满足引理2.25的条件,则u(s,x)=E|Xts,r|2是方程的粘性解.
第三章,主要讨论了G-随机微分方程均方稳定的Lyapunov判断准则及其在含不确定系数的随机系统稳定性判断中的应用.具体内容有：
定理3.6如果存在V∈C1,2(R+Rn;R),满足如下两条件：
(i)对任意(t,x)∈R+×Rn有fV(t,x)≤0,
(ii)存在常数c1,c2>0,使得C1｜x｜2≤V(t,x)≤C2｜x｜.则G-随机微分方程(3.2.1)是均方稳定的.
定理3.7如果存在非负函数V∈C1,2(R+×Rn;R)满足下而两个条件：
(i)存在常数A>0,使得(?)V(t.x)≤-λV(t.x).
(ii)存在常数c1,c2>0使得,对任意(t,x)∈R+×Rn,C1｜x｜(t,x)≤C2｜x｜2.,则G-随机微分方程(3.2.1)是均方指数渐近稳定的.
对如下形式的线性随机微分方程,其稳定性有相应的代数判据.其中B,Hij,Cj∈Rn×n",而Lyapunov,函数V(x)具有形式V(x)=xTPx,p∈S+n(R).记分块矩阵H,C如下
命题3.13设尸∈S+n(R),满足下列一对线性矩阵不等式其中α取-1或1,则P满足(3.3.16).从而线性G-随机微分方程(3.3.15)是均方指数渐近稳定的.
并在一定的条件下讨论了G-随机微分方程Lyapunov准则的必要性.
定理3.16设G-随机微分方程(3.3.15)的系数b,hij,σ满足引理3.14的的条件如果G-随机微分方程(3.3.15)是均方指数渐近稳定的,则存在V∈C12(R+×Rn；R)满足不等式(3.3.2)和(3.3.5).
在G-随机微分方程Lyapunov判断准则的基础上,我们还讨论了其如下含不确定参数的随机系统的稳定性首先,构造G雨数,令G：Sd(R)→R按如下定义则存在着G-期望及G-正态随机变量η(?)N(0,∑),使得G(A)=1/2E
定理3.21设b,hij,σj满足引理3,14的条件,则不确定随机系统(3.4.3)一致均方指数渐近稳定的允分必要条件是G-随机微分方程(3.4.6)是均方指数渐近稳定的.
第四章,讨论了以H∞范数作为主要指标的G-随机系统鲁棒性问题,具体包括G-随机微分方程的稳定化和基于状态反馈的H∞控制设计.具体内容有：
对如下含外界干扰的G-随机系统其中v∈MG2(R+；Rnv)为外界干扰项,z∈,Rnz为观测项,定义算子(?)MG2(R+；Rnv)→MG2(R+；Rnz)如下(?)v=z(·.0,v).算子的范数(?)为如正的H∞范数
定理4.4如果存和γ>0,∈C1,2(r+×Rn;R),使得,对任意(t,x.v)∈R+×Rn×Rnv,有HvoV(t,x):=(?)vV(t,x)+mT(t,x,v)m(t,x,v)-γ2｜v｜2≤0及(3.3.2),则(4.2.1)在R+上是外部稳定的.
对如下形式的含外界干扰的G-随机系统还有如下结论：
定理4.5设G-随机系统(4.2.11)的系数b.hij,σj满足引理3.14的条件,则下列三个条件是等价的：
(i)系统(4.4.1)是内部稳定的；
(ii)存在V∈C1,2(R+×Rn;R),Λ,c1,c2>0,使得(?)0V(t,x)≤-λV(t,x), c1｜x｜2≤V(t,x)≤c2｜x｜2,
(iii)存在M>0,使E｜x(x,x0,0)｜2≤M｜x0｜2.且存在二阶导数(?)xx2xV有界的V满足(4.2.12)和(4.2.13).
定理4.6如果存在V∈C1,2(R+×Rn;R)及γ>0,使得,对任意(t,x)∈R+×Rn,有其中及(3.3.2),则系统(4.2.11)在R+上是外部稳定的且‖(?)‖≤γ.
对如下的由G-随机微分方程来描述的控制系统,考虑其稳定化最优控制.有下列结论.
