## Abstract

The present paper is devoted to the Riesz basis property of the mode shapes for an aircraft wing model in an inviscid subsonic airflow. The model has been developed in the Flight Systems Research Center of the University of California at Los Angeles in collaboration with NASA Dryden Flight Research Center. The model has been successfully tested in a series of flight experiments at Edwards Airforce Base, CA, and has been extensively studied numerically. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. 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 a finite—meromorphic operator—valued function of the spectral parameter. Its poles are precisely the aeroelastic modes. In the author's previous works, it has been shown that the set of aeroelastic modes asymptotically splits into two disjoint subsets called the *β*-branch and the *δ*-branch, and precise spectral asymptotics with respect to the eigenvalue number have been derived for both branches. The asymptotical approximations for the mode shapes have also been obtained. In the present work, the author proves that the set of the mode shapes forms an unconditional basis (the Riesz basis) in the Hilbert state space of the system. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.

## 1. Introduction

The objective of this paper is to prove the Riesz basis property of the set of the mode shapes, which occurs in the initial boundary-value problem arising in modelling of vibrations of an aircraft wing in a subsonic airflow. The present paper, on the one hand, is a continuation of an ongoing research (Balakrishnan & Edwards 1980; Balakrishnan 1998*b*, 2001; Dierolf *et al*. 2000; Shubov 2000, 2001*a*–,*c*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*,*b*) and, on the other hand, is an independent piece of work. In order to simplify reading of the paper, all necessary results from the previous works are collected in §3.

An ultimate goal of aircraft wing modelling is to design a flutter control mechanism. Flutter is a structural dynamical instability, which consists of violent vibrations of a solid structure with rapidly increasing amplitude when the structure interacts with gas or fluid flow. It usually results either in serious damage to the structure or in its complete destruction. Flutter occurs when the parameters characterizing fluid–structure interaction reach certain critical values. In engineering practice, flutter must be avoided either by design of the structure or by introducing control mechanisms capable of suppressing harmful vibrations. Flutter is known as an inherent feature of fluid–structure interaction and, thus, it cannot be eliminated completely. However, the critical conditions for the flutter onset can be shifted to the safe range of the operating parameters. This is exactly the goal of designing flutter control mechanisms.

The most well-known cases of flutter are related to the flutter in aircraft wings, tails and control surfaces. Flutter is an in-flight event that happens beyond some speed-altitude combinations. High speed aircrafts are most susceptible for flutter although no speed regime is truly immune from flutter. Flutter instabilities occur in a variety of different engineering and even biomedical situations. For example, in aeronautic engineering, flutter of helicopter, propeller and turbine blades is a serious problem. Flutter also affects electric transmission lines, high-speed hard disk drives and long-spanned suspension bridges. Flutter of cardiac tissue and blood vessel walls is of a special concern to medical practitioners.

Flutter is an extremely complex physical phenomenon, whose complete theoretical explanation is an open problem. At the present moment, there exist only a few models of fluid–structure interaction involving flutter development for which precise mathematical formulations are available. We believe that analytical treatment of flutter problem is an important component of this area of research. Such treatment can provide insights not available from purely computational or experimental results. This is certainly important for designing flutter control mechanisms.

Ideally, a complete picture of a fluid–structure interaction should be described by a system of partial differential equations containing both the equations governing the mechanical vibrations of an elastic structure and the hydrodynamic equations governing the motion of gas or fluid flow. The system of equations of motion should be supplied with appropriate boundary and initial conditions. The structural and hydrodynamic parts of the system must be coupled in the following sense. The hydrodynamic equations define a pressure distribution on the elastic structure. This pressure distribution in turn defines the so-called aerodynamic loads, which appear as forcing terms in structural equations. On the other hand, the parameters of the elastic structure enter the boundary conditions for the hydrodynamic equations.

The above picture is mathematically very complicated, and to make a particular problem tractable, we introduce simplifying assumptions. We assume that the model describes a wing of *high aspect ratio* (i.e. the length of a wing is much greater than its width, though both quantities are finite) in a *subsonic*, *inviscid*, *incompressible* airflow. In the current model, the hydrodynamic part is not present explicitly. In fact, the hydrodynamic equations have been solved *explicitly* and aerodynamic loads are represented in the form of forcing terms in the structural equations as time convolution-type integrals with very complicated kernels. Thus, the model is described by a linear system of integro-differential equations. The very notion of spectral analysis for such systems is a new mathematical challenge. Our results on this model include the following (see Shubov 2000, 2001*a*–,*c*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*,*b*). We treat the system of equations of motion as a single evolution–convolution equation in the Hilbert state space of the model. (The integral convolution part of this equation vanishes if a speed of an air stream is equal to zero, and we obtain the equation of motion for the so-called *ground vibrations* (Balakrishnan 1998*a*,*b*; Shubov & Peterson 2003).)

We have introduced the notion of the generalized resolvent operator for the above evolution–convolution equation and represented the solution of the original initial boundary-value problem in the frequency domain in terms of this generalized resolvent. The generalized resolvent is

*an analytic finite-meromorphic operator-valued function*of the spectral parameter. We define the aeroelastic modes as the poles of the generalized resolvent; the corresponding mode shapes are defined in terms of the residues at the poles.We have found the asymptotic formulae for the aeroelastic modes. (To the best of our knowledge, these are the first such formulae in the literature on aeroelasticity, which are related to a linear model.) The entire set of aeroelastic modes splits into two branches, which are asymptotically close to the eigenvalues of the structural part of the system. We have also derived the asymptotic formulae for the aeroelastic mode shapes (Shubov 2000, 2001

*a*,*c*; Shubov & Balakrishnan 2004*a*).In the present paper, we show that the set of all mode shapes forms a non-orthogonal basis (a Riesz basis) in the state space of the system. (The set of the generalized eigenvectors of the structural part of the system has a similar property (Shubov 2001

*b*,*c*; Shubov & Balakrishnan 2004*a*).)Using the Riesz basis property of the mode shapes in our forthcoming paper, we will present the solution of the original initial boundary-value problem in the form of expansion with respect to the mode shapes and the integral along the negative real semi-axis, which is formally associated with the continuous spectrum.

The precise mathematical formulation of the model is given in §2. The model is quite complicated and its physical origin is not obvious from its formulation. For readers' convenience, we have introduced below a short subsection, which contains a description of the main physical assumptions used in the derivation of the model.

### (a) The origin of the model

The physical model which is studied in the present paper has been developed at the Flight Systems Research Center of UCLA in collaboration with NASA Dryden Flight Research Center. The model has been tested in a series of flight experiments at Edwards, Air force Base, CA, and was extensively studied numerically (Balakrishnan 1998*b*, 2001). The flight test results have shown an agreement with predictions of the model for at least lower frequency aeroelastic modes. So, the practical importance of the model has a solid confirmation.

A detailed derivation of the model is not among the objectives of this paper. However, we provide below a brief description of the general structure of any wing model, and then we list physical assumptions which lead to our particular model.

#### (i) The general structure of wing models

The general structure of wing models typically contains the following ingredients.

##### Structural part of a model

*System of equations governing the vibrations of the wing*. In the model studied in this paper, the nonlinear elastic phenomena have been neglected. Thus, the main system is linear and it has the following form:(1.1)Here,

*X*(*η,**t*) is a vector function representing structural dynamic variables (bending, torsion, etc.). Depending on a particular model,*η*may be either a one-dimensional (*η*=*x*) or two-dimensional (*η*=(*x*,*y*)) variable, which characterizes the position of a point on the wing, and*t*denotes time. The ‘overdot’ denotes the time derivative,*M*and*D*are matrix coefficients which split into a sum of two terms: one of them depends on elastic characteristics of the wing structure, and the other depends on the density and speed of the airstream.*K*is a fourth-order*matrix differential operator*, whose coefficients also split into sum of two terms similar to*M*and*D*. The right-hand side term (*η*,*t*) represents the*aerodynamic loads*, i.e. the generalized forces and moments exerted on the wing by the airflow. System (1.1) has a closed form only if one represents as an operator acting on the dynamic variables . Obtaining such a representation is the most difficult part in the design of a particular wing model, requiring an investigation of the aerodynamical part of the model.*Boundary conditions for system**(1.1)*. It is assumed that the left end of a wing is attached to an aircraft; the right end of a wing is either free or, e.g. reflects the action of self-straining actuators (family of specific dynamical boundary conditions).

##### Aerodynamical part of a model

*The system of hydrodynamic equations governing the airflow around the wing*. In general, this is the Euler (inviscid case) or the Navier–Stokes (viscous case) system supplied with the continuity equation (for the incompressible or compressible case). Both of the above systems are too complicated. To make a model tractable, one has to introduce a set of simplifying (but physically realistic) assumptions and to deal with simplified hydrodynamic equations.The set of the boundary conditions for the airflow on the wing surface and at infinity.

