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ResearchCite this article: Shubov MA. 2014 Onfluttering modes for aircraft wing model insubsonic air flow. Proc. R. Soc. A 470: 20140582.http://dx.doi.org/10.1098/rspa.2014.0582

Received: 30 July 2014Accepted: 8 October 2014

Subject Areas:applied mathematics, differential equations,mathematical modelling

Keywords:aeroelastic modes, non-selfadjoint differentialoperator, stability, flutter, circle of instability

Author for correspondence:Marianna A. Shubove-mail: [email protected]

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rspa.2014.0582 orvia http://rspa.royalsocietypublishing.org.

On fluttering modes for aircraftwing model in subsonic airflowMarianna A. Shubov

Department of Mathematics and Statistics, University of NewHampshire, 33 Academic Way, Durham, NH 03824, USA

The paper deals with unstable aeroelastic modesfor aircraft wing model in subsonic, incompressible,inviscid air flow. In recent author’s papers asymptotic,spectral and stability analysis of the model hasbeen carried out. The model is governed by asystem of two coupled integrodifferential equationsand a two-parameter family of boundary conditionsmodelling action of self-straining actuators. TheLaplace transform of the solution is given in termsof the ‘generalized resolvent operator’, which is ameromorphic operator-valued function of the spectralparameter λ, whose poles are called the aeroelasticmodes. The residues at these poles are constructedfrom the corresponding mode shapes. The spectralcharacteristics of the model are asymptotically closeto the ones of a simpler system, which is called thereduced model. For the reduced model, the followingresult is shown: for each value of subsonic speed,there exists a radius such that all aeroelastic modeslocated outside the circle of this radius centred at zeroare stable. Unstable modes, whose number is alwaysfinite, can occur only inside this ‘circle of instability’.Explicit estimate of the ‘instability radius’ in terms ofmodel parameters is given.

1. IntroductionWe study the problem of stability of the aeroelastic modesfor a high aspect ratio aircraft wing model in a subsonicair flow. As is well known, a flexible wing vibrating in anair flow may lose stability owing to flutter. We recall thatflutter, which is known as a very dangerous aeroelasticdevelopment, is the onset, beyond some speed–altitudecombinations, of unstable and destructive vibrations ofa lifting surface in an air stream [1–16]. We consider themodel of an aircraft wing in an incompressible inviscidsubsonic air flow, whose mathematical formulation can

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be found in references [1,5–9,11–13,16–19]. This is a two-dimensional strip model, which appliesto bare wings of high aspect ratio [1,2,7,8,10]. The structure is modelled by a uniform cantileverbeam, which bends and twists. In addition, the action of self-straining actuators is modelled usinga currently accepted approach [3–5,20,21]. Probably the most important type of aircraft flutterresults from coupling between bending and torsional motions of a wing with large aspect ratio.The model we consider has been designed to treat flutter caused by this type of coupling.

This work is a continuation of the research completed in references [11–19], where detailedasymptotic and spectral analysis of the model has been carried out. To keep the paper self-contained, we provide in §§3 and 4 a precise formulation of the model and all the necessaryresults from our previous works. In §2, we give a brief description of the abstract form of themodel and introduce the ‘reduced model’ which, as we show, is appropriate for describing thefluttering modes. We also define the aeroelastic modes for both full and reduced models and givea preliminary statement of the main result of the paper. The result consists of the following. Theset of the aeroelastic modes of the reduced model may contain only a finite number of unstable(fluttering) modes that belong to the ‘instability circle’ centred at the origin. An explicit estimateon the radius of this circle is obtained. Full statement of the main result and its proof is given in§6 (theorem 6.1).