定理4.14设V∈C1,2(R+×Rn;R)及u0(t,x)∈u满足正列条件则,在控制u=u,0(t,x)下,系统(4.3.1)是均方指数渐近稳定的且u0(t,x)为满足最优问题(4.3.2)的最优化稳定控制,同时还有Js,x0(u0)=V(s,x0).
我们还讨论了如正形式G-随机系统的状态反馈H∞控制设计问题
定理4.18设γ>0,如果存在V∈C1,2(R+×Rn；R),满足其中则为系统(4.4.5)在R+上的状态反馈H∞控制,这里函数λij(t,x)定义为,对任意固定(t,x)∈R+×Rn,使得下式成立的λij为其值：
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 form
Moreover, 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, and
Proposition2.28Suppose the G-SDEs satisfy the conditions of Lemma2.25. then u(s,x)=E|Xts,x|2is the viscosity solution of
In 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)
(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 stable
Furthermore, 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
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.
引文
[1]L. ARNOLD, Stochastic differential equations: theory and applications, John Wiley and Sons, 1974.
[2]P.R. BEESACK, Gronwall inequalities, Carleton Univ. Math. Notes, 11, 1975.
[3]R. BELLMAN, Stability Theory of Differential Equations, MeGraw Hill, New York, 1953.
[4]R.K. BOEL, D. VARAIYA, Optimal control of jump processes, SIAM J. Control Optim., 15(1977): 92 119.
[5]R. BUOKDAIIN, J. Ma, Stochastic viscosity solutions for nonlinear stochastic, partial differential equations, I, Stochastic Process. Appl., 93(2001): 181 204.
[6]R. BUCKDAHN, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations, II, Stochastic Process. Appl., 93(2001): 205 228.
[7]B.S. CHEN, Y. Lin and Y. Chuano, Robust H∞ observer-based tracking control of stochastic immune systems under environmental disturbances and measurement noises, Asian J. Control, 13(2011): 667 690.
[8]X. CHEN, Dynkin's formula under the G-Expectation, Statistics and Proba-bility Letters, 80(2010): 519-526.
[9]Z. CHEN, Continuous properties of g-martingales, Chin. Ann. of Math., 22B80(2001): 115-128.
[10]Z. Chen, L. Epstein, Ambiguity, risk and assert returns in continous time, Econometrica, 70(2002): 1403 1443.
[11]Z. CHEN, T. CHEN, M. DAVISON, Choquet expectation, and Peng's g-expectation, Ann. Prob., 33(2005): 1179 1199.
[12]L. DENIS, M. Hu, AND S. Peng, Function spaces and capacity re-lated to a sublinear expectation: application to G-Brownian motion pathes, arXiv:0802.1240vl[math.PR] 9 Feb. 2008.
[13]J.L. DOOB, Classical Potential theory and its Probabilistic Counterpart, Springer-Verlag, New York, 1984.
[14]J.C. DOYLE, K. GLOVER, P.P. KRANCIS, FRANCIS, State-space so-lutions to standard and control problem, IEEE Trans. Automat. Control, 34(1989): 881-897.
[15]P.GAHINET, P.APKARIAN, A linear matrix inequality approach to H∞ con-trol, Int. J. Robust Nonlin., 4(1994): 421-448.
[16]F. GAO, Pathwise properties and homeomorphic flows for stochastic differen-tial equations driven by G-Brownian motion, Stochastic Processes and their Apllications, 119(2009): 3356-3382.
[17]F. GAO, H. JIANG, Large deviation for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and their Apllications, 120(2010): 2212-2240.
[18]E. GERSHON, D. J. N. LIMEBEER, U. SHAKED, AND I. YAESH, Robust H∞ filtering of stationary continuous-time linear systems with stochastic uncer-tainties, IEEE Trans. Automat. Control, 46 (2001): 1788-1793.
[19]E. GERSHON, U. SHAKD,H∞ output-feedback of discrete-time systems with state-multiplicative noise, Automatica, 44(2008): 574-579.
[20]G. CHOQUET, Theory of Capacities, Annales de Institut Fourier, 5(1953): 131-195.
[21]R. GETOOR, J.L. Doob: Foundations of Stochastic Processes and Probabilistic Potential Theory, The Annals of Probability, 37(2009): 1647-1663.
[22]I.I.GIHMAN AND A.V. SROROHOD, The theory of stochastic processes Ⅲ, Translated from the Russian by S.Kotz, Springer-Verlag, Berlin, 1979.
