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.

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.

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

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

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]

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.

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.

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.

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.

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.

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.

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.

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.

Adaptive Stabilization for ODE Systems Coupled with Parabolic PDES

This paper is concerned with the adaptive stabilization for ODE systems coupled with parabolic PDEs. The presence of the uncertainties/unknonws and the coupling between the subsystems makes the system under investigation essentially different from those of the existing literature,and hence induces more technique obstacles in control design. Motivated by the related literature, an invertible infinite-dimensional backstepping transformation with appropriate kernel functions is first introduced to change the original system into a new one, from which the control design becomes much convenient. It is worthwhile pointing out that, since the kernel equations for which the kernel functions satisfy are coupled rather than cascaded, the desirable kernel functions are more difficult to derive than those of the closely related literature. Then, by Lyapunov method and a dynamics compensated technique, an adaptive stabilizing controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converging to zero. Finally, a simulation example is provided to validate the proposed method.

Some extremal properties of multivariate polynomial splines in the metric L_p(R^d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems.It is proved that given any stable linear time invariant fractional order system,there exists a positive definite functional with respect to the system state,and the first order time derivative of that functional is negative definite.A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

The Space of Maps from a Locally Compact Space to a Banach Space

The space of continuous maps from a topological space X to a topological space Y is denoted by C（X,Y）with the compact-open topology.In this paper we prove that C（X,Y）is an absolute retract if X is a locally compact separable metric space and Y a convex set in a Banach space.From the above fact we know that C（X,Y）is homomorphic to Hilbert space l2 if X is a locally compact separable metric space and Y a separable Banach space;in particular,C（Rn,Rm） is homomorphic to Hilbert space l2.