##### Coupling between structural and aerodynamical parts of a model. The aforementioned two parts of a model have to be coupled in the following way

The boundary conditions for the airflow on the wing surface involve the structural dynamic variables

*X*(*η*,*t*).The solution of the hydrodynamic equations that govern the airflow allows us to determine the

*pressure difference*Δ*p*(*η*,*t*) between the lower and upper wing surfaces. The aerodynamic loads (*η*,*t*) in (1.1) can be expressed in an explicit form in terms of Δ*p*(*η*,*t*). Therefore, the loads can, in principle, be expressed in the form of a certain operator acting on .

Combination of all of the above ingredients provides the closed system of equations and boundary conditions that govern a particular model.

#### (ii) Discussion on physical assumptions for a general class of wing models

We start with two-dimensional wing models in a subsonic airflow. A wing is represented as a rectangle in the *xy*-plane: in a three-dimensional airflow. The *x*-axis is directed along the wing-span and the *y*-axis along a one-dimensional cross-section.

##### Physical assumptions about an airflow

*The flow is non-viscous*. Therefore, the stream velocity(**v***x*,*y*,*z*,*t*) satisfies the Euler equation. Because this equation is so complicated, it is simplified based on the assumptions given below.*The flow is compressible*. The latter means that the flow velocity(**v***x*,*y*,*z*,*t*) and density*ρ*(*x*,*y*,*z*,*t*) are related by the continuity equation, .*The flow is isentropic*. Thus, the pressure*p*and the density*ρ*are related by the law*p*=*kρ*^{γ}(where*γ*=*c*_{p}/*c*_{v},*c*_{p}and*c*_{v}are specific heats at constant pressure and volume, respectively, and*k*=exp(*s*/*c*_{v}), with*s*being a specific entropy).*The flow is irrotational (i.e. the vorticity*∇×=0**v***)*,*or equivalently*,*it is potential*. This fact means that=∇**v***Φ*, where*Φ*(*x*,*y*,*z*,*t*) is a potential function. (The majority of aerodynamic results, which are significant for aeroelasticity, depends on this assumption. Such a condition holds for any initially irrotational flow field possessing a unique pressure–density relation*p*=*p*(*ρ*)—‘barotropic fluid’.)*The flow is a small perturbation of a uniform stream*(freestream) with speed*u*directed along the positive*y*-axis. Its potential can be represented in the form , where*φ*is the small perturbation of the undisturbed potential*uy*.

##### Reduced hydrodynamic equation

The above assumptions when applied to the compressible Euler equation lead to a linear hyperbolic equation for the perturbation potential *φ*. We do not present the aforementioned equation here (it can be found e.g. in Fung 1993; Bisplinghoff *et al*. 1996). We mention only that the coefficients in this equation involve the sound speed *c* and the Mach number *M*=*u*/*c*. We stress at this point that the model we consider is *linear*. There are two reasons for this: (i) we neglected the nonlinear elastic phenomena and (ii) the reduced hydrodynamic equation is a linear hyperbolic equation for the perturbation potential *φ*. The latter assumption is widely recognized to be quite natural for subsonic flows. Nonlinear wing models deal mainly with transonic and supersonic regimes. Indeed, the minimal complexity of equation which should be used for transonic flow requires an additional quadratic nonlinear term in the aforementioned equation for *φ*. Linear subsonic equations *cannot* give an account for shock waves. We assume that at *t*=0, the flow is an unperturbed freestream, i.e. .

##### Boundary conditions

*Flow tangency condition*:(1.2)where*w*(*x*,*t*) is the normal velocity of the wing surface (downwash), which is determined in terms of the structural dynamic variables,*X*(*x*,*y*,*t*). Equation (1.2) means that there is no relative movement between the wing surface and the airflow; i.e. the airflow is attached to the wing.*Kutta–Joukowski condition*(Ashley & Landahl 1985; Fung 1993; Bisplinghoff*et al*. 1996). We do not present this condition but just mention that it is a non-stationary (dynamic) boundary condition for the so-called acceleration potential*ψ*=*φ*_{t}+*uφ*_{y}.*Far-field conditions*. These conditions require that the perturbation potential*φ*and its gradient ∇*φ*tend to zero at spatial infinity.

##### Coupling between the hydrodynamic equation and structural equation (1.1)

The downwash function

*w*(*x*,*y*,*t*) from (1.2) can be given by an explicit expression in terms of the structural dynamic variables*X*(*x*,*y*,*t*) from (1.1).The aerodynamic loads (

*x*,*y*,*t*) from (1.1) can be given by explicit formulae in terms of the pressure difference Δ*p*(*x*,*y*,*t*) between the lower and upper wing surfaces. In turn, , where*ψ*(*x*,*y*, 0+,*t*) is the limit value of the acceleration potential on the wing surface.

#### (iii) The model considered in the present paper

The model considered in the present paper has been derived from the above described class of models as a result of combination of the following limit cases.

The flow was assumed to be

*incompressible*, which means that the hyperbolic equation for*φ*has been replaced with the Laplace equation, Δ*φ*=0. (The incompressibility assumption*ρ*=const. leads to the limits*c*→∞,*M*→0,*Mc*=*u*, with*c*being the sound speed.) However, the problem is still non-stationary since time enters the boundary conditions (1.2) and Kutta–Joukowski conditions.The wing was assumed to be a long and slender one-dimensional structure of a

*high aspect ratio*,*l*/*b*≫1, i.e.*b*→0. In that case, wing vibrations are governed by system (2.2) in §2 below. The downwash in (1.2) can be expressed by explicit formulae in terms of the bending and torsion angle. The lift (force per unit span) and pitching moment (per unit span) can also be given by standard explicit formulae in terms of the pressure difference Δ*p*. With these remarks, we conclude the description of the physical origin of the model considered in the present paper.

### (b) Plan of presentation

In §2, we present the model equations and the family of the boundary conditions. Then we give the reformulation of the problem in the abstract setting (see equation (2.24) below) and explain that it is meaningful to consider the eigenvalues (the aeroelastic modes) and the eigenfunctions (the mode shapes) of the model. In §3, we provide formulation of the main results from our previous works (Shubov 2000, 2001*a*–,*c*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*,*b*) on this model and also formulate the main result of the present paper (theorem 3.8). As has already been mentioned, the set of the aeroelastic modes splits into the *β*-branch and the *δ*-branch, whose behaviour is different. In §4, we prove that the *β*-branch mode shapes form a Riesz basis in their closed linear span. The main reason for the latter fact is the asymptotic analysis results, from which it follows that the *β*-branch mode shapes are quadratically close to the *β*-branch eigenfunctions of the structural part of the model. However, asymptotic analysis results do not provide the same relationship between the corresponding *δ*-branches. Sections 5–11 are devoted to the proof that *δ*-branch mode shapes form a Riesz basis in their closed linear span in the state space. The proof for the *δ*-branch is technically much more difficult. To carry out this proof, we have to utilize some information on Fredholm operator-valued analytic functions. In §5, we give a necessary review of the properties of such functions. In §6, we present the corollaries of the results of §5 related to the operator-valued functions appearing in the present wing model. In §7, we prove a technical result on the normalization condition for the mode shapes of the adjoint operator-valued function, which is important for §§8 and 9. In §8, we prove that the entire set of the mode shapes is *minimal*, i.e. that no mode shape belongs to the closed linear span of the remaining mode shapes. Section 9 contains technical results for the *δ*-branch mode shapes, which are necessary for the proof of the main result of the paper. In §9 particularly, we derive the *improved* estimates for the distances between the *δ*-branches of the aeroelastic modes and the eigenvalues of the structural part differential operator. In §10, those improved estimates are used to show that the *δ*-branch mode shapes are quadratically close to the *δ*-branch eigenfunctions corresponding to the structural operator. In §11, we complete the proof of the main result of the paper.