[23]I.I.GIHMAN AND A.V. SROROHOD, Stochastic differential equations, Springer-Verlag, Berlin, 1972.
[24]K. Glover, J.C. Doyle, State-space formulae for all stabilizing controllers that satisfy an H (infinity)-noun bound and relations to risk sensitivity, Sys-tems and Control Letters, 11(1988): 107-172.
[25]C.GRAHAM, Mckean-Vlasov Ito-Skorohod equations and nonlinear diffusion with discrete jumps, Stochastic process and their applications, 40(1992): 69 82.
[26]T.H. GRONWALL, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 20(1919): 293-296.
[27]R.Z.HAS'MINSKII, Stochastic stability of differential equations translated by D.louvish, Sijthoff and Noordhoff, 1981.
[28]J.W. Helton,"Orbit structure of the Mobius transformation semigroup action on H-infinity (broadband matching)". Adv. in Math. Suppl. Stud. 3(1978): 129-197.
[29]D. HINRICHSEN AND A. J. Pritchard, Stochastic H∞, SIAM J. Control Optim., 36(1998): 1504 1538.
[30]M. Hu AND S. Peng, On Representation Theorem of G-Expectations and Paths of G-Brownian Motion , Acta Mathematicae Applicatae Sinica, English Series 25(2009): 539-546.
[31]G.A. Hunt, Markoff Processes and Potentials Ⅰ, Illinois J. Math., 1(1957): 44-93.
[32]G.A. Hunt, Markoff Processes and Potentials Ⅲ, Illinois J. Math., 2(1958): 151 213.
[33]R.E. Kalman, Control of randomly varying linear dynamical systems, Proc. Sympos. Appl. Math., 13 Amer. Math. Soc, Provindence, R.I., 287 298, 1962.
[34]V.L. KHARITONOV, Asymptotical stability of an equilibrium position of a fam-ily of systems of linear differential equations (in Russian), Differential Urav-nen, 14(1978): 2086-2088.
[35]H.J. KUSMNER, Stochastic stability and control, Academic Press, 1967.
[36]J. LASALLE AND S. LEFSCHETZ, Stability by Liapunov's Direct Method with Applications, Academic Press, NewYork,1961.
[37]X.LI AND S.PENG, Stopping times and related Ito's calculus with G-Brownian motion, arXiv:0910.3871v2 [math.PR] 6 Apr 2011.
[38]F. LIN, Robust control of nonlinear systems:compensating for uncertainty, International Journal of control,56(1992):1453-1459.
[39]F. LIN, An optimal control approach to robust control design, International Journal of control,73(2000):177-186.
[40]F. LIN, Robust Control Design: An Optimal Control Approach, John Wiley & Sons,2007.
[41]J.LI AND S.PENG, Stochastic optimization theory of backward stochastic dif-ferential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations, Nonlinear Anal. Theor.,70(2009):1776-1796.
[42]X.LIN, R.ZHANG, H∞ Control for Stochastic Systems with Poisson jumps, J. Syst. Sci. Complex,24(2011):683-700.
[43]X.LIN, W.ZHANG, AND X.WANG, The Output Feedback H∞ Control De-sign for the Linear Stochastic System Driven by Both Brownian Motion and Poisson Jumps: A Nonlinear Matrix Inequality Approach, Asian Journal of Control, DOI:10.1002/asjc.644,2013.
[44]A.M. LYAPUNOV, The general problem of the stability of motion (in Russian, 1892), translated by A. T. Fuller, Tayor and Francis,1992.
[45]X. MAO, Exponential stability of stochastic differential equations, Marcel Dekker,1994.
[46]X. MAO, Stochastic Versions of the LaSalle Theorem, Journal of differential equations,153(1999):988-1001.
[47]R.C.MERTON, Option pricing when underlying stock returns are discontinu-ous, Journal of Financial Economics,3(1976):125-144.
[48]B.OKSENDAL, Stochastic Differential equations: An Introduction with appli-cations, Springer-Verlag, NewYork,2005.
[49]E. PARDOUX AND S. PENG, Adapted Solution of a Backward stochastic dif-ferential equation, Systems Control Lett.,14(1990):55-61.
[50]S. PENG, A General Stochastic Maximum Principle for Optimal Control. Prob-lems, SIAM J. Control and Optimization,28(1990):966-979.
[51]S. PENG, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim.,30(1992):284 304.