2. Abstract form of the model and the main resultThe model is governed by a system of two coupled partial integrodifferential equations with atwo-parameter family of boundary conditions. The integral parts of the equations represent theforces and moments exerted to the wing owing to the air flow. The above-mentioned system withboundary conditions can be represented as a single-operator evolution–convolution equation ina Hilbert space H of four-component vector-valued functions of one spatial variable. Namely itcan be given in the form of

Ψ (t) = iLβδΨ (t) +∫ t

0F(t − τ )Ψ (τ ) dτ , Ψ |t=0 =Ψ0. (2.1)

Here, Ψ (t) ∈ H is the state of the system at time t, Lβδ is an unbounded non-selfadjointoperator in H (a 4 × 4 matrix differential operator of fourth-order with the domain defined bythe boundary conditions). This operator depends on two complex parameters β and δ whichenter the boundary conditions as control gains. F(t) ≡ F(t; u) is a matrix-valued convolution kernelwhich depends on the speed u of the air stream. The overdot represents the time derivative.iLβδ is the dynamics generator for the structural part of the model, and the convolution integralrepresents the aerodynamic loads. Note equation (2.1) is not an evolution equation. It does nothave a dynamics generator and does not define any semi-group in the standard sense.

By applying the Laplace transformation to both sides of equation (2.1), we obtain

(λI − iLβδ − λF(λ))Ψ (λ) = (I − F(λ))Ψ0, (2.2)

where I is the identity operator, and F(λ) is the Laplace transform of F(t). At those points λ, wherethe operator (λI − iLβδ − λF(λ)) has a bounded inverse, the solution of equation (2.2) can be givenexplicitly as

Ψ (λ) = (λI − iLβδ − λF(λ))−1(I − F(λ))Ψ0. (2.3)

To find the time–space representation of the solution, one has ‘to calculate’ the inverse Laplacetransform of Ψ . To carry out this step, it is necessary to investigate the ‘generalized resolventoperator’

R(λ) = (λI − iLβδ − λF(λ))−1. (2.4)

As shown in references [11–15,17–19], R(λ) is an operator-valued meromorphic function on thecomplex plane with a branch-cut along the negative real semi-axis. The poles of R(λ) are calledthe aeroelastic modes. (The branch-cut is associated with ‘the continuous spectrum’.)



3


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In references [11–15,19], the author has derived explicit asymptotic formulae for the aeroelasticmodes. It turned out (see theorem 4.3) that the whole set of aeroelastic modes, which we denote{λn}, splits asymptotically into two distinct branches; each branch approaches its own verticalasymptote. Both branches have only two points of accumulation ±i∞. The aeroelastic modesdepend on the speed u of the airstream: λn = λn(u). Flutter occurs if with u increasing at least oneof the modes crosses the imaginary axis from the left into the right half-plane, i.e. �λn(u)> 0 forsome n.

If the speed of the air stream u = 0, then the convolution (aerodynamic load) term in (2.1)vanishes and (2.1) takes the form

Ψ (t) = iLβδΨ (t), Ψ (0) =Ψ0. (2.5)

Equation (2.5) describes the ground vibrations of the wing in the absence of the airflow. Lβδ

is an unbounded non-selfadjoint operator in H with a compact resolvent. This operator has adiscrete spectrum of complex eigenvalues {λ(0)

n }n∈Z′ (Z′ = Z \ {0}), which is symmetric with respectto the imaginary axis. The numbers iλ(0)

n are the eigenmodes of the ground vibrations. Because theground vibrations occur when u = 0, it is obvious that iλ(0)

n = λn(0). The operator Lβδ is dissipative( �(LβδΨ ,Ψ )H ≥ 0 for any Ψ ∈ D(Lβδ)) and, therefore, its spectrum is located in the upper half-plane: �λ(0)

n ≥ 0 for all n. (By (·, ·)H , we denote the energy inner product in H defined in (4.1).)Accordingly, the ground-vibration modes iλ(0)

n belong to the left half-plane: �(iλ(0)n ) ≤ 0, i.e. all

these modes are stable.It has been shown by the author in references [14,17] (see lemma 4.1) that the Laplace transform

F(λ) of the matrix kernel F(t) from (2.1) admits the following decomposition:

λF(λ) = M + N(λ), (2.6)

where M is a constant 4 × 4 matrix independent of the spectral parameter λ and N(λ) isasymptotically small: ‖N(λ)‖ → 0 as |λ| → ∞. Both terms in the right-hand side of (2.6) dependon the airstream speed u. Denote by G(t) the inverse Laplace transform of N(λ). Then, the mainmodel equation (2.1) can be represented in the form

Ψ (t) = (iLβδ + M)Ψ (t) +∫ t

0G(t − τ )Ψ (τ ) dτ . (2.7)

Let us discard the convolution term in (2.7). Then, we obtain equation (2.8)

Ψ (t) = KβδΨ (t), where Kβδ = iLβδ + M. (2.8)

We call (2.8)—the equation of the ‘reduced model’. The dynamics generator of the reducedmodel is the operator Kβδ = i(Lβδ − iM). We denote its spectrum by {λn}n∈Z′ and point outthat the eigenvalues depend on u: λn = λn(u). This spectrum, as well as the spectrum of iLβδ ,is symmetric with respect to the real axis. That is why we use the index n ∈ Z

′ = Z\{0} numberingthe eigenvalues. It has been shown by the author in reference [13] that the eigenmodes of thereduced model have the same asymptotic representation as λn and iλ(0)

n . The differences in theasymptotic approximations of all three sets {λn}n∈Z′ , {iλ(0)

n }n∈Z′ and {λn}n∈Z′ are in the remainderterms. We note that, unlike Lβδ , the operator Lβδ − iM is no longer dissipative. So, it mayhave the eigenvalues in the lower half-plane. Accordingly, the dynamic generator Kβδ mayhave unstable (fluttering) eigenvalues in the right half-plane: �λn(u)> 0. In other words, as waspointed out in reference [13], the perturbation term M is responsible for the appearance of thefluttering modes.

The main result of this paper deals with the eigenvalues {λn} of the reduced model (2.8). Thisresult consists of the following (see theorem 6.1 for the precise statement).

For each value of the airstream speed u, there exists R(u)> 0 such that all eigenmodes of thereduced model (2.8) satisfying |λn|>R(u) are stable: �λn < 0. An explicit estimate for R(u) interms of the model parameters is given. This result means that there exists a ‘circle of instability’



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of radius R(u) such that unstable (fluttering) eigenmodes may occur only inside this circle, andthe number of unstable modes is always finite.

To the best of our knowledge, no similar estimates obtained analytically for a physicallyrealistic wing model are available in the literature on aeroelasticity.

The paper is organized as follows. In §3, we present a precise description of the wing modelas a system of two coupled integrodifferential equations with a set of boundary conditions. In §4,we give a reformulation of the original initial-boundary problem to the form of a single-operatorevolution–convolution equation (2.1). In particular, we introduce the ground-vibrations energyfunctional E0[Ψ ] naturally associated with equation (2.5). This functional defines the norm in thestate space H : ‖Ψ ‖2

H = E0[Ψ ]. In §5, we also recall the statements of the main spectral resultsobtained in the author’s papers [11–19]. Paragraphs 5 and 6 contain the main results of the paper.In §5, we derive a new non-local energy functional E [Ψ ; t] naturally associated with the equationof the reduced model (2.8). The two energy functionals satisfy the inequality E0[Ψ (t)] ≤ E [Ψ ; t] forany function Ψ =Ψ (t) ∈ H . In §6, we prove the main result of the paper, theorem 6.1. The proof isbased on a special type estimate that we derive for the time derivative of the energy E (t) = E [Ψ ; t]of the reduced model (Ψ =Ψ (t) is a solution of the reduced model equation (2.8)). More precisely,we introduce a notion of a strongly stable mode. Namely if (λk,Ψk) is an eigenpair of the operatorKβδ , then it is said that λk is strongly stable if Ek(t)< 0, where Ek(t) is the new energy functionalevaluated on the solution eλktΨk of the reduced model equation (2.8). We show that if a mode λk isstrongly stable, then it is stable in the usual sense: �λk < 0. From the aforementioned estimate forE (t), we derive that all modes such that |λk|>R(u) are strongly stable and, therefore, are stable.