## 2. Statement of problem. Operator setting in energy space

To describe mathematical model, let us introduce the dynamical variables (see Balakrishnan 1998*a*; Balakrishnan 2001; Shubov 2001*a*)(2.1)where *h*(*x*, *t*) is the bending and *α*(*x*, *t*) is the torsion angle. The model, which we will investigate, can be described by the following linear system:(2.2)

We recall that the notation ‘.’ (overdot) denotes the differentiation with respect to *t*. We use the subscripts ‘s’ and ‘a’ to distinguish the structural and aerodynamical parameters, respectively. All 2×2 matrices in equation (2.2) are given by the following formulae:(2.3)where *m* is the density of the flexible structure (mass per unit length), *S* is the mass moment, *I* is the moment of inertia, *ρ* is the density of air, *u* is the stream speed, *a* is the linear parameter of the structure, −1≤*a*≤1 (*a* is a relative distance between the elastic axis of a model wing and its line of centre of gravity)(2.4)(2.5)where *E* is the bending stiffness, *G* is the torsion stiffness. We mention here that, in general, the matrix *D*_{s} may have non-zero entries. This matrix is accountable for the distributed damping in the flexible structure. Even though *D*_{s}≡[0] in the present paper, we keep this matrix in its *standard place*. The case of a non-trivial matrix *D*_{s} is technically much more complicated and will be studied in the future. The right-hand side of system (2.2) can be represented as the following system of two convolution-type integral operations:(2.6)(2.7)(2.8)

We emphasize that the kernels in both integral operations (2.6) and (2.7) are independent of the spatial variable *x*. However, the functionals *f*_{1} and *f*_{2} from (2.6) and (2.7) depend on *x* due to the fact that the function *g* from (2.8) depends on *x*.

The aerodynamical functions *C*_{i}, *i*=1, …, 5, are defined in the following ways (Balakrishnan & Edwards 1980; Ashley & Landahl 1985; Bisplinghoff *et al*. 1996; Balakrishnan 1998*a*, 2001):(2.9)where *K*_{0} and *K*_{1} are the modified Bessel functions of zero and first orders, respectively (Magnus *et al*. 1966; Watson 1966). It is known that the self-straining control actuator action can be modelled by the following boundary conditions (Chen *et al*. 1987; Tzou & Gadre 1989; Yang & Lee 1994; Cox & Zuazua 1995; Balakrishnan 1997, 1998*a*, 1999, 2001; Shubov 2000, 2001*a*–,*c*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*,*b*)(2.10)(2.11)where is the closed right half-plane. We use the prime for the derivative with respect to *x*. We point out now that the assumption that the boundary parameters *β* and *δ* vary in the right half-plane will lead to the fact that the energy of vibrations of an aircraft wing dissipates when an aircraft is on the ground, i.e. when *f*_{1}=*f*_{2}=0. This assumption corresponds to the physical reality. However, technically analysis can be extended to any complex *β* and *δ* in a straightforward manner.

The boundary conditions at *x*=−*L* are(2.12)

Let the initial state of the system be given as follows:(2.13)

Additionally, we assume that the parameters satisfy the following two conditions:(2.14)

We remark that the second condition in (2.14) has clear physical interpretation: it means that the flow speed must be below the ‘divergence’ or static aeroelastic instability speed for the system (Fung 1993; Bisplinghoff *et al*. 1996). Now we describe this energy space. Let be the set of four-component vector-valued functions (superscript T means the transposition) obtained as a closure of smooth functions satisfying the boundary conditions(2.15)in the following energy norm:(2.16)where the parameters , , , and a positive constant Δ, which we need in what follows, are defined by the formulae(2.17)As shown in Shubov (2001), under conditions (2.14), norm (2.16) is well defined. To rewrite the original initial boundary-value problem in the space , we have to complete preliminary steps.

Let be the kernels in the convolution operations (2.6) and (2.7), i.e.(2.18)

(2.19)The problem defined by equation (2.2) and conditions (2.10)–(2.13) can be represented in the form(2.20)_{βδ} is the following matrix differential operator in :(2.21)defined on the domain(2.22)where *H*^{i}, *i*=1, 2, 4, are the standard Sobolev spaces (Adams 1975). Notice, all material parameters in (2.21) and (2.22) are constant. is a linear integral operator in given by the formula(2.23)

In (2.23), the star ‘*’ stands for the convolution operation and the kernels and are defined in (2.18) and (2.19).

### (a) Important remark

We emphasize that equation (2.20) is not an *evolution equation*. It does not have a dynamics generator and does not define any semi-group in the standard sense. However, the notion of the spectral analysis is well-understood. Now we briefly explain it. The aircraft wing model considered in the present paper can be described by the evolution–convolution type equation of the form(2.24)Here, *Ψ*(.)∈, with being the state space of the system; *Ψ* is a four-component vector-valued function, *A* (*A*=_{βδ}) is a matrix differential operator, and *F*(*t*) is a matrix-valued function. Let us take the Laplace transform of both parts of equation (2.24). The formal solution in the Laplace representation can be given by the formula(2.25)where *Ψ*_{0} is the initial state, i.e. *Ψ*(0)=*Ψ*_{0}, and the symbol ‘^{∧}’ is used to denote the Laplace transform. It is an extremely non-trivial problem ‘to calculate’ the inverse Laplace transform of equation (2.25) in order to have the representation of the solution in the space–time domain. To do this, it is necessary to investigate the ‘generalized resolvent operator’(2.26)

In the case of our wing model, (*λ*) is an operator-valued meromorphic function on the complex plane with a branch-cut along the negative real semi-axis. The poles of (*λ*) are called the eigenvalues, or *the aeroelastic modes*. The branch-cut corresponds to the continuous spectrum. We note that the Laplace transform representation for the solution of problem (2.20), corresponding to (2.25), has the following form:(2.27)

Our ultimate goal is to find the solution of the problem in the space–time domain. To this end, we have ‘to calculate’ the inverse Laplace transform of . It will be done by accomplishing the contour integration in the complex *λ*-plane. In this connection, the properties of the ‘generalized resolvent operator’(2.28)are of special importance for us. As has already been mentioned, in our works (Shubov 2000, 2001*a*,*b*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*), we have analysed the differential part and the role of the convolution part of the system. In particular, we have shown that the convolution part does not ‘destroy’ the main characteristics of the discrete spectrum, the spectrum which is produced by the differential part of the problem. Namely, we have proven that the aeroelastic modes are asymptotically close to the discrete spectrum of the operator i_{βδ} and have calculated the rate at which the aeroelastic modes approach that spectrum. Second, we have also proven that there may be only a finite number of the aeroelastic modes having positive real parts. The latter fact is highly non-trivial from the mathematical point of view and very important from the engineering point of view since it means that for a given stream speed *u*, there exists at most a finite number of *unstable aeroelastic mode shapes*. Third, in the present paper we show that the set of mode shapes forms a Riesz basis in the energy space. The latter fact will be crucial for the proof of the convergence of an infinite series and an improper integral which occur when we perform an aforementioned contour integration.

Finally, we describe the kind of *control problem* that will be considered in connection with the flutter suppression. In our specific wing model, both the matrix differential operator and the matrix integral operator contain entries depending on the speed *u* of the surrounding airflow. Therefore, the aeroelastic modes are functions of . The wing is stable if for all *k*. However, if *u* is increasing, some of the modes move to the right half-plane. The flutter speed for the *k*th mode is defined by the relation . To understand the flutter phenomenon, it is not sufficient to trace the motion of aeroelastic modes as functions of a speed of airflow. It is necessary to have an efficient representation for the solution of our boundary-value problem, containing the contributions from both the discrete and the continuous parts of the spectrum. It is known that flutter cannot be eliminated completely. To successfully suppress flutter, one should design self-straining actuators (i.e. in mathematical language, to design is to select parameters in the boundary conditions, which are the control gains *β* and *δ* in formulae (2.10) and (2.11)), in such a way that flutter does not occur in the desired speed range. *This is a highly non-trivial boundary control problem*.

## 3. Formulation of necessary results from papers (Shubov 2000, 2001*a*–,*c*; Shubov & Peterson 2003; Shubov & Balakrishnan 2004*a*,*b*). Statement of main result

In this section, we provide formulation of the main result of the paper. However, to do so, we need several results from our previous works.

### (a) Asymptotic and spectral properties of matrix differential operator

_{βδ}*is a closed linear operator in**, whose resolvent is compact, and therefore, the spectrum is discrete (**Istratescu 1981**;**Weidmann 1987**;**Gohberg & Krein 1996**)*.*Operator*_{βδ}*is non-self-adjoint unless*.*If**and**, then this operator is dissipative, i.e*.*for Ψ∈*(_{βδ}).*The adjoint operator**is given by the matrix differential expression**(2.21)**on the domain obtained from**(2.22)**by replacing the parameters β and δ with*and*, respectively*.

In theorem 3.2, we provide asymptotical representation for the spectrum of the operator _{βδ}. First, we recall that a vector *Φ* in a Hilbert space *H* is an *associate vector* of a non-self-adjoint operator *A* of an order *m* corresponding to an eigenvalue *λ*, if *Φ*≠0 and(3.1)

If *m*=0, then *Φ* is an eigenvector. The set of all associate vectors and eigenvectors together will be called the set of *the root vectors* (Gohberg & Krein 1996).