[52]S. PENG, Backward SDE and related G-expectations. Backward stochastie dif-ferential equations, in El N. Karoui and L. Mazliak, eds. Pitman Res. Notes Math. Ser. Longman Harlow,364(1997):141 159.
[53]S. PENG, Nonlinear expectations and nonlinear Markov chains,,26B(2005), 159184.
[54]S. PENG, G-expectation, G-Brownian motion and related stochastic calculus of ltd's type, Proceedings of the 2005 Abel Symposium,2007:541 567.
[55]S. PENG, G-Brownian Motion and Dynamic Risk Measure, wider Volatility Uncertainty, arXiv:0711.2834vl [math.PR],19 Nov 2007.
[56]S. PENG, Multi-dimensional G-Brownian motion and related stochastic calcu-lus under G-expectation, Stoch. Proc. Appl.,,118(2008):2223 2253.
[57]S. PENG, Nonlinear Expectations and Stochastic Calculus under Uncertainty,, 2010, preprint.
[58]S. PENG, Z. Wu, Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control, SIAM J. Control and Opti-mization,37(1999):825-843.
[59]L. S. PONTRYAGIN, V. G. BOLTYANSKII, R. V. GAMKRELIDZE, AND E. F. MISHCHENKO, The Mathematical Theory of Optimal Processes, Interscience Publishers, John Wiley and Sons, New York,1962.
[60]R.RISHEL, A minimum principle for controlled jump processes, Lecture Notes in Economics and Mathematical Systems,107(1975):493-508.
[61]N. ROUCHE, P. HABETS AND M. LLOY, Stability theory by Lyapunov's direct method, Springer-Verlag,1977.
[62]J. SHI, Z. WU, Relationship between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions, Appl. Math. Optim.,63(2011):151-189.
[63]S.TANG AND X.LI, Necessary condition for optimal control of stochastic sys-tems with random jumps, SIAM J.Control Optim.,32(1994):1447-1475.
[64]A. TANNENBAUM, Feedback stabilization of linear dynamical plants with un-certainty in the gain factor, International Journal of Control 32(1980):1-16.
[65]M.G. TODOROV, M.D. FRAGOSO, Output Feedback H∞ Control of Continuous-Time Infinite Markovian Jump Linear Systems via LMI Meth-ods, SIAM J. Control Optim.,42(2008):950-974.
[66]G. WANG, Q. ZHANG AND V. SREERAM, Robust H∞ control of norm bounded uncertain systems via Markovian approach, Asian J. Control,13(2011):956-965.
[67]Z.Wu, Fully Coupled FBSDE with Brownian Motion and Poisson Process in Stopping Time Duration, J.Aust.Math.Soc.,74(2003):249-266.
[68]Z.Wu, Z. Yu, Dynamic programming principle for one kind of stochastic re-cursive optimal control problem and Hamilton-Jacobi-Bellman equation, SIAM J. Control Optim.,47(2008):2616-2641.
[69]Z. Wu, F. ZHANG, Stochastic Maximum Principle for Optimal Control Prob-lems of Forward-Backward Systems Involving Impulse Controls, IEEE Trans. Auto. Control,56(2011):1401-1406.
[70]C.S.WU AND B.S.CHEN, Unified design for H2/H∞, and mixed control of spacecraft, J. Guidance Control Dyn.,22(1999):884-896.
[71]C.S.WU AND B.S.CHEN, Adaptive attitude control of spacecraft:Mixed H2/H∞ approach, J. Guidance Control Dyn.,24(2001):755-766.
[72]J. Xu, B. Zhang, Martingale characterization of G-Brownian motion, Stoch. Proc. Appl., 119(2009): 232-248.
[73]J. Yong, X. Y. Zhou, Stochastic Controls: Hamiltonian systems and HJB equations, Springer-Verlag, 1999.
[74]D. ZHANG AND Z. Chen, Exponential stability for stochastic differential equa-tion driven by G-Brownian motion, Applied Mathematics Letters, 25(2012): 1906-1910.
[75]D. ZHANG AND Z. Chen, Stability theorem for stochastic differential equations driven by G-Brownian motion. Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matcmatica, 19(2011): 205 221.
[76]W. Zhang and B. S. Chen, State feedback Hoc control for a class of non-linear stochastic systems, SIAM J. Control Optim., 44 (2006): 1973-1991.