3. Statement of problem: aeroelastic modelTo give a mathematical formulation of the model, let us introduce the dynamic variables

X(x, t) =(

h(x, t)α(x, t)

), 0 ≤ x ≤ L<∞, t ≥ 0, (3.1)

where h(x, t) is the bending at location x and time moment t, and α(x, t) is the torsion angle at (x, t).The model can be described by the following linear system [3,5,11–15,18,19]:

(Ms − Ma)X(x, t) + (Ds − uDa)X(x, t) + (Ks − u2Ka)X(x, t) =[

f1(x, t)f2(x, t)

], (3.2)

where the overdot is used for the derivative with respect to time. We use the subscripts ‘s’ and‘a’ to distinguish structural and aerodynamic parameters. All 2 × 2 matrices in equation (3.2) aregiven by the formulae

Ms =[

m SS I

], Ma = πρ

[−1 aa −(a2 + 1

8 )

], Ds − uDa = −πρu

[0 −11 0

], (3.3)

where m is the density of the structure (mass per unit length), S is the mass moment, I is themoment of inertia, ρ is the density of air, u is the speed of an air stream and a is a structuralparameter such that |a|< 1.

Ks =

⎡⎢⎢⎣E

d4

dx4 0

0 −Gd2

dx2

⎤⎥⎥⎦ , Ka = πρ

[0 00 1

], (3.4)



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where E is the bending stiffness and G is the torsion stiffness. The right-hand side of system (3.2)(the aerodynamic forces and moments) can be represented as the following convolution integrals:

f1(x, t) = −2πρ∫ t

0[uC2(t − σ ) − C3(t − σ )]g(x, σ ) dσ , (3.5)

f2(x, t) = −2πρ∫ t

0

[12

C1(t − σ ) − auC2(t − σ )

+ aC3(t − σ ) + uC4(t − σ ) + 12

C5(t − σ )]

g(x, σ ) dσ (3.6)

and g(x, t) = uα(x, t) + h(x, t) + ( 12 − a)α(x, t). (3.7)

Note the kernels in both integrals (3.5) and (3.6) are independent on the spatial variable x.However, f1 and f2 depend on x, because the function g from (3.7) does. The aerodynamicfunctions Ci, i = 1, . . . , 5, are defined in the following ways [2,5–7]:

C1(λ) ≡∫∞

0e−λtC1(t) dt = u

λ

e−λ/u

K0(λ/u) + K1(λ/u), �λ> 0, C2(t) =

∫ t

0C1(σ ) dσ ,

C3(t) =∫ t

0C1(t − σ )(uσ −

√u2σ 2 + 2uσ ) dσ , C4(t) = C2(t) + C3(t)

and C5(t) =∫ t

0C1(t − σ )((1 + uσ )

√u2σ 2 + 2uσ − (1 + uσ )2) dσ .

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3.8)

In formulae (3.8), C1(λ) is the Laplace transform of the function C1(t); K0 and K1 are modifiedBessel functions of the zero- and first-orders, respectively [22]. Formulae (3.8) can be found inreference [6]. As is known, the self-straining control actuator action can be modelled by thefollowing boundary conditions [5,19–21,23]:

Eh′′(L, t) + βh′(L, t) = 0, h′′′(L, t) = 0 (3.9)

andGα′(L, t) + δα(L, t) = 0, β, δ ∈ C

+, (3.10)

where the prime denotes the derivative with respect to x. In equation (3.10), C+ denotes the open

right half-plane of the complex plane; β and δ are called the control parameters. At the left end,the structure is fixed, i.e. the following boundary conditions are imposed

h(0, t) = h′(0, t) = α(0, t) = 0. (3.11)

Let the initial state of the system be given by

h(x, 0) = h0(x), h(x, 0) = h1(x), α(x, 0) = α0(x), α(x, 0) = α1(x). (3.12)

We consider the problem defined by equation (3.2) and conditions (3.9)–(3.12) under theassumptions

det

[m SS I

]> 0, 0< u<

√πG

2L√ρ

. (3.13)

Remark 3.1. The second condition in (3.13) physically means that the flow speed must be belowthe ‘divergence’ or static instability speed for the system. Indeed, if we consider the static problemcorresponding to system (3.2) and conditions (3.9)–(3.11), then we can check that the componentα satisfies the following Sturm–Liouville problem:

Gα′′(x) + πρu2α(x) = 0, α(0) = α′(L) = 0.