*The operator*_{βδ}*has a countable set of the complex eigenvalues. If*(3.2)*then the set of the eigenvalues is located in a strip parallel to the real axis*.*The entire set of the eigenvalues asymptotically splits into two different subsets. We call them the β-branch and the δ-branch and denote these branches by**and**, respectively. If**and**, then each branch is asymptotically close to its own horizontal line in the closed upper half-plane. If**and**, then both horizontal lines coincide with the real axis. If**, then the operator*_{βδ}*is self-adjoint and, thus, its spectrum is real. The entire set of the eigenvalues may have only two points of accumulation: +∞ and −∞ in the sense that**and**as n→±∞ (see formulae**(3.3) and (3.4)**below)*.*The following asymptotical representation is valid for the β-branch as |n|→∞:*(3.3)*with Δ being defined in**(2.17)*.*A complex-valued sequence {ξ*_{n}} is bounded above in the following sense:*, C*(*ω*)*→*0*as ω→*0.

*This branch may have a finite number of multiple eigenvalues of a finite algebraic multiplicity each. For such an eigenvalue, the geometric multiplicity may be less than the corresponding algebraic multiplicity, i.e. in addition to the eigenvector or eigenvectors, there may be the associate vectors.*

*The following asymptotical representation is valid for the δ-branch of the spectrum:*(3.4)

*In* *(3.4)*, ‘ln’ *means the principal value of the logarithm. If β and δ stay away from zero, i.e. |β|≥β*_{0}>0 *and |δ|≥δ*_{0}>0*, then the estimate O*(*|n|*^{−1/2}) *in* *(3.4)* *is uniform with respect to both parameters. In this branch, there may be only a finite number of multiple eigenvalues of a finite multiplicity each. Therefore, only a finite number of the associate vectors may exist*.

The next result is concerned with the properties of the root vectors of the operator _{βδ}. We recall that two sequences of vectors {*φ*_{n}} and {*Χ*_{n}} in a Hilbert space *H* are said to be *biorthogonal* if for every *m* and *n*, we have . It is known (Young 1980; Gohberg & Krein 1996) that for a given sequence {*φ*_{n}}, a biorthogonal sequence {*Χ*_{n}} exists if and only if {*φ*_{n}} is *minimal* (this means that each element of the sequence does not belong to the closed linear span of the others). If the biorthogonal sequence {*Χ*_{n}} exists and unique, then the original sequence {*φ*_{n}} is minimal and complete. The set which is biorthogonal to the root vectors of the main differential operator _{βδ} has been described in Shubov (2000).

A basis in a Hilbert space is a Riesz basis if it is a linear isomorphic image of an orthonormal basis, i.e. if it is obtained from an orthonormal basis by means of a bounded and boundedly invertible operator (Young 1980).

*The set of the root vectors of the operator* _{βδ} *forms a Riesz basis in the energy space* .

The Riesz basis property of the root vectors has been proven in Shubov (2000) based on the *functional model* for non-self-adjoint operators by Sz.-Nagy & Foias (1970), the main elements of which have been reproduced in §7 of Shubov (2000). The next statement (see Shubov & Balakrishnan 2004*a*,*b*) deals with the asymptotic distribution of the aeroelastic modes (the poles of the generalized resolvent).

*The set of the aeroelastic modes is countable and does not have accumulation points on the complex plane*.*There might be only a finite number of multiple poles of a finite multiplicity each. There exists R>*0*such that all aeroelastic modes, whose distance from the origin is greater than R, are simple poles of the generalized resolvent. The value of R depends on the speed u of an air stream, i.e. R=R*(*u*).*The set of the aeroelastic modes splits asymptotically into two series, which we call the β-branch and the δ-branch. Asymptotical distribution of the β- and the δ-branches of the aeroelastic modes can be obtained from the asymptotical distribution of the spectrum of the operator*_{βδ}.*Namely, if**are the β-branch aeroelastic modes, then**and the asymptotical distribution of the set**is given by the right-hand side of formula**(3.3)*.*Similarly, if**are the δ-branch aeroelastic modes, then the asymptotical distribution of the set**is given by the right-hand side of formula**(3.4)*.

### (b) Structure and properties of matrix integral operator

To prove theorem 3.5, we have to exploit very detailed asymptotic analysis. However, the asymptotical information is not sufficient to derive such ‘geometric’ properties of the set of the mode shapes as *minimality*, *completeness* and *the Riesz basis property* in the state space . To address the aforementioned properties, we have to analyse the matrix integral operator.

We now provide all necessary information on the Laplace transform of the convolution-type matrix integral operator (2.23).

*Let* *be the Laplace transform of the kernel of matrix integral operator* *(2.23)*. *The following formula is valid for* *:*(3.5)*where*(3.6)(3.7)*and T is the Theodorsen function defined by the formula*(3.8)*K*_{0} *and K*_{1} *are the modified Bessel functions (**Magnus et al. 1966**;* *Watson 1966**;* *Abramowitz & Stegun 1972**)*.

We recall (Magnus *et al*. 1966; Watson 1966) that the modified Bessel functions can be defined for by , where is the Hankel function of the first kind and of the order *n*, *n*=0, 1. The functions *K*_{0} and *K*_{1} can be represented as absolutely convergent series (Magnus *et al*. 1966; Watson 1966) when *z* varies on the complex plane with the branch-cut along the negative real semi-axis. The following asymptotic representations hold for the Theodorsen function:(3.9)The remainder terms in both formulae (3.9) are analytic functions of *z* for *z*≠0. Thus, the Theodorsen function is a bounded analytic function on the complex plane with the aforementioned branch-cut. Using (3.9), we can give a new form of the generalized resolvent defined in (2.28). Namely, let us introduce a new function *V* by the formula(3.10)

Taking into account that *z*−*λ*/*u*, we can write as the following sum:(3.11)where the matrix is defined by the formula(3.12)with *A* and *B* being given by(3.13)

The matrix-valued function is defined by the formula(3.14)where for *A*_{1}(*λ*) and *B*_{1}(*λ*) we have(3.15)

Therefore, the generalized resolvent (2.28) can be written in the form(3.16)

*is a bounded linear operator in*.*The operator*_{βδ}*defined by the formula*(3.17)*is an unbounded non-self-adjoint operator in**with compact resolvent (in particular, the operator**exists and is compact). This operator has a purely discrete spectrum. The asymptotical representation for the spectrum of the operator*_{βδ}*coincide with the asymptotical representation for the spectrum of the operator*_{βδ}*and, therefore, this asymptotical representation for the spectrum is presented in**theorem 3.2*.*In contrast to the operator*_{βδ}*, the operator*_{βδ}is*not dissipative for any boundary parameters. This means that even though the asymptotical representation for the spectrum of the operator*_{βδ}*is the same as for the spectrum of the operator*_{βδ}*, one cannot claim that the spectrum of the operator*_{βδ}*has to be in the closed upper half-plane of the complex plane. A finite number of the eigenvalues of*_{βδ}*can be in the lower half-plane (see**Shubov & Balakrishnan 2004b**). In addition,*_{βδ}*is also a Riesz spectral operator, i.e. the set of the root vectors of*_{βδ}*forms a Riesz basis of*.*is an analytic matrix-valued function on the complex plane with the branch-cut along the negative real semi-axis. For each λ,**(λ) is a bounded operator in**with the following estimate for its norm:*(3.18)*where**is an absolute constant, the precise value of which is immaterial for us*.

Finally, we give a formulation of *the main result of the paper*, which will be crucial for the ‘calculation’ of the inverse Laplace transform of the solution. As it will be shown in our forthcoming paper, the solution of problem (2.20) can be represented in the form of a sum of an infinite series and an improper integral. The absolute convergence of the integral can be easily seen. The aforementioned infinite series represents a series with respect to the mode shapes. Its unconditional convergence is guaranteed by the result formulated below.

*The set of the mode shapes forms a Riesz basis in the energy space* .

## 4. Riesz basis property of *β*-branch mode shapes

We begin this section with definition 4.1 (Gohberg & Krein 1996; Young 1980).

Let be a Riesz basis in a separable Hilbert space *H*. Then, for any orthonormal basis , there exists a bounded and boundedly invertible operator * R* such that(4.1)

The operator **R**^{−1} is called an orthogonalizer of the Riesz basis .

