Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoing and incoming transformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.     

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.     

Riesz Basis Generation of Abstract Second-Order Partial Differential Equation Systems with General Non-Separated Boundary Conditions

Riesz basis analysis for a class of general second-order partial differential equation systems with nonseparated boundary conditions is conducted. Using the modern spectral analysis approach for parameterized ordinary differential operators, it is shown that the Riesz basis property holds for the general system if its associated characteristic equation is strongly regular. The Riesz basis property can then be readily established in a unified manner for many one-dimensional second-order systems such as linear string and beam equations with collocated or noncollocated boundary feedbacks and tip mass attached systems. Three demonstrative examples are presented.     

Asymptotic analysis of nonselfadjoint operators generated by coupled Euler‐Bernoulli and Timoshenko beam model

In the current paper, we present a series of results on the asymptotic and spectral analysis of coupled Euler‐Bernoulli and Timoshenko beam model. The model is well‐known in the different branches of the engineering sciences, such as in mechanical and civil engineering (in modelling of responses of the suspended bridges to a strong wind), in aeronautical engineering (in predicting and suppressing flutter in aircraft wings, tails, and control surfaces), in engineering and practical aspects of the computer science (in suppressing bending‐torsional flutter of a new generation of hard disk drives, which is expected to pack high track densities (20,000+TPI) and rotate at very high speeds (25,000+RPM)), in medical science (in bio mechanical modelling of bloodcarrying vessels in the body, which are elastic and collapsible). The aforementioned mathematical model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions representing the action of the self‐straining actuators. This linear hyperbolic system is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators and a dynamics generator of the semigroup is our main object of interest. We formulate and proof the following results: (a) the dynamics generator is a nonselfadjoint operator with compact resolvent from the class ??p with p > 1; (b) precise spectral asymptotics for the two‐branch discrete spectrum; (c) a nonselfadjoint operator, which is the inverse of the dynamics generator, is a finite‐rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof that the root vectors of the dynamics generator form a complete and minimal set. In our forthcoming paper, we will use the spectral results to prove that the dynamics generator is Riesz spectral, which will allow us to solve several boundary and distributed controllability problems via the spectral decomposition method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basis Property of a Rayleigh Beam with Boundary Stabilization

A Rayleigh beam equation with boundary stabilization control is considered. Using an abstract result on the Riesz basis generation of discrete operators in Hilbert spaces, we show that the closed-loop system is a Riesz spectral system; that is, there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis in the state Hilbert space. The spectrum-determined growth condition, distribution of eigenvalues, as well as stability of the system are developed. This paper generalizes the results in Ref. 1.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations

In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy.     

A semigroup method for delay equations with relatively bounded operators in the delay term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations.     

Operator waves in Hilbert space and related partial differential equations

The success of the theory of characteristic functions of nonselfadjoint operators and its applications to the System Theory [1–17] is the inspiration for attempts towards creating a general theory in the much more complicated case of several commuting nonselfadjoint operators. In this paper we study the close relations between sets of commuting operators in Hilbert space and related systems of partial differential equations. At the same time a generalization of the classical Cayley Hamilton Theorem, in the case of two commuting operators, is obtained which leads to unexpected connections with the theory of algebraic curves.     

Asymptotic Representations for Root Vectors of Nonselfadjoint Operators and Pencils Generated by an Aircraft Wing Model in Subsonic Air Flow

This paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential equations and a two parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space.     

A Lemma On C 0 -semigroups And Applications

In this note we characterize a large class of C ₀-semigroups which can be applied to prove the existence and the uniqueness of the solutions of many systems of partial differential equations. In fact, we apply our result to a strongly damped wave equation, damped vibration of a string equation and reaction diffusion systems. Finally, we formulate an open problem.     

A Semigroup Method for Delay Equations with Relatively Bounded Operators in the Delay Term

We consider well-posedness and stability of abstract partial differential equations with unbounded operators in their delay terms. We show that the problem is equivalent to an abstract Cauchy problem in a product Banach space, give sufficient conditions on the well-posedness, make a complete spectral analysis and give robust stability criteria. The results are applied to the problem of small delays and to damped plate equations. September 12, 2000     

A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems

The Liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations. Extensions to infinite dimensional settings have been studied by several authors (see e.g. Haraux, 1991 [11], and Komornik and Zuazua, 1990 [17] and references therein). Results on optimal energy decay rates under general conditions of the feedback is far from being complete. The purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation. We present a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for these systems, and on comparison principles based on the dissipation property, and interpolation inequalities (in the infinite dimensional case) for lower bounds of the energy. We stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi-discretized nonlinear damped vibrating PDE's. A part of this approach has been introduced in Alabau-Boussouira (2004, 2005) [1] and [2]. In the present paper, we identify a new, simple and explicit criteria to select a class of nonlinear feedbacks, for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in Alabau-Boussouira (2004, 2005) [1] and [2]. Moreover, we prove optimality of the decay rates for this class, in the finite dimensional case. This class includes a wide range of feedbacks, ranging from very weak nonlinear dissipation (exponentially decaying in a neighborhood of zero), to polynomial, or polynomial-logarithmic decaying feedbacks at the origin. In the infinite dimensional case, we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation. This principle relies on suitable interpolation inequalities. It allows us to give lower bounds for the energy of smooth initial data for the one-dimensional wave equation with a distributed polynomial damping, which improves Haraux (1995) [12] lower estimate of the energy for this case. We also establish lower bounds in the multi-dimensional case for sufficiently smooth solutions when such solutions exist. We further mention applications of these various results to several classes of PDE's, namely: the locally and boundary damped multi-dimensional wave equation, the locally damped plate equation and the globally damped coupled Timoshenko beams system but it applies to several other examples. Furthermore, we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems. We illustrate these results on the one-dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples. These optimal rates are not uniform with respect to the discretization parameter. We also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin.     

On operators with closed analytic core

As shown by Mbekhta [9] and [10], the analytic core and the quasi-nilpotent part of an operator play a significant role in the local spectral and Fredholm theory of operators on Banach spaces. It is a basic fact that the analytic core is closed whenever 0 is an isolated point of the spectrum. In this note, we explore the extent to which the converse is true, based on the concept of support points. Our results are exemplified in the case of decomposable operators, Riesz operators, convolution operators, and semi-shifts.     

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter . Corresponding to different values of , the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators and to establishing the exact and approximate controllability of the system governed by the damped wave equation.

     

Commutative and noncommutative representations of matrix-valued herglotz-nevanlinna functions

Operator realizations of matrix-valued Herglotz-Nevanlinna functions play an important and essential role in system theory, in the spectral theory of bounded nonselfadjoint operators, and in interpolation problems. Here, a generalization for realization results of the Brodskiǐ-Livsic type is given for Herglotz-Nevanlinna functions whose spectral measures are compactly supported.     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Large-time behavior of solutions to degenerate damped hyperbolic equations

We investigate the asymptotic behavior of solutions to damped hyperbolic equations involving strongly degenerate differential operators. First we establish the existence of a global attractor for the damped hyperbolic equation under consideration. Then we prove the finite dimensionality of the global attractor.     

A note on the spectral theorem

We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)^–1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.     

Riesz–Jacobi Transforms as Principal Value Integrals

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz–Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz–Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.