This problem has non-trivial solutions only for a countable set of speeds, {un}∞n=1, givenexplicitly by the following formula: πρu2

n = (((2n − 1)/2L)π )2G, n = 1, 2, 3, . . .. The smallest valuecorresponding to n = 1 is exactly the aeroelastic static divergence speed.



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To introduce the energy of the system, we have to complete some preliminary steps. Let C1(t)and C2(t) be the kernels in the convolution integrals in (3.5) and (3.6), that is

C1(t) = −2πρ(uC2(t) − C3(t)) (3.14)

and

C2(t) = −2πρ( 12 C1(t) − auC2(t) + aC3(t) + uC4(t) + 1

2 C5(t)). (3.15)

Denoting M = Ms − Ma, D = Ds − uDa, K = Ks − u2Ka, we rewrite equation (3.2) in the form

MX(x, t) + DX(x, t) + KX(x, t) = (F X)(x, t), t ≥ 0, (3.16)

with the matrix integral operator F being given by the formula

F · =

⎡⎢⎢⎢⎣

∫ t

0C1(t − σ )

ddσ

· dσ∫ t

0C1(t − σ )

[u +

(12

− a)

ddσ

·]

dσ

∫ t

0C2(t − σ )

ddσ

· dσ∫ t

0C2(t − σ )

[u +

(12

− a)

ddσ

·]

dσ

⎤⎥⎥⎥⎦ . (3.17)

M, D and K are usually called the spatial operators and F—the time operator. Note system (3.16)is not singular, because det M> 0. Indeed, if

m = m + πρ, S = S − aπρ, I = I + πρ(a2 + 18 ), (3.18)

then using (3.13) one obtains

�≡ det M = mI − S2 = (mI − S2) + [πρI + πρm(a2 + 18 ) + (πρ)2(a2 + 1

8 ) + 2πρaS]> 0. (3.19)

Rewriting system (3.16) component-wise, we have

mh(x, t) + Sα(x, t) + πρuα(x, t) + Eh′′′′

(x, t) =∫ t

0C1(t − σ )g(x, σ ) dσ

and Sh(x, t) + Iα(x, t) − πρuh(x, t) − (Gα′′ + πρu2α(x, t)) =∫ t

0C2(t − σ )g(x, σ ) dσ ,

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(3.20)

with g being defined in (3.7). By setting C1 = C2 = 0 and completing some standard steps forsystem (3.20), we obtain that the energy of vibrations can be introduced by the following formula(see [11,19]):

E0(t) = 12

∫L

0[E|h′′(x, t)|2 + G|α′(x, t)|2 + m|h(x, t)|2 + I|α(x, t)|2

+ S(α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)) − πρu2|α(x, t)|2] dx. (3.21)

Lemma 3.2 ([11,19]). (i) If the problem parameters satisfy conditions (3.13), then the system energyE0(t) evaluated on smooth functions subjected to the boundary conditions (3.11) is always non-negativeand is equal to zero if and only if h(x, t) = α(x, t) = 0, x ∈ (0, L), t ≥ 0. (ii) If �β ≥ 0 and �δ ≥ 0, then theenergy of the system governed by the differential part of (3.20) dissipates: E0(t) ≤ 0 (the differential partmeans that the right-hand sides of (3.20) have been replaced with zeros).