As is well known (Gohberg & Krein 1996; Young 1980), the basis determines ‖* R*‖ and ‖

**R**^{−1}‖ uniquely, i.e. if and are two different orthonormal bases, then there exists a unitary operator

*U*

_{0}such that . If

*and*

**R**

**R**^{−1}are isomorphisms such that(4.2)then ‖

*‖=‖*

**R***′‖ and . Thus, for a given Riesz basis , it is reasonable to introduce a constant(4.3)*

**R**Now we are in a position to formulate the main result of this section.

*The β-branch of the mode shapes* *forms a Riesz basis in its closed linear span in* .

First, we recall that in our works (Shubov 2001*b*; Shubov & Balakrishnan 2004*b*) it has been shown that the remainder terms in the asymptotical representations for both the *β*-branch eigenfunctions of the operator _{βδ} and the *β*-branch mode shapes behave as *O*(|*n*|^{−1}) as |*n*|→∞, where *n* is either the number of an eigenfunction or the number of a mode shape. The latter fact certainly means that the *β*-branch mode shapes are quadratically close to the *β*-branch eigenfunctions of the operator _{βδ}, i.e.(4.4)

Since there might be only a finite number of associate vectors, we can introduce a numeration in (4.4), making no difference between the eigenvectors and the associate vectors. Proceeding with the proof, let *ϵ*>0 be a small number and *N* be chosen in such a way that(4.5)Let be an arbitrary *l*^{2}-sequence. Setting , we have(4.6)Due to (4.5), we immediately obtain from (4.6) that(4.7)where is an absolute constant (see formula (4.3)). Similarly from (4.5), we obtain(4.8)

Combining (4.7) and (4.8), and using statement 3 of theorem 2.1 from ch. VI of Gohberg & Krein (1996) we obtain that the set forms a Riesz basis in its closed linear span. Let _{N} be a closed linear span of the infinite subset of the *β*-branch mode shapes . The remaining subset of the *β*-branch contains a finite number of the mode shapes as well as a finite number of *the associate mode shapes*. We will make no difference between them in the rest of the proof.

To complete the proof of this theorem, we show that addition of a finite number of the remaining mode shapes cannot destroy the Riesz basis property. Let us take an arbitrary vector from the remaining finite subset and denote it by , |*m*|<*N*, and let _{1} be a one-dimensional subspace spanned by . Consider the following direct sum:(4.9)

Let us show now that the set forms a Riesz basis in _{N−1}. Let be an arbitrary *l*^{2}-sequence and let us denote(4.10)It is clear that the only estimate to be shown is the estimate from below(4.11)where *C*_{1} does not depend on the sequence {*b*_{n}}. We have(4.12)

To proceed, we will use the notion of the *minimal angle φ*(, ) between two subspaces and (Gohberg & Krein 1996) and we have(4.13)

However,(4.14)

It is shown (see theorem 7.1 below) that the set is a minimal in _{N−1} and obviously complete. Therefore, there exists a unique biorthogonal sequence with the property . Returning to (4.13), we have(4.15)

Setting *d*_{1}=(1−*d*^{2})^{1/2}, 0≤*d*_{1}<1, we have from (4.12)(4.16)Estimate (4.16) is desired and it means that forms a Riesz basis in . By repeating the same procedure for the vectors, , we complete the proof.

The theorem is completely shown. ▪

## 5. Definitions and properties of analytic Fredholm operator-valued functions

In this section, we consider operators and operator-valued functions acting in a separable Hilbert space *H*. We denote the set of all bounded operators in *H* by . We say (Gohberg & Sigal 1971; Marcus 1988; Gohberg *et al*. 1990; Lutgen 2001) that is an operator-valued function defined for if for each *λ*, is a closed linear operator in *H*. The resolvent set of is the set of *λ*∈*Ω* such that has a bounded inverse operator. The spectrum of is the complement of the resolvent set in *Ω*. Let be the spectrum of the fixed operator . Clearly, the following fact holds: if, and only if, . The point spectrum of denoted by , is the set of *λ*∈*Ω* such that . Such points are called *eigenvalues*, non-trivial vectors in the kernel are called the corresponding *eigenvectors*, and the dimension of the kernel is called the *geometric multiplicity*. If the domain of is independent of *λ* and if for each *f* in this domain, the function is analytic in the usual strong sense, then we can define the *chains of* and *canonical systems of eigen- and associated vectors* as well as a concept of *algebraic multiplicity* of an eigenvalue *λ*_{0}. We will provide all necessary definitions later.

In the present paper, we deal with a special class of analytic operator-valued functions, i.e. with analytic Fredholm operator-valued functions.

A bounded linear operator ** A** in a Hilbert space

*H*is called a Fredholm operator if its range Im

*(5.1)are finite. The number defined as(5.2)is called an index of*

**is closed and the numbers***A***. For a Fredholm operator, the index is always finite.**

*A*Let be an analytic operator-valued function. We say that *λ*_{0}∈*Ω* is an eigenvalue of a finite type of if (i) is a Fredholm operator, (ii) for some non-zero *f*∈*H*, and (iii) is an invertible operator for all *λ* in a punctured disk 0<|*λ*−*λ*_{0}|<ϵ around *λ*_{0}.

We notice here that if *λ*_{0} is a finite-type eigenvalue, then .

The following important result holds (see Gohberg *et al*. 1990, p. 203).

*Assume that for some λ*_{0}∈*Ω, an analytic operator-valued function* *is a Fredholm operator with* *. Then there exists ϵ*>0 *such that for all* *, the operator-valued function* *can be factored as*(5.3)*where E*(*.*) *and F*(*.*) *are analytic Fredholm operator-valued functions having bounded inverse operators at each point of the disk |λ−λ _{0}|≤ϵ. The operator-valued function D*(

*.*)

*has the form*(5.4)

*where P*

_{0},

*P*

_{1}, …,

*P*

_{r}

*are mutually disjoint projections of rank one; the projection*(

*I*−

*P*

_{0})

*has a finite rank and k*

_{1}≤

*k*

_{2}≤⋯≤

*k*

_{r}

*are positive integers. The sum*(5.5)

*is called the algebraic multiplicity of*

*at λ*

_{0}.

We notice that has only one linearly independent normalized eigenvector *φ*_{0} corresponding to the eigenvalue *λ*_{0} if, and only if, the projection (*I*−*P*_{0}) is one-dimensional. Indeed, assume that there exists a unique vector *φ*_{0} such that , i.e. from (5.3) we have . Since (*F*(*λ*_{0}))^{−1} exists and is bounded, we have that . Since (*E*(*λ*_{0}))^{−1} exists and is bounded, we obtain that if *φ*_{0} is an eigenvector of , then *g*_{0}=*F*(*λ*_{0})*φ*_{0} is annihilated by *P*_{0}, which means that dim(*I*−*P*_{0})=1. On the other hand, assume that dim(*I*−*P*_{0})=1, i.e. there exists a unique and normalized vector *g*_{1}∈*H* such that *P*_{0}*g*_{1}=0. Since *g*_{0} is a non-trivial vector and (*F*(*λ*_{0}))^{−1} is a bounded operator, then is an eigenvector of at *λ*_{0}. With this short proof, we conclude the remark.

We also mention (Marcus 1988) that there might exist *m* linearly independent *associate vectors φ*_{1}, *φ*_{2}, …, *φ*_{m}, which satisfy the following equation:(5.6)If, for an eigenvalue *λ*_{0}, there exists only one eigenvector *φ*_{0} and a finite chain of the associate vectors, we will call the collection the chain of the root vectors corresponding to *λ*_{0} and denote this chain of the root vectors by .

We notice here that in general there can be a finite number of multiple eigenvalues associated to more than one eigenvector, which means that such an eigenvalue is related to more than one chain of the associate vectors. However, as has been shown in (Shubov & Balakrishnan 2004*a*,*b*), the distant aeroelastic modes in the individual *β*- and *δ*-branches are simple and, moreover, the distant *β*- and *δ*-branch aeroelastic modes cannot coincide. Therefore, without loss of generality, we will not consider the case when one eigenvalue can be associated to multiple eigenvectors and correspondingly to the multiple chains of the associate vectors. The next result is of a special importance for us (Gohberg *et al*. 1990).

*Let* *be an analytic Fredholm operator-valued function. If for some λ∈Ω, the operator* *has bounded inverse, then the spectrum* *is discrete, i.e. it contains at most countably many eigenvalues; each eigenvalue has a finite algebraic multiplicity with no accumulation points in Ω. Furthermore,* *, is a meromorphic function on Ω whose poles are exactly the eigenvalues of* *. The principal part of the Laurent expansion around each pole has only finitely many non-zero terms, the coefficients in these terms being of finite rank operators.*

In what follows, we need the generalization of the Rouche's theorem to the analytic operator-valued functions. To formulate it, we provide several more definitions.

