理学硕士学位论文一类线性差分方程组解的稳定性分析郭亮哈尔滨工业大学2004年7月图书分类号：O241.84U.D.C.: 517.962.2理学硕士学位论文一类线性差分方程组解的稳定性分析硕士研究生：郭亮导师：薛小平教授申请学位级别：理学硕士学科、专业：基础数学所在单位：理学院数学系答辩日期：2004年7月授予学位单位：哈尔滨工业大学Classified Index：O241.84U.D.C.: 517.962.2A Dissertation for the Degree of Master of ScienceSTABILITY OF LINEAR DIFFERENCEEQUATIONSCandidate：Guo LiangSupervisor：Prof. Xue XiaopingAcademic Degree Applied for：Master of Science Speciality：Pure Mathematics Affiliation：Department of Mathematics Date of Defence：July, 2004Degree-Conferring-Institution：Harbin Institute of Technology哈尔滨工业大学理学硕士学位论文摘要差分方程是和微分方程相平行的一个数学理论，它不但在数学各分支内应用甚广，而且由于电子计算机的迅速发展和广泛使用，它已成为现代控制理论、通讯理论等科技领域内的一个基本数学工具。
用差分方程描述动力系统稳定性的研究是李雅普诺夫稳定性理论的近代内容。
Lyapunov函数法、LaSalle不变原理、比较原理虽然是研究离散系统稳定性的一般方法，但应用这些方法构造V函数技巧性强，因此无一般规律可言。
如果能够根据系统本身的参数，得出一系列简单实用的离散系统稳定性代数判据，这就会使一些离散系统稳定性问题得到简化，更加简洁实用。
姓名：张书华简历：1993年9月至1996年7月在中国科学校系统科学研究所林群院士的指导下，攻读计算数学专业博士学位，1997年9月至1999年12月在加拿大Alberta大学从事博士后研究工作，现为天津财经大学基础部教授。
在教学方面，张书华多年来一直从事本科生和研究生的教学工作，曾经教授过“数学分析”、“实变函数”、“数值分析”、“应用数学基础”、“专业外语”、“高等数学”、“线性代数”、“概率与数理统计”、“微积分”等多门课程，并用英语讲授过“Calculus with Applications”课程。
他还主持过“概率与数理统计多媒体课件研发”的教研项目，其多媒体课件曾获“天津商学校优秀多媒体课件评比”二等奖。
现正为本科生讲授公共基础课“微积分I（选教）”与“微积分II”及信息系本科生专业课“实变函数”与“专业外语”。
在科研方面，张书华自1993年以来一直从事偏微分方程高效数值解的研究工作，在偏积分—微分方程及相关方程的有限元高精度算法及数理金融学等方面完成了一系列工作，其中包括证明了H. Brunner 教授等提出的一个猜想，并解决了由他提出的两个公开问题。
近年来，在此方面出版英文专著1部，发表论文30余篇 (11篇被SCI检索)，其中大部分论文发表在国际权威杂志和国内核心期刊上，且其中部分工作曾获天津市科技进步奖自然科学二等奖。
2002年至2003年在加拿大从事合作研究期间曾指导博士研究生1名。
他是国内外许多杂志的审稿人，还是在伊朗德黑兰召开的国际研讨会“Efficient Techniques for Numerical Solutions of Coupled PDE’s and Applications to Reservoir Simulations”的大会特邀报告人之一。
他现为天津市计算数学会理事。
一、完成与在研项目1．参与完成加拿大自然科学与工程委员会(Natural Sciences and Engineering Research Council of Canada)战略基金两项：(1) On-line Real-time intelligent systems for accident prevention (OGP0009406).(2) Integrated intelligent systems for modeling and monitoring for petroleum and environmental impact (OGP0001926).2．参与完成加拿大自然科学与工程委员会基金一项：Finite element methods for mathematical equations arising in viscoelastic, thermoelasticity, and micro electronic device (OGP0006940)3．主持完成国家留学回国人员基金项目一项：偏微分方程的高精度算法(留[1999]363).4．参与国家973在研项目一项：电力系统中微分-代数方程的并行算法(G1998020322).5．主持教育部985行动计划“应用数学中的前沿问题”在研项目一项：积分-微分方程的高效数值方法(J200013).6．主持天津市教委科技发展基金在研项目一项：积分-微分方程的高效数值方法7．参与天津教委科技发展基金在研项目一项：期权定价的数值方法8．主持天津财经大学科研发展基金在研项目一项：积分-微分方程的高效混合元方法二、近期发表的主要论著目录1. Galerkin Methods of Higher Accuracy for Partial Integro-Differential Equations and Volterra Type Equations (专著), Research Information Ltd, 2000.2. An immediate analysis for global superconvergence for integrodifferential equations, Appl. Math., 42 (1998), 1-21.3. A direct global superconvergence analysis for Sobolev and viscoelasticity type equations, Appl. Math., 42 (1998), 23-34.4. Extrapolation and defect correction for diffusion equations with boundary integralconditions, ACTA Mathematica Scientia, 17(1998), 405-412. (SCI检索)5. Non-standard Galerkin methods of high accuracy for parabolic problems, Transa. Tianjin Univ., 3(1998), 66-70.6. Methods for improving approximate accuracy for hyperbolic integro-differential equations, Syst. Sci. & Math. Sci., 10(1999), 282-288.7. Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations, J. Comput. Math., 16(1999), 51-62. (SCI检索)8. High accuracy analysis for first order hyperbolic equations by Petrov-Galerkin methods, Syst. Sci. & Math. Sci., 11(1999), 134-143.9. An acceleration method for integral equations by using interpolation post-processing, Advances in Comput. Math., 9(1999), 117-129. (SCI检索)10. High accuracy analysis for integrodifferential equations, ACTA Mathematicae Applicatae Sinica, 14(1999), 202-211.11. Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods, J. Integral Equations Appl., 10(1999), 375-396.12. A new numerical method for monitoring environmental impact, IFAC, 15(1999), 85-90.13. Numerical solutions for second-kind Volterra integral equations by Galerkin methods, Appl. Math., 45 (2000), 19-39.14. Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations, Appl. Math., 45(2000), 241-263.15. Petrov-Galerkin methods for linear Volterra integro-differential equations, SIAM J. Numer. Anal., 38(2000), 937-963. (SCI检索)16. Extrapolation and a-posteriori error estimators of Petrov-Galerkin methods for non-linear Volterra integro-differential equations, J. Comput. Math., 19 (2001), 407-422. (SCI检索) 17.Optimal error estimates and superconvergence in maximum norm of mixed finite element methods for nonFickian flows in porous media, Discrete Continuous Dynamics Systems, 9(2001), 178-208. (SCI检索)18. Petrov-Galerkin methods for Voletrra integro-differential equations, Dynamics Continuous Discrete Impulsive Systems, 8(2001), 405-424. (SCI检索)19. Sharp L^2-error estimates and superconvergence of mixed finite element methods for non-Fickian flows in porous media, SIAM J. Numer. Anal., 40(2002), 1538--1560. (SCI检索) 20. Global superconvergence analysis for Galerkin finite element methods ofintegro-differential and related equations, Dynamics Continuous Discrete Impulsive Systems, 9(2002), 389-405. (SCI检索)21. Higher accuracy analysis for the Wilson element solution to Sobolev type equations, 32(2002), Math. Practice Theory, 85-90.22. Asymptotic expansion and extrapolation of Galerkin finite element methods for parabolic partial differential equations, Dynamics Continuous Discrete Impulsive Systems, 10(2003), 13-24. (SCI检索)23. Determination of an unknown coefficient in a parabolic inverse problem, Dynamics Continuous Discrete Impulsive Systems, 10(2003), 35-44. (SCI检索)24. Analysis for Global superconvergence of the mixed element solution to Sobolev type equation, Math. Practice Theory, 33(2003), 85-90.25. An integral formula and computation for the solution of American options, preprint.26. A new representation for the valuation of American options, preprint.27. Numerical analysis for the pricing of American put options on zero-coupon bonds, preprint.28. Probabilistic numerical method for partial differential equations and its applications in the valuation of options, preprint.29. Superconvergence and a posteriori estimation of finite element methods for a nonlocal problem in American option valuation, preprint.30. Supply chain coordination under channel rebates, preprint.三、社会兼职现为天津市计算数学会理事。
FINITE DIMENSIONAL REDUCTION OF NONAUTONOMOUS DISSIPATIVESYSTEMSAlain MiranvilleUniversit´e de Poitiers Collaborators:Long time behavior of equations of the formy′=F(t,y)For autonomous systems:y′=F(y)In many situations,the evolution of the sys-tem is described by a system of ODEs:y=(y1,...,y N)∈R N,F=(F1,...,F N)Assuming that the Cauchy problemy′=F(y),y(0)=y0,is well-posed,we can define the family of solv-ing operators S(t),t≥0,acting on a subset φ⊂R N:S(t):φ→φy0→y(t)This family of operators satisfiesS(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0We say that it forms a semigroup onφQualitative study of such systems:goes back to Poincar´eMuch is known nowadays,at least in low di-mensionsEven relatively simple systems can generate very complicated chaotic behaviorsThese systems are sensitive to perturbations: trajectories with close initial data may diverge exponentially→Temporal evolution unpredictable on ti-me scales larger than some critical value→Show typical stochastic behaviorsExample:Lorenz systemx′=σ(y−x)y′=−xy+rx−yz′=xy−bzObtained by truncature of the Navier-Stokes equationsGives an approximate description of a layer of fluid heated from belowSimilar to what is observed in the atmosphereFor a sufficiently intense heating:sensitive dependence on the initial conditions,repre-sents a very irregular convection→Butterfly effectVery often,the trajectories are localized in some subset of the phase space having a very complicated geometric structure(e.g.,locally homeomorphic to the product of R m and a Cantor set)→Strange attractor(Ruelle and Takens)Main feature of a strange attractor:dimen-sionSensitivity to initial conditions:>2(dimen-sion of the phase space≥3,say,3)Contraction of volumes:its volume is equal to0→noninteger,strictly between2and3→Fractal dimensionExample:Lorenz system:dim F A=2.05...Distributed systems:systems of PDEsφis a subset of an infinite dimensional func-tion space(e.g.,L2(Ω)or L∞(Ω))Solution:y:R+→φt→y(t)x→y(t,x)If the problem is well-posed,we can define the semigroup S(t):S(t):φ→φy0→y(t)The analytic structure of a PDE is much more complicated than that of an ODE:the global well-posedness can be a very difficult problemSuch results are known for a large class of PDEs→it is natural to investigate whether the notion of a strange attractor extends to PDEsSuch chaotic behaviors can be observed in dissipative PDEsChaotic behaviors arise from the interaction of•Energy dissipation in the higher part of the Fourier spectrum•External energy income in the lower part•Energyflux from the lower to the higher modesThe trajectories are localized in a”thin”in-variant region of the phase space having a very complicated geometric structure→the global attractor1.The global attractor.S(t)semigroup acting on E:S(t):E→E,t≥0S(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0 Continuity:x→S(t)x is continuous on E,∀t≥0A set A⊂E is the global attractor for S(t)if(i)it is compact(ii)it is invariant:S(t)A=A,t≥0(iii)∀B⊂A,lim t→+∞dist(S(t)B,A)=0dist(A,B)=supa∈A infb∈Ba−b EEquivalently:∀B⊂φbounded,∀ǫ>0,∃t0= t0(B,ǫ)s.t.t≥t0implies S(t)B⊂UǫThe global attractor is uniqueIt is the smallest closed set enjoying(iii)It is the maximal bounded invariant setTheorem:(Babin-Vishik)We assume that S(t)possesses a compact attracting set K, i.e.,∀B⊂E bounded,lim t→+∞dist(S(t)B,K)=0Then S(t)possesses the global attractor A.The global attractor is oftenfinite dimen-sional:the dynamics,restricted to A isfinite dimensionalFractal dimension:Let X be a compact setdim F X=lim supǫ→0+ln Nǫ(X)ǫNǫ(X):minimum number of balls of radius ǫnecessary to cover XIf Nǫ(X)≤c(1Theorem:(H¨o lder-Ma˜n´e theorem)Let X⊂E compact satisfy dim F X=d and N>2d be an integer.Then almost every bounded linear projector P:E→R N is one-to-one on X and has a H¨o lder continuous inverse.This result is not valid for other dimensions (e.g.,the Hausdorffdimension)If A hasfinite fractal dimension,then,fixing a projector P satisfying the assumptions of the theorem,we obtain a reduced dynamical system(S),S= P(A),which isfinite dimensional(in R N)and H¨o lder continuousDrawbacks:(S)cannot be realized as a system of ODEs which is well-posedReasonable assumptions on A which would ensure that the Ma˜n´e projectors are Lipschitz are not knownComplicated geometric structure of A and AThe lower semicontinuitydist(A0,Aǫ)→0asǫ→0is more difficult to prove and may not hold It may be unobservable:∂y∂x2+y3−y=0,x∈[0,1],ν>0y(0,t)=y(1,t)=−1,t≥0A={−1}There are many metastable”almost station-ary”equilibria which live up to t⋆≡eν−12.Inertial manifolds.A Lipschitzfinite dimensional manifold M⊂E is an inertial manifold for S(t)if(i)S(t)M⊂M,∀t≥0(ii)∀u0∈E,∃v0∈M s.t.S(t)u0−S(t)v0 E≤Q( u0 E)e−αt,α>0,Q monotonicM contains A and attracts the trajectories exponentiallyConfirms in a perfect way thefinite dimen-sional reduction principle:The dynamics reduced to M can be realized as a Lipschitz system of ODEs(inertial form)Perfect equivalence between the initial sys-tem and the inertial formDrawback:all the known constructions are based on a restrictive condition,the spectral gap condition→The existence of an inertial manifold is not known for several important equations, nonexistence results for damped Sine-Gordon equations3.Exponential attractors.A compact set M⊂E is an exponential at-tractor for S(t)if(i)It hasfinite fractal dimension(ii)S(t)M⊂M,∀t≥0(iii)∀B⊂E bounded,dist(S(t)B,M)≤Q( B E)e−αt,α>0,Q monotonicM contains AIt is stillfinite dimensional and one has a uni-form exponential control on the rate of at-traction of trajectoriesIt is no longer smoothDrawback:it is not unique→One looks for a simple algorithm S→M(S)Initial construction:non-constructible and valid in Hilbert spaces onlyConstruction in Banach spaces:Efendiev, Miranville,Zelik→Exponential attractors are as general as global attractorsMain tool:Compact smoothing property on the difference of2solutionsLet S:E→E.We consider the discrete dynamical system generated by the iterations of S:S n=S◦...◦S(n times)Theorem:(Efendiev,Miranville,Zelik)We consider2Banach spaces E and E1s.t.E1⊂E is compact.We assume that•S maps theδ-neighborhood Oδ(B)of a bounded subset B of E into B•∀x1,x2∈Oδ(B),≤K x1−x2 ESx1−Sx2 E1Then the discrete dynamical system gener-ated by the iterations of S possesses an ex-ponential attractor M(S)s.t.(i)M(S)⊂B,is compact in E anddim F M(S)≤c1(ii)S M(S)⊂M(S)(iii)dist(S k B,M(S))≤c2e−c3k,k∈N,c3>0 (iv)The map S→M(S)is H¨o lder continu-ous:∀S1,S2,dist sym(M(S1),M(S2))≤c4 S1−S2 c5,c5>0, wheredist sym(A,B)=max(dist(A,B),dist(B,A))S =supSh Eh∈Oδ(B)Furthermore all the constants only depend on B,E,E1,δand K and can be computed explicitly.Remarks:1)We have a mapping S→M(S)and,due to the H¨o lder continuity,we can construct continuous families of exponential attractors2)Exponential attractors for a continuous semigroup S(t):Prove that∃t⋆>0s.t.S⋆=S(t⋆)satisfies the assumptions of the theorem→M⋆for S⋆If(x,t)→S(t)x is Lipschitz(or H¨o lder)on B×[0,t⋆],setS(t)M⋆M=∪t∈[0,t⋆]We again have a mapping S(t)→M(S)which is H¨o lder continuous3)For damped hyperbolic equations:asymp-totically smoothing property4.Finite dimensional reduction of nonau-tonomous systems.Systems of the form∂yDrawback:the uniform attractor has infinite dimension in general.Example:∂yThe family{A(t),t∈R}is a pullback attrac-tor for U(t,τ)if(i)A(t)is compact in E,∀t∈R(ii)U(t,τ)A(τ)=A(t),∀t≥τ(iii)∀B⊂E bounded,dist(U(t,t−s)B,A(t))=0lims→+∞Remarks:1)The pullback attractor is unique2)If the system is autonomous,we recover the global attractor3)In general,A(t)hasfinite fractal dimen-sion,∀t∈RDrawback:The forward convergence does not hold in generalExample:y′=f(t,y),where f(t,y)=−y if y≤0,(−1+2t)y−ty2 if t∈[0,1],and y−y2if t≥1Then A(t)={0},∀t∈R,but every trajectory starting from a neighborhood of0leaves this neighborhood never to enter it againThe forward convergence does not hold be-cause the rate of attraction is not uniform in t→This can be solved by constructing ex-ponential attractorsWe can construct a family{M(t),t∈R}, called nonautonomous exponential attractor, s.t.(i)dim F M(t)≤c1,∀t∈R,c1independent of t(ii)U(t,τ)M(τ)⊂M(t),∀t≥τ,(iii)∀B⊂E bounded,dist(U(t,τ)B,M(t+τ))≤Q( B E)e−αt,t∈R,t≥τ,α>0,Q monotonic(iii)implies the pullback attraction,but also the forward attraction→(i)and(iii)yield a satisfactoryfinite di-mensional reduction principle for nonautono-mous systemsRemarks:1)The time dependence is arbitrary2)The map U(t,τ)→{M(t),t∈R}is also H¨o lder continuous。