4. Main spectral results from papers [11–14,17–19]

(a) Operator evolution–convolution form of the modelWe consider the solution of the problem given by system (3.2) and conditions (3.9)–(3.11) inthe energy space H , which is a Hilbert space of Cauchy data. The norm in H is inducedby the expression for the energy (3.21). Namely let H be the set of four-component complex



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vector-valued functions Ψ = (h, h,α, α)T ≡ (ψ0,ψ1,ψ2,ψ3)T, (where the superscript ‘T’ stands fortransposition) obtained as a closure of smooth functions satisfying the boundary conditions

ψ0(0) =ψ ′0(0) =ψ2(0) = 0

in the following energy norm

‖Ψ ‖2H = 1

2

∫L

0[E|ψ ′′

0 (x)|2 + G|ψ ′2(x)|2 + m|ψ1(x)|2 + I|ψ3(x)|2

+ S(ψ1(x)ψ3(x) + ψ1(x)ψ3(x)) − πρu2|ψ2(x)|2] dx, (4.1)

with m, S and I being defined in (3.18). To rewrite the original initial-boundary-value problem as

a single equation in the space H , we apply 2 × 2 matrix M−1 = (1/�)[

m −S−S I

]to equation (3.16)

and have

X(x, t) + (M−1D)X(x, t) + (M−1K)X(x, t) = (M−1F X)(x, t). (4.2)

It can be verified directly that the problem defined by equation (4.2) and conditions (3.9)–(3.12)can be written in the form of the evolution–convolution equation (2.1) presented in §2. In thatequation, Lβδ is a non-selfadjoint matrix differential operator in H , which is defined by thefollowing matrix differential expression

Lβδ = −i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

−EI�

d4

dx4 −πρuS�

− S�

(G

d2

dx2 + πρu2

)−πρuI

�

0 0 0 1

ES�

d4

dx4πρum�

m�

(G

d2

dx2 + πρu2

)πρuS�

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.3)

on the domain

D(Lβδ) = {Ψ ∈ H :ψ0 ∈ H4(0, L), ψ1 ∈ H2(0, L), ψ2 ∈ H2(0, L),

ψ3 ∈ H1(0, L); ψ1(0) =ψ ′1(0) =ψ3(0) = 0; ψ ′′′

0 (L) = 0;

Eψ ′′0 (L) + βψ ′

1(L) = 0, Gψ ′2(L) + δψ3(L) = 0}, (4.4)

where Hi, i = 1, 2, 4 are the standard Sobolev spaces [24].The matrix kernel F(t) of the convolution integral in (2.1) is given by the formula

F(t) = 1�

⎡⎢⎢⎢⎢⎣

1 0 0 0

0 IC1(t) − SC2(t) 0 0

0 0 1 0

0 0 0 SC1(t) + mC2(t)

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

0 0 0 0

0 1 u12

− a

0 0 0 0

0 1 u12

− a

⎤⎥⎥⎥⎥⎥⎦ . (4.5)

(b) The integral part of the modelWe describe the structure of the convolution part of the model.

Lemma 4.1 ([12–15,19]). Let F(λ) be the Laplace transform of the kernel (4.5). The followingdecomposition holds λF(λ) = M + N(λ) (see (2.6).) The constant matrix M is given by the formula

M =

⎡⎢⎢⎢⎢⎣

0 0 0 0

0 A uA ( 12 − a)A

0 0 0 0

0 B uB ( 12 − a)B

⎤⎥⎥⎥⎥⎦ , (4.6)



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where

A = −πρu�

[I −

(12

− a)

S]

, B = πρu�

[S −

(12

− a)

m]

. (4.7)

The matrix-valued function N(λ) is given by the formula

N(λ) =

⎡⎢⎢⎢⎢⎣

0 0 0 0

0 A1(λ) uA1(λ) ( 12 − a)A1(λ)

0 0 0 0

0 B1(λ) uB1(λ) ( 12 − a)B1(λ)

⎤⎥⎥⎥⎥⎦ , (4.8)

where

A1(λ) = −Γ (λ)[

I +(

12

+ a)

S]

, B1(λ) = Γ (λ)[

S +(

12

+ a)

m]

and Γ (λ) = 2πρu�

[T(λ

u

)− 1

2

].