(i) A Cauchy domain is a disjoint union in of a finite number of non-empty open connected sets Δ_{1}, Δ_{2}, …, Δ_{r}, such that *i*≠*j*, and for each *j*, the boundary of the set Δ_{j} consists of a finite number of non-intersecting closed rectifiable Jordan curves, which are oriented in such a way that Δ_{j} belongs to the inner domains of the curves. (ii) *Γ* is called a Cauchy contour if *Γ* is the oriented boundary of a bounded Cauchy domain. (iii) Let *G* be a Cauchy domain with and let be a bounded operator-valued analytic function of *λ*, *λ∈Ω*. Then is said to be normal with respect to *Γ*, , if has a bounded inverse for *λ∈Γ* and is a Fredholm operator for each *λ*∈*G*.

The following generalization of the Rouche's theorem holds (Gohberg & Sigal 1971; Gohberg *et al*. 1990).

*Let W*(*.*) *and* *be analytic operator-valued functions and let W be normal with respect to a contour Γ. If*(5.7)*then the operator-valued function*(5.8)*is also normal with respect to Γ and*(5.9)*where* *and* *are the algebraic multiplicities of U and W relative to the contour Γ, i.e.*(5.10)(5.11)*where* *are the eigenvalues of finite types of U inside Γ and* *are the eigenvalues of W inside Γ.*

In what follows, denotes the derivative of with respect to *λ*. The following result provides the formula for the total multiplicity with respect to the contour *Γ*.

*Let* *be an analytic operator-valued function and assume that* *is normal with respect to the contour Γ. Then*(5.12)*where* ‘tr’ *denotes the trace of the operator.*

We notice that for a finite-rank operator *F*, we have(5.13)with *λ*_{j}(*F*) being an eigenvalue of *F*. In formula (5.13), the multiplicities of the eigenvalues have been taken into account, and summation in fact goes along a finite number of terms.

An operator-valued function *G* is called finitely meromorphic at *λ*_{0} if G has a pole at *λ*_{0} and the coefficients of the principal part of its Laurent expansion at *λ*_{0} are operators of finite ranks, i.e. in some punctured neighbourhood of *λ*_{0}, the following expansion is valid:(5.14)which converges in the operator norm on , such that *G*_{−1}, …, *G*_{−q} are finite-rank operators. In that case, we write *ΞG*(.) for the principal part of *G* at *λ*_{0}. Thus,(5.15)

Note that *ΞG* is analytic on and its values are finite rank operators. We will need the following statement (Gohberg & Sigal 1971).

*Let G*_{1} *and G*_{2} *be* *-valued operator functions, which are finitely meromorphic at λ*_{0}*. Then G*_{1}*G*_{2} *and G*_{2}*G*_{1} *are finitely meromorphic at λ*_{0} *and*(5.16)

To conclude this section, we address the question of the Riesz basis property of the set of functions. The following result is valid (Lutgen 2001).

*If* *is a Riesz basis in a separable Hilbert space H and if a sequence of vectors* *is such that*(5.17)*then the set* *is also a Riesz basis in H. The set* *is a Riesz basis of a defect N with N*=*m*_{2}−*m*_{1}.

## 6. Analytic Fredholm operator-valued functions in wing model problem

In what follows, we use the theory of analytic Fredholm operator-valued functions. To this end, we have to replace the operator valued functions and (*λI*−i_{βδ}), which are unbounded, with some Fredholm operator-valued functions, which have to be bounded. Let and let be defined by(6.1)

Based on statement (ii) of theorem 3.7, both *W* and *Z* are well defined. As follows from theorem 3.7, the spectrum of the operator **K**_{βδ} consists of normal eigenvalues; only a finite number are multiple eigenvalues having finite multiplicities. We know also that there exists a positive integer *N*, such that all eigenvalues of **K**_{βδ} denoted by are simple. Regarding the operator-valued function , we know that there exists a countable set of points on the complex *λ*-plane, at which does not have an inverse. These points are exactly the aeroelastic modes; at the points *λ*, which are not the aeroelastic modes, the operator exists and is compact (Shubov & Balakrishnan 2004*a*,*b*). We also know that for |*n*|≥*N*, all aeroelastic modes correspond to the simple poles of the generalized resolvent operator; i.e. the set represents simple poles of (*λ*). In the next statement, we establish the relationship between the spectral characteristics of the operator **K**_{βδ} and the operator-valued function *W*(*λ*), and also the relationship between the spectral characteristics of the operator-valued functions *Z*(*λ*) and . Clearly, the asymptotical representation for the spectrum of the operator **K**_{βδ} coincides with that of the set of the aeroelastic modes (see theorem 3.5).

*λ*_{0}*is an eigenvalue of Z*(*.*)*and g is the corresponding eigenvector if, and only if, λ*_{0}*is an aeroelastic mode and**is the corresponding mode shape.**The spectrum of the operator-valued function W*(*.*)*coincides with the spectrum of the operator***K**_{βδ}*and the corresponding root spaces are identical.*

We start with statement (i). Let *λ*_{0} be an aeroelastic mode and *φ* be the corresponding mode shape, i.e. . Since, (Shubov & Balakrishnan 2004*a*,*b*), we obtain(6.2)Equation (6.2) means that *λ*_{0} and *g* are the eigenvalue and eigenvector of *Z*(.). The inverse statement can be shown in a similar way. We do not discuss the relationship between the associate functions corresponding to multiple poles of the operator-valued functions and [*Z*(.)]^{−1}. (Note there may be only finite numbers of associate vectors in both cases.)

Proof of statement (ii). As we know, **K**_{βδ} might have a finite number of multiple eigenvalues of finite multiplicity each. Let us denote these multiple eigenvalues by making no difference between the branches, i.e. let us represent the spectrum in the form(6.3)

For each multiple eigenvalue, there exist one or several linearly independent eigenvectors and each eigenvector might have associate vectors. The entire collection of all eigenvectors and all associate vectors corresponding to one and the same eigenvalue *μ* forms *the root space* corresponding to this *μ*. In order not to overcomplicate the presentation below, let us assume that *μ*_{n} is a multiple eigenvalue with multiplicity *κ*_{n}, i.e. there exists one eigenvector and (*κ*_{n}−1) associate vectors. Let be the root vectors, and is either an eigenvector or an associate vector of the operator **K**_{βδ} corresponding to *μ*_{n}. The adjoint operator has as its eigenvalue with the same multiplicity *κ*_{n}. Let be the set of the corresponding root vectors of the operator . It is possible to enumerate the root vectors in such a way that the following biorthogonality condition holds (Shubov 2000, 2001*c*):(6.4)

The resolvent of **K**_{βδ} can be represented as the following strongly convergent series:(6.5)where *N* is the number of multiple eigenvalues of **K**_{βδ} with the multiplicities *κ*_{1}, *κ*_{2}, …, *κ*_{N}. Using the definition of *W*(.) from (6.1) and the spectral decomposition (6.5), the operator-valued function *W*(.) can be written as the following strongly convergent series:(6.6)Due to the fact that both the set and the set are two biorthogonal Riesz bases, the validity of statement (ii) follows immediately from (6.6). The generalization to the case when the root space contains several linearly independent eigenvectors and the corresponding chains of the associate vectors can be done in a straightforward fashion.

The theorem is completely shown. ▪

In the conclusion of this section, we prove the following statement.

*Z*(*λ*) *is an analytic Fredholm operator-valued function.*

According to definition 5.1, the only facts to verify are given in formula (5.2). Let us start with *n*(*Z*). For a given , we have that . If *λ* coincides with an aeroelastic mode, then is a finite-dimensional subspace. Due to the fact that , we obtain that Ker{*Z*(*λ*)} is also a finite-dimensional subspace. A similar fact on *n*(*Z*^{*}(*λ*)) can be easily obtained by using the results of our paper (Shubov & Balakrishnan 2004*a*,*b*). It is clear that if *λ* is not an aeroelastic mode, then ind(*Z*(λ))=0.

The lemma is shown. ▪

## 7. Normalization of set of adjoint mode shapes

Now we prove the normalization condition for the eigenfunctions of the operator-valued functions *Z*(*λ*) and *W*(*λ*). The eigenvectors of *Z*(*λ*) are connected to the mode shapes through the relation , where is the eigenvector of *Z*(*λ*) and *Φ*_{n} is the mode shape. Since this set of mode shapes splits asymptotically into two branches, the same fact is valid for the set of the eigenvectors; i.e. we can denote the set of the eigenvectors of *Z*(*λ*) by . As we know, there exists an open neighbourhood centred at an aeroelastic mode such that *Z*(*λ*) has a bounded inverse in this neighbourhood with one punctured point. Let us take, for example, the *δ*-branch simple aeroelastic mode. The case of the *β*-branch aeroelastic mode can be treated similarly. As follows from Gohberg & Sigal (1971), the adjoint operator-valued function [*Z*(*λ*)]^{*} has its eigenvalue at the point (Gohberg & Sigal 1971). Let be the corresponding eigenvector, i.e.(7.1)According to theorem 7.1 of Gohberg & Sigal (1971, p. 621), the operator-valued function [*Z*(*λ*)]^{−1} admits the following representation when :(7.2)We recall that is an open neighbourhood of the point such that for all , , *Z*(*λ*)^{−1} exists. In (7.2), is a holomorphic operator-valued function in .