[77]W. Zhang, H. Zhang and B. S. Chen, Stochastic H2/H∞ control with (x,u,v)-dependent noise: Finite horizon case, Automation., 42 (2006): 1891 1898.
[78]G.Zames, Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Trans. Automatic Control, 26(1981): 301 320.
[79]G.Zames, B.A. Francis, On H∞ optimal sensitivity theory for SISO feed-back systems, IEEE Trans. Automatic Control, 29(1984): 9-16.
[2]P.R. BEESACK, Gronwall inequalities, Carleton Univ. Math. Notes, 11, 1975.
[3]R. BELLMAN, Stability Theory of Differential Equations, MeGraw Hill, New York, 1953.
[4]R.K. BOEL, D. VARAIYA, Optimal control of jump processes, SIAM J. Control Optim., 15(1977): 92 119.
[5]R. BUOKDAIIN, J. Ma, Stochastic viscosity solutions for nonlinear stochastic, partial differential equations, I, Stochastic Process. Appl., 93(2001): 181 204.
[6]R. BUCKDAHN, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations, II, Stochastic Process. Appl., 93(2001): 205 228.
[7]B.S. CHEN, Y. Lin and Y. Chuano, Robust H∞ observer-based tracking control of stochastic immune systems under environmental disturbances and measurement noises, Asian J. Control, 13(2011): 667 690.
[8]X. CHEN, Dynkin's formula under the G-Expectation, Statistics and Proba-bility Letters, 80(2010): 519-526.
[9]Z. CHEN, Continuous properties of g-martingales, Chin. Ann. of Math., 22B80(2001): 115-128.
[10]Z. Chen, L. Epstein, Ambiguity, risk and assert returns in continous time, Econometrica, 70(2002): 1403 1443.
[11]Z. CHEN, T. CHEN, M. DAVISON, Choquet expectation, and Peng's g-expectation, Ann. Prob., 33(2005): 1179 1199.
[12]L. DENIS, M. Hu, AND S. Peng, Function spaces and capacity re-lated to a sublinear expectation: application to G-Brownian motion pathes, arXiv:0802.1240vl[math.PR] 9 Feb. 2008.
[13]J.L. DOOB, Classical Potential theory and its Probabilistic Counterpart, Springer-Verlag, New York, 1984.
[14]J.C. DOYLE, K. GLOVER, P.P. KRANCIS, FRANCIS, State-space so-lutions to standard and control problem, IEEE Trans. Automat. Control, 34(1989): 881-897.
[15]P.GAHINET, P.APKARIAN, A linear matrix inequality approach to H∞ con-trol, Int. J. Robust Nonlin., 4(1994): 421-448.
[16]F. GAO, Pathwise properties and homeomorphic flows for stochastic differen-tial equations driven by G-Brownian motion, Stochastic Processes and their Apllications, 119(2009): 3356-3382.
[17]F. GAO, H. JIANG, Large deviation for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and their Apllications, 120(2010): 2212-2240.
[18]E. GERSHON, D. J. N. LIMEBEER, U. SHAKED, AND I. YAESH, Robust H∞ filtering of stationary continuous-time linear systems with stochastic uncer-tainties, IEEE Trans. Automat. Control, 46 (2001): 1788-1793.
[19]E. GERSHON, U. SHAKD,H∞ output-feedback of discrete-time systems with state-multiplicative noise, Automatica, 44(2008): 574-579.
[20]G. CHOQUET, Theory of Capacities, Annales de Institut Fourier, 5(1953): 131-195.
[21]R. GETOOR, J.L. Doob: Foundations of Stochastic Processes and Probabilistic Potential Theory, The Annals of Probability, 37(2009): 1647-1663.
[22]I.I.GIHMAN AND A.V. SROROHOD, The theory of stochastic processes Ⅲ, Translated from the Russian by S.Kotz, Springer-Verlag, Berlin, 1979.
[23]I.I.GIHMAN AND A.V. SROROHOD, Stochastic differential equations, Springer-Verlag, Berlin, 1972.
[24]K. Glover, J.C. Doyle, State-space formulae for all stabilizing controllers that satisfy an H (infinity)-noun bound and relations to risk sensitivity, Sys-tems and Control Letters, 11(1988): 107-172.
[25]C.GRAHAM, Mckean-Vlasov Ito-Skorohod equations and nonlinear diffusion with discrete jumps, Stochastic process and their applications, 40(1992): 69 82.