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(4.9)

T(z) is the Theodorsen function defined by the formula

T(z) = K1(z)K0(z) + K1(z)

, (4.10)

with K0 and K1 being the modified Bessel functions of the zero- and first-orders, respectively [22]. It is ananalytical function on the complex plane with the branch-cut along the negative real semi-axis.

Lemma 4.2 ([12–15,19]). M is a bounded linear operator in H . N(λ) is an analytic matrix-valuedfunction on the complex plane with the branch-cut along the negative real semi-axis. For each λ, N(λ) is abounded operator in H with the following estimate for its norm:

‖N(λ)‖ ≤ C(1 + |λ|)−1, (4.11)

where C is an absolute constant, whose precise value is immaterial for us.

As was mentioned in §2, the model governed by the evolution equation (2.8) with thedynamics generator Kβδ = iLβδ + M is called the reduced model and will be our main objectof interest.

(c) Spectral asymptoticsThe following facts are among the results obtained by the author in references [11–15,17–19].

— The convolution part of the problem does not ‘destroy’ the main characteristics of thediscrete spectrum produced by the differential part. Namely the set of the aeroelasticmodes is asymptotically close to the set of the eigenvalues of the operator iLβδ . Recallthat the operator iLβδ is the dynamics generator for the model describing wing vibrationswhen an aircraft is on the ground (not in-flight), and the convolution part is missing.In turn, identification of the ground-vibration frequencies is always the first step in anyaeroelastic analysis, because it is an experimentally known fact that the ‘flutter frequency’is typically very close to one of the lower ground-vibration frequencies.

— The set of the generalized eigenvectors of the operator Lβδ forms a Riesz basis in thestate space H . The set of the mode shapes forms a Riesz basis in its closed linear span inH . This result is crucial for an unconditional convergence of a series with respect to theresidues at the poles of R(λ).



9


...................................................

As has already been mentioned, an aeroelastic mode is a pole of the generalized resolvent (2.4).Equivalently, it is a value of λ, for which the equation

(λI − iLβδ − M − N(λ))Φ = 0 (4.12)

has a non-trivial solution in the energy space H , and the corresponding solution, Φ, is the modeshape.

Theorem 4.3 ([11–14,19]). (i) The set of the aeroelastic modes is countable and does not haveaccumulation points on the complex plane C. There can be only a finite number of multiple poles of afinite multiplicity each. For a multiple mode, in addition to the aeroelastic mode shape or shapes, there maybe the associate mode shapes.

(ii) Let the boundary control parameter δ satisfy the condition

|δ| =√

GI. (4.13)

The set of the aeroelastic modes splits into two asymptotically disjoint sets, which we call the β-branchand the δ-branch. Let {λβn }n∈Z′ be the β-branch aeroelastic modes. The following asymptotic approximationholds as |n| → ∞:

λβn = i(sgn n)

π2

L2

√EI�

(|n| − 1

4

)2+ κn(ω), ω= |δ|−1 + |β|−1. (4.14)

The complex-valued sequence {κn}n∈Z′ is bounded above in the following sense:

supn∈Z′

{|κn(ω)|} = C(ω), C(ω) → 0 as ω→ 0.

Let {λδn}n∈Z′ be the δ-branch aeroelastic modes. The following asymptotic approximation holds as|n| → ∞:

λδn = iπn

L√

I/G− 1

2L√

I/Glnδ +

√GI

δ −√

GI+ O

(1√|n|)

. (4.15)

Here, ‘ln’ means the principal value of logarithm. If both β and δ stay away from zero, i.e. |β| ≥ β0 > 0 and|δ| ≥ δ0 > 0, then the estimate O(|n|−1/2) is uniform with respect to the boundary parameters.

The following statement is concerned with the spectral results for the dynamics generatorKβδ = iLβδ + M (see (2.8)) of the reduced model.

Theorem 4.4 ([13,14]). The operator Kβδ = iLβδ + M has a two-branch discrete spectrum. Let thesebranches be denoted by {λβn }n∈Z′ and {λδn}n∈Z′ and called the β-branch and the δ-branch, respectively.The asymptotic distribution of the eigenvalues of