Now we can use theorem 5.8. By formula (5.12), we have(7.3)Here, *Γ* is a smooth closed contour in enclosing .

In our case, *Z*(*λ*) is an analytic function, which does not have any poles. Thus, *Z*′(*λ*) is also analytic in . Therefore, formula (7.3) can be written as(7.4)

In (7.4), we have used formula (5.15), theorem 5.10, and the fact that is simple. Using formula (7.2), we obtain from (7.4)(7.5)We notice thatis a rank-one operator in . Therefore, its trace is equal to the eigenvalue. Obviously, is the eigenfunction of *Π*. The corresponding eigenvalue can be found from the equation . Using (7.5), we conclude(7.6)

Arguing the same way, we obtain the normalization condition for the *β*-branch(7.7)

Summarizing all of the above, we arrive at the following statement.

*Let*(7.8)*be the set of aeroelastic modes, where* *is the set of aeroelastic modes corresponding to multiple poles of the generalized resolvent operator and* *is the set of aeroelastic modes corresponding to the simple poles of the generalized resolvent.*

*Let* *be the set of eigenvectors of the operator-valued function Z*(*λ*) *corresponding to the set* , *and let* *be the corresponding set of the eigenvectors of the adjoint operator-valued function* [*Z*(λ)]^{*}*. Then the following normalization conditions are valid:*(7.9)

## 8. Minimality of set of mode shapes

To prove the main result of this section (see theorem 8.2 below), we have to recall the notion of the *Riesz integral* (Gohberg *et al*. 1990; Gohberg & Krein 1996). Let *Γ* be a simple rectifiable contour, which encloses some region and lies entirely within a resolvent set of an analytical Fredholm operator-valued function *Z*(*λ*). Thus, the inverse *Z*(*λ*)^{−1} is also an analytic operator-valued function on *Γ*. Assuming that *Γ* has a positive orientation with respect to *G*_{Γ}, let us form an integral(8.1)

*(i)* *is the projection, i.e. it is a bounded operator in* *and such that* . *(ii) Let λ*_{1} *and λ*_{2} *be two different modes and Γ*_{1} and *Γ*_{2} *be two boundaries for the domains G*_{1} *and G*_{2} *containing λ*_{1} *and λ*_{2}*, respectively. If G*_{1}∪*Γ*_{1} *and G*_{2}∪*Γ*_{2} *do not have common points, then the following orthogonality relation holds:*(8.2)

The proof of this theorem can be found in Gohberg & Sigal (1971) and Gohberg *et al*. (1990). Using theorem 8.1, we can prove the main result of the section.

*The set of all aeroelastic mode shapes is minimal in* .

As we already know (Shubov 2001*c*; Shubov & Balakrishnan 2004*a*,*b*), the generalized resolvent operator (*λ*) does not exist if, and only if, *λ* is an aeroelastic mode. Otherwise, (*λ*) exists and is a compact operator in . Theorem 5.5 indicates that the set of the aeroelastic modes is, at most, countable and there might be multiple poles of (*λ*). Representation (7.8) shows that both sets and consist of isolated simple poles of the generalized resolvent operator. Thus, each aeroelastic mode can be surrounded by a circle of small radius in such a way that there is one and only one mode inside a circle. For the current proof, we need not distinguish the branches of the modes, i.e. let be an entire set of the aeroelastic modes and be the set of the corresponding mode shapes. Let us enumerate the aeroelastic modes in a special way. Namely, if a number of multiple modes is odd, then we assume that there exists *N* such that forms a set of simple modes. If the number of multiple modes is even, then we assume that there exists *N* such that the set forms the set of multiple modes and the remaining part forms the set of simple modes. To complete the proof, we use the contradiction argument. Namely, let us assume that the entire set of aeroelastic modes is not minimal; i.e. there exists an infinite sequence of non-trivial complex points such that(8.3)Clearly, there exists *c*_{m}≠0, such that *m*>*N*, and(8.4)

Let be a Riesz projection associated to a contour surrounding a simple pole *λ*_{m}. Let us apply to both parts of equation (8.16). We have(8.5)

The theorem is completely proven. ▪

## 9. Refined asymptotic estimate for *δ*-branch aeroelastic modes

In this section, we show that the asymptotic estimate given in statement (*ii*) of theorem 3.5 for the *δ*-branch aeroelastic modes can be refined in the following way. From that statement and formula (3.4), we can conclude that as |*n*|→∞. The latter result is probably the best that can be obtained based on the techniques of the asymptotic analysis. To carry out the aforementioned refinement, we use the techniques of the operator theory and of the complex analysis. The main result of this section is the following.

*For the δ-branch aeroelastic modes, the following asymptotic estimate is valid when |n|→∞:*(9.1)*with* *being the δ-branch of the eigenvalues of the operator K*

_{βδ}

*. The precise value of the constant C*

_{2}

*in*

*(9.1)*

*is immaterial for us*.

As follows from the asymptotic formulae for the spectrum of the operator **K**_{βδ}, there could be the following two cases:In the first case, we would say that the *δ*- and *β*-branches are separated, in the second case, the branches are not separated. It is clear that in case (b), there exist two infinite subsequences of the *β*- and the *δ*-branches of the operator **K**_{βδ} such that(9.2)

In the proof below, we deal with case (b), which is technically much more difficult then the case (a), and at the end of the proof, we briefly address the proof for case (a). Let us write formula (6.3) for the spectrum of the operator **K**_{βδ} in the form(9.3)

Let us split the set **S**_{δ} into two subsets and according to the following rule (figure 1):(9.4)where *C*_{3} is an absolute constant, whose value will be identified later, and let(9.5)

Let us use the notation *B*_{r}(*μ*_{0}) and *Γ*_{r}(*μ*_{0}) for a disc and a circle of radius *r* centred at the point *μ*_{0}, i.e.(9.6)

In terms of (9.6), by setting , we obtain that a given if and only if the disc contains one eigenvalue of the operator **K**_{βδ}. The fact that means that the disc contains precisely two eigenvalues of **K**_{βδ}; namely contains one eigenvalue , which is the centre of the disc, and one eigenvalue from the *β*-branch of the spectrum of **K**_{βδ}. Note that if , then . We can easily see that if , then(9.7)If , then we have(9.8)and(9.9)

In what follows, we also need the notation(9.10)for the *β*- and the *δ*-branch aeroelastic modes. If *ρ*(**K**_{βδ}) denotes the resolvent set of the operator **K**_{βδ}, then for with , the following formula holds (see definitions (6.1)):(9.11)

Let us fix some with *n*_{0}≥*N*≫1 and estimate when . Note that there exists *m*_{0} large enough and such that . Due to decomposition (6.5), we can write(9.12)

If *n*≠*n*_{0} and *m*≠*m*_{0}, then the following estimates can be easily verified for :(9.13)where Using the Riesz basis property of the root vectors of the operator **K**_{βδ}, we obtain from (9.12) for any *F*(9.14)Let us use the notation for the set of the root vectors of the operator (9.15)Since is a Riesz basis biorthogonal to the Riesz basis of the root vectors of **K**_{βδ}, we can introduce for the Riesz basis (9.15) the notion similar to and have from (9.14)(9.16)Taking into account that , we obtain the following estimate for the norm of the resolvent operator when :(9.17)Having estimate (9.17), we can easily obtain the following estimate for :(9.18)We already know that . Let us choose *C*_{3} in such a way that(9.19)with being defined in (9.13). With this choice of *C*_{3}, estimate (9.18) becomes(9.20)and we can apply the Rouche's theorem for operator-valued functions (theorem 5.7), which yields an important conclusion: if , then in the disc with and *C*_{3} satisfying (9.19), there exist precisely two pairs of points and .

So far, we have considered the most difficult case when . In the simpler case, when , we can obtain a similar result, i.e. in each disc there exists one, and only one, aeroelastic mode . Therefore, estimate (9.1) holds for both cases.