[26]T.H. GRONWALL, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 20(1919): 293-296.
[27]R.Z.HAS'MINSKII, Stochastic stability of differential equations translated by D.louvish, Sijthoff and Noordhoff, 1981.
[28]J.W. Helton,"Orbit structure of the Mobius transformation semigroup action on H-infinity (broadband matching)". Adv. in Math. Suppl. Stud. 3(1978): 129-197.
[29]D. HINRICHSEN AND A. J. Pritchard, Stochastic H∞, SIAM J. Control Optim., 36(1998): 1504 1538.
[30]M. Hu AND S. Peng, On Representation Theorem of G-Expectations and Paths of G-Brownian Motion , Acta Mathematicae Applicatae Sinica, English Series 25(2009): 539-546.
[31]G.A. Hunt, Markoff Processes and Potentials Ⅰ, Illinois J. Math., 1(1957): 44-93.
[32]G.A. Hunt, Markoff Processes and Potentials Ⅲ, Illinois J. Math., 2(1958): 151 213.
[33]R.E. Kalman, Control of randomly varying linear dynamical systems, Proc. Sympos. Appl. Math., 13 Amer. Math. Soc, Provindence, R.I., 287 298, 1962.
[34]V.L. KHARITONOV, Asymptotical stability of an equilibrium position of a fam-ily of systems of linear differential equations (in Russian), Differential Urav-nen, 14(1978): 2086-2088.
[35]H.J. KUSMNER, Stochastic stability and control, Academic Press, 1967.
[36]J. LASALLE AND S. LEFSCHETZ, Stability by Liapunov's Direct Method with Applications, Academic Press, NewYork,1961.
[37]X.LI AND S.PENG, Stopping times and related Ito's calculus with G-Brownian motion, arXiv:0910.3871v2 [math.PR] 6 Apr 2011.
[38]F. LIN, Robust control of nonlinear systems:compensating for uncertainty, International Journal of control,56(1992):1453-1459.
[39]F. LIN, An optimal control approach to robust control design, International Journal of control,73(2000):177-186.
[40]F. LIN, Robust Control Design: An Optimal Control Approach, John Wiley & Sons,2007.
[41]J.LI AND S.PENG, Stochastic optimization theory of backward stochastic dif-ferential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations, Nonlinear Anal. Theor.,70(2009):1776-1796.
[42]X.LIN, R.ZHANG, H∞ Control for Stochastic Systems with Poisson jumps, J. Syst. Sci. Complex,24(2011):683-700.
[43]X.LIN, W.ZHANG, AND X.WANG, The Output Feedback H∞ Control De-sign for the Linear Stochastic System Driven by Both Brownian Motion and Poisson Jumps: A Nonlinear Matrix Inequality Approach, Asian Journal of Control, DOI:10.1002/asjc.644,2013.
[44]A.M. LYAPUNOV, The general problem of the stability of motion (in Russian, 1892), translated by A. T. Fuller, Tayor and Francis,1992.
[45]X. MAO, Exponential stability of stochastic differential equations, Marcel Dekker,1994.
[46]X. MAO, Stochastic Versions of the LaSalle Theorem, Journal of differential equations,153(1999):988-1001.
[47]R.C.MERTON, Option pricing when underlying stock returns are discontinu-ous, Journal of Financial Economics,3(1976):125-144.
[48]B.OKSENDAL, Stochastic Differential equations: An Introduction with appli-cations, Springer-Verlag, NewYork,2005.
[49]E. PARDOUX AND S. PENG, Adapted Solution of a Backward stochastic dif-ferential equation, Systems Control Lett.,14(1990):55-61.
[50]S. PENG, A General Stochastic Maximum Principle for Optimal Control. Prob-lems, SIAM J. Control and Optimization,28(1990):966-979.
[51]S. PENG, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim.,30(1992):284 304.
[52]S. PENG, Backward SDE and related G-expectations. Backward stochastie dif-ferential equations, in El N. Karoui and L. Mazliak, eds. Pitman Res. Notes Math. Ser. Longman Harlow,364(1997):141 159.
[53]S. PENG, Nonlinear expectations and nonlinear Markov chains,,26B(2005), 159184.
[54]S. PENG, G-expectation, G-Brownian motion and related stochastic calculus of ltd's type, Proceedings of the 2005 Abel Symposium,2007:541 567.
[55]S. PENG, G-Brownian Motion and Dynamic Risk Measure, wider Volatility Uncertainty, arXiv:0711.2834vl [math.PR],19 Nov 2007.
[56]S. PENG, Multi-dimensional G-Brownian motion and related stochastic calcu-lus under G-expectation, Stoch. Proc. Appl.,,118(2008):2223 2253.
[57]S. PENG, Nonlinear Expectations and Stochastic Calculus under Uncertainty,, 2010, preprint.
[58]S. PENG, Z. Wu, Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control, SIAM J. Control and Opti-mization,37(1999):825-843.
[59]L. S. PONTRYAGIN, V. G. BOLTYANSKII, R. V. GAMKRELIDZE, AND E. F. MISHCHENKO, The Mathematical Theory of Optimal Processes, Interscience Publishers, John Wiley and Sons, New York,1962.
[60]R.RISHEL, A minimum principle for controlled jump processes, Lecture Notes in Economics and Mathematical Systems,107(1975):493-508.
[61]N. ROUCHE, P. HABETS AND M. LLOY, Stability theory by Lyapunov's direct method, Springer-Verlag,1977.
[62]J. SHI, Z. WU, Relationship between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions, Appl. Math. Optim.,63(2011):151-189.
[63]S.TANG AND X.LI, Necessary condition for optimal control of stochastic sys-tems with random jumps, SIAM J.Control Optim.,32(1994):1447-1475.
[64]A. TANNENBAUM, Feedback stabilization of linear dynamical plants with un-certainty in the gain factor, International Journal of Control 32(1980):1-16.
[65]M.G. TODOROV, M.D. FRAGOSO, Output Feedback H∞ Control of Continuous-Time Infinite Markovian Jump Linear Systems via LMI Meth-ods, SIAM J. Control Optim.,42(2008):950-974.
[66]G. WANG, Q. ZHANG AND V. SREERAM, Robust H∞ control of norm bounded uncertain systems via Markovian approach, Asian J. Control,13(2011):956-965.
[67]Z.Wu, Fully Coupled FBSDE with Brownian Motion and Poisson Process in Stopping Time Duration, J.Aust.Math.Soc.,74(2003):249-266.
[68]Z.Wu, Z. Yu, Dynamic programming principle for one kind of stochastic re-cursive optimal control problem and Hamilton-Jacobi-Bellman equation, SIAM J. Control Optim.,47(2008):2616-2641.
[69]Z. Wu, F. ZHANG, Stochastic Maximum Principle for Optimal Control Prob-lems of Forward-Backward Systems Involving Impulse Controls, IEEE Trans. Auto. Control,56(2011):1401-1406.
[70]C.S.WU AND B.S.CHEN, Unified design for H2/H∞, and mixed control of spacecraft, J. Guidance Control Dyn.,22(1999):884-896.
[71]C.S.WU AND B.S.CHEN, Adaptive attitude control of spacecraft:Mixed H2/H∞ approach, J. Guidance Control Dyn.,24(2001):755-766.
[72]J. Xu, B. Zhang, Martingale characterization of G-Brownian motion, Stoch. Proc. Appl., 119(2009): 232-248.
[73]J. Yong, X. Y. Zhou, Stochastic Controls: Hamiltonian systems and HJB equations, Springer-Verlag, 1999.
[74]D. ZHANG AND Z. Chen, Exponential stability for stochastic differential equa-tion driven by G-Brownian motion, Applied Mathematics Letters, 25(2012): 1906-1910.
[75]D. ZHANG AND Z. Chen, Stability theorem for stochastic differential equations driven by G-Brownian motion. Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matcmatica, 19(2011): 205 221.
[76]W. Zhang and B. S. Chen, State feedback Hoc control for a class of non-linear stochastic systems, SIAM J. Control Optim., 44 (2006): 1973-1991.
[77]W. Zhang, H. Zhang and B. S. Chen, Stochastic H2/H∞ control with (x,u,v)-dependent noise: Finite horizon case, Automation., 42 (2006): 1891 1898.
[78]G.Zames, Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Trans. Automatic Control, 26(1981): 301 320.
[79]G.Zames, B.A. Francis, On H∞ optimal sensitivity theory for SISO feed-back systems, IEEE Trans. Automatic Control, 29(1984): 9-16.