An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical ＇σ/σx＇-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.

Invertibility of Infinite-Dimensional Hamiltonian Operators and Its Application to Plate Bending Equation

The results of invertibility and spectrum for some different classes of infinite-dimensional Hayniltonian operators, after a brief classification by domains. are given. By the above results, the associated infinite-dimensional Hamiltonian operator with simple supported rectangular plate is proved to be invertible. Furthermore, by a certain compactness, we find that the spectrum of this operator consists only of isolated eigenvalues with finite geometric multiplicity, which will play a significant role in finding the analytical and numerical solution based on Hamiltonian system for a class of plate bending equations.

Spectral Description of a Class of Infinite-Dimensional Hamiltonian Operators and Its Application to Plane Elasticity Equations Without Body Force

In this paper,the results of spectral description and invertibility of upper triangle infinite-dimensionalHamiltonian operators with a diagonal domain are given.By the above results,it is proved that the infinite-dimensionalHamiltonian operator associated with plane elasticity equations without the body force is invertible,and the spectrumof which is non-empty and is a subset of R.

Realization of the Infinite-Dimensional 3-Algebras in the Calogero-Moser Model

We investigate realization of the infinite-dimensional 3-algebras in the classical Calogero-Moser model. In terms of the Lax matrix of the Calogero Moser model and the Nambu 3-brackets in which the variables are the coordinates qi, and canonically conjugate momenta pi and the coupling parameter β are an extra auxiliary phase-space parameter, we present the realization of the Virasoro-Witt, w∞ and SDi f f （T2） 3-algebras, respectively.

Completeness of the System of Root Vectors of 2×2 Upper Triangular Infinite-Dimensional Hamiltonian Operators in Symplectic Spaces and Applications

The authors investigate the completeness of the system of eigen or root vectors of the 2×2 upper triangular infinite-dimensional Hamiltonian operator H 0.First,the geometrical multiplicity and the algebraic index of the eigenvalue of H0 are considered.Next,some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H0 are obtained.Finally,the obtained results are tested in several examples.

H~∞-IDENTIFICATION OF INFINITE-DIMENSIONAL LINEAR STOCHASTIC SYSTEMS

Ｈ~∞－ＩＤＥＮＴＩＦＩＣＡＴＩＯＮＯＦＩＮＦＩＮＩＴＥ－ＤＩＭＥＮＳＩＯＮＡＬＬＩＮＥＡＲＳＴＯＣＨＡＳＴＩＣＳＹＳＴＥＭＳＷＥＩＣｈｅｎ；ＧＵＯＬｅｉ（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａＢｅｉｊｉｎｇ?..

Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations

Based on a recent result on linking stochastic differential equations on R^d to （finite-dimensional） Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensionl stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.

A Nonlinear Small-Gain Theorem for Large-Scale Infinite-Dimensional Systems

This paper develops a large-scale small-gain result for dynamic networks composed of infinite-dimensional subsystems. It is assumed that the subsystems are input-to-output stable（IOS）and unboundedness observable（UO）, and the large-scale infinite-dimensional system can be proved to be IOS and UO if the proposed small-gain condition is satisfied.

ASYMPTOTIC SIMILARITY OF INFINITE-DIMENSIONAL LINEAR SYSTEMS AND APPLICATIONS TO STABILITY

In this note a generalization of the concept of similarity called asymptotic similarity for infinite-dimensional linear systems is introduced. We show that this asymptotic similarity preserves the spectrum and the exponential growth bound.

Geodesic uniqueness problem in infinite-dimensional Teichmuller spaces

LET[μ] be a point in a Teichmuller space T（Γ） and [μ]≠[0]. When T（Γ） is finite-di-mensional, the extremal Beltrami differential in [μ]is unique and the geodesic segment α:[tμ0] （0≤t≤1） is the unique geodesic segment joining [0] and [μ], where μ0 is the uniqueextremal Beltrami differential in [μ]. However, when T（Γ） is infinite-dimensional, [μ]

Representation of Infinite-Dimensional Forward Price Models in Commodity Markets

We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by Filipovi´c(Consistency problems for Heath–Jarrow–Morton interest rate models,2001).The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite-dimensional Lévy process.It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein–Uhlenbeck processes,or more generally,as a sum of certain stationary processes.These results link the possibly infinite-dimensional forward dynamics to classical commodity spot models.We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert–Schmidt operators are essentially integral operators.The covariance operator of the Lévy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators,and we analyse in several examples the consequences on model dynamics and their probabilistic properties.Also,we represent the forward price for contracts delivering over a period in terms of an integral operator,a case being relevant for power and gas markets.In several examples,we reduce our general model to existing commodity spot and forward dynamics.