The theorem is completely shown. ▪

## 10. Quadratic closeness of mode shapes and eigenfunctions of operator **K**_{βδ}

**K**

To prove the main statement of this section (theorem 10.2), we need some technical result.

*The following estimate holds:*(10.1)

As we know (Shubov 2000; Shubov & Balakrishnan 2004*b*), the set of the mode shapes is almost normalized in . Thus, to prove (10.1) it suffices to show that(10.2)with some absolute constant *C*_{4} being independent of *m*. Since the set forms a Riesz basis in its closed linear span, the proof of (10.2) will be complete if we show that(10.3)Now let us turn to the normalization condition (7.7) and have(10.4)or(10.5)

Let us show now that , where is an adjoint aeroelastic mode, is an eigenvector of an adjoint operator-valued function [*Z*(.)]^{*}, and *α*_{m} is a numerical coefficient. Indeed, we have . The latter fact means that (10.2) is valid if we show that . Returning to (10.5), we have(10.6)Due to (3.21), we have(10.7)

According to Shubov (2000) and Shubov & Balakrishnan (2004*b*), the sequence is almost normalized and, therefore, combining (10.6) with (10.7), we obtain that . The latter fact leads to the almost normalization of the sequence .

The lemma is proven. ▪

*Let the set of the aeroelastic modes be numbered as follows. If* *is the set of multiple modes, then the entire set of the aeroelastic modes can be represented in the form*(10.8)*where*(10.9)

*The aeroelastic modes in the sets Λ*^{β} *and Λ*^{δ} *correspond to the simple roots of the generalized resolvent operator. Let the numeration in* *(9.3)* *for the set of the eigenvalues of the operator K*

_{βδ}

*be done in such a way that*(10.10)

*with C*

_{5}

*being some absolute constant. Then the sets of the corresponding mode shapes and eigenvectors of the operator*

**K**_{βδ}

*are quadratically close, i.e.*(10.11)

Before we prove the theorem, we make the following statement.

We notice that though the set contains the aeroelastic modes corresponding to the multiple roots of the generalized resolvent operator, the same fact may not be true for the set . Namely, the set may contain both multiple and simple eigenvalues of *K*_{βδ}. It may also happen that several multiple eigenvalues should be added to *δ*- or *β*-branch. In the latter case, a multiple eigenvalue should be repeated as many times as its multiplicity. The splitting (9.3) has to be done in such a way that relations (10.10) hold.

Let us take some small *ϵ*>0 and split the set *Λ*^{δ} into two subsets and , where(10.12)

(10.13)

Let , then we can see that(10.14)

As follows from theorem 9.1, we also have for any and sufficiently large *n*_{0} that the following estimate holds:(10.15)Let and |*n*_{0}| be sufficiently large. It can be easily shown that(10.16)

(10.17)Due to theorem 8.1, we also have(10.18)and(10.19)

We will use relations (10.12)–(10.19) for the estimates related to the operator-valued functions *Z*(.)and *W*(.). Based on theorem 6.1, we can see that if and are an aeroelastic mode and a mode shape for the operator-valued function , then and(10.20)are an eigenvalue and a corresponding eigenfunction of the operator-valued function *Z*(.). We note that *Z*(.) has a bounded inverse for the punctured disc *B*_{1} defined as(10.21)

The function *W*(.) has a bounded inverse if *λ* is in the punctured disc *B*_{2} defined as(10.22)

As follows from Gohberg & Sigal (1971), the analytic Fredholm operator-valued functions [*Z*(.)]^{−1} and [*W*(.)]^{−1} act on ∈ according to the following formulae:(10.23)and(10.24)

In formulae (10.23) and (10.24), and are holomorphic operator-valued functions in ; and are the eigenvectors of the adjoint operator-valued function corresponding to the eigenvalues and , respectively. The following normalization and orthogonality conditions are valid:(10.25)

Let us consider the following difference:(10.26)Integrating (10.26) along the circle and using (10.25), we obtain(10.27)Note that in (10.27) the fact that and are holomorphic operator-valued functions has been taken into account. To continue the evaluation of (10.27), let us assume that(10.28)Thus, we continue (10.27) and have(10.29)Note, we have taken into account thatUsing (10.20) and (10.28), we can rewrite (10.29) as(10.30)From (10.30), it follows immediately that(10.31)For , the integrand in (10.31) can be evaluated as follows:(10.32)

As follows from (9.16) for a given , the norms are bounded above uniformly with respect to *n*_{0} when . Since , we obtain from (10.32) the estimate(10.33)where *C*_{6} is an absolute constant, the precise value of which is immaterial for us. Combining (10.31) with (10.33), we obtain with another absolute constant *C*_{7} that(10.34)

Using lemma 10.1, we obtain from (10.34) that the following relation is valid:(10.35)where and are defined by formulae (10.25) and (10.28). Taking into account (10.28), we can see that the set of functions is quadratically close to the Riesz basis . Since the sequence is almost normalized, i.e. , there is no need to use a new notation for the function . Thus, without any misunderstanding, combining (10.35) with (4.4), we obtain (10.11).

The theorem is completely proven. ▪

## 11. Riesz basis property of mode shapes

To establish the Riesz basis property of the mode shapes, we have to prove the following remaining result.

*If*(11.1)*is the set of the generalized eigenvectors of the operator K*

_{βδ}

*and*(11.2)

*is the set of the mode shapes, then M*=

*N*.

To prove the result, we consider two cases, i.e. (*a*) *N*<*M* and (*b*) *N*>*M*.

*Case* (*a*). If *N*<*M*, then we obviously have(11.3)Let us show that estimate (11.3) implies that there exists a bounded and boundedly invertible operator of the form , with being a compact operator , which relates the two families of functions according to the rule(11.4)Indeed, let be a linear mapping defined on a dense set of finite linear combinations of the vectors from the Riesz basis of the generalized eigenvectors of the operator **K**_{βδ}(11.5)

Let us prove that is a bounded operator. We have(11.6)We notice that the last step in estimate (11.6) is valid due to (11.3). Taking into account the Riesz basis property of the generalized eignevectors of the operator **K**_{βδ}, we obtain thatwhich means that the operator is bounded. In the proof that , we will not distinguish the *β*- and *δ*-branches, i.e. assume that we have two sequences of functions and such that the first sequence forms a Riesz basis in , the second sequence is minimal and is quadratically close to the first one, i.e.(11.7)We have already shown that there exists a bounded operator such that and , *m*=1, 2, …, *N*, . Now we show that , . Let us assume that a normalized sequence converges to zero weakly, i.e.(11.8)

It suffices to show that the sequence of images converges to zero strongly, i.e.(11.9)

From (11.8), we get the following two facts: (*a*) , with the constant *C*_{8} being independent of *p*; and (*b*) for each , as *p*→∞. To prove (11.9), let us fix an arbitrary small *ϵ*>0 and show that there exists *P* such that(11.10)Due to estimate (11.7), we can find a positive integer *N*_{1} such that(11.11)Thus, we have(11.12)Using (11.11) and the property (*a*), we obtain(11.13)For a given integer *N*_{1}, due to property (*b*), we can find *P* such that for all *p*≥*P*(11.14)Using (11.14) we estimate(11.15)Combining (11.13) and (11.15), we obtain the desired result, i.e. , .

We now show that the inverse operator also exists and, therefore, has to be bounded. Using contradiction argument, let us assume that the equation have a non-trivial solution *F*∈. Let us expand *F* with respect to the Riesz basis and have(11.16)

We also obtain from our assumptions that(11.17)

Due to the minimality of the system , the latter equation leads to *f*_{m}=*f*_{n}=0 and thus, *F*=0, which means that the set forms a Riesz basis in . According to our assumption, there exists a vector , *N*+1≤*s*≤*M*, which can be expanded with respect to the Riesz basis of mode shapes, i.e.(11.18)

However, (11.18) contradicts to the fact that the set of the mode shapes is minimal in .

The case (*a*) is proven.

*Case* (*b*). If *M*<*N*, then by repeating the proof for case (*a*), we obtain that there exists a bounded and boundedly invertible operator , which relates the two systems(11.19)

Relations (11.19) certainly mean that the entire set of the mode shapes forms a Riesz basis in its closed linear span in . Due to the completeness of the set of the mode shapes in (Shubov & Balakrishnan 2004*a*), we obtain that the aforementioned closed linear span coincides with the entire space .

The theorem is completely shown. ▪

## Acknowledgements

Partial support by the National Science Foundation grants ECS no. 0080441, DMS no. 0072247, and the Advanced Research Program-01 of Texas grant 0036-44-045 is highly appreciated by the author.

## Footnotes

- Received April 13, 2005.
- Accepted September 13, 2005.

- © 2005 The Royal Society