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter H∈(1/4,1/2)

A two-dimensional （2D） stochastic incompressible non-Newtonian fluid driven by the genuine cylindrical fractional Brownian motion （FBM） is studied with the Hurst parameter ∈ （1/4,1/2） under the Dirichlet boundary condition. The existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids are obtained by the estimate on the and the identity of the infinite double series spectrum of the spatial differential operator in the analytic number theory. The existence of the mild solution and the random attractor of a random dynamical system are then obtained for the stochastic non-Newtonian systems with ∈ （1/2,1） without any additional restriction on the parameter H.

Completeness of system of root vectors of upper triangular infinitedimensional Hamiltonian operators appearing in elasticity theory

This paper deals with a class of upper triangular infinite-dimensional Hamilto- nian operators appearing in the elasticity theory. The geometric multiplicity and algebraic index of the eigenvalue are investigated. Furthermore, the algebraic multiplicity of the eigenvalue is obtained. Based on these properties, the concrete completeness formulation of the system of eigenvectors or root vectors of the Hamiltonian operator is proposed. It is shown that the completeness is determined by the system of eigenvectors of the operator entries. Finally, the applications of the results to some problems in the elasticity theory are presented.

STRING STABILITY OF INFINITE INTERCONNECTED SYSTEMS

The notion of string stability of a countably infinite interconnection of a class of nonlinear system was introduced. Intuitively, string stability implies uniform boundedness of all the stares of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded. Vector V-function method used to judge the stability is generalized for infinite interconnected system and sufficient conditions which guarantee the asymptotic string stability of a class of interconnected system are given. The stability regions obtained here are much larger than those in previous papers. The method given here overcomes some difficulties to deal with stability of infinite nonlinear interconnected system in previous papers.

Controlling a time-delay system using multiple delay feedback control

In this paper multiple delay feedback control （MDFC） with different and independent delay times is shown to be an efficient method for stabilizing fixed points in finite-dimensional dynamical systems. Whether MDFC can be applied to infinite-dimensional systems has been an open question. In this paper we find that for infinite-dimensional systems modelled by delay differential equations, MDFC works well for stabilizing （unstable） steady states in long, moderate- and short-time delay regions, in particular for the hyperchaotic case.

Symplectic-like Difference Schemes for Generalized Hamiltonian Systems

The nature of infinite-dimensional Hamiltonian systems are studied for the purpose of further study on some generalized Hamiltonian systems equipped with a given Poisson bracket. From both theoretical and practical viewpoints, we summarize a general method of constructing symplectic-like difference schemes of these kinds of systems. This study provides a new algorithm for the application of the symplectic geometry method in numerical solutions of general evolution equations.

Optimality Conditions and Algorithms for Direct Optimizing the Partial Differential Equations

New form of necessary conditions for optimality (NCO) is considered. They can be useful for design the direct infinite- dimensional optimization algorithms for systems described by partial differential equations (PDE). Appropriate algo-rithms for unconstrained minimizing a functional are considered and tested. To construct the algorithms, new form of NCO is used. Such approach demonstrates fast uniform convergence at optimal solution in infinite-dimensional space.

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

For the approximation in L_(p)-norm,we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots.For p=1,∞,we obtain its values.By these results we know that for the Sobolev classes,the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.At the same time,the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

How big are the increments of 1~p-valued Gaussian processes?

be a sequence of independent Gaussian processes with σk2 (h)The large increments for Y(·) with boundedσ (p, h ) are investigated. As an example the large increments of infinite-dimensional fractional Ornstein-Uhlenbeck process in 1p are given. The method can also be applied to certain processes with stationary increments.

On d-Dimensional Lattice (co)sine n-Algebra

We present the(co)sine n-algebra which is indexed by the d-dimensional integer lattice.Due to the associative operators,this generalized(co)sine n-algebra is the higher order Lie algebra for the n even case.The particular cases are the d-dimensional lattice sine 3 and cosine 5-algebras with the special parameter values.We find that the corresponding d-dimensional lattice sine 3 and cosine 5-algebras are the Nambu 3-algebra and higher order Lie algebra,respectively.The limiting case of the d-dimensional lattice(co)sine n-algebra is also discussed.Moreover we construct the super sine n-algebra,which is the super higher order Lie algebra for the n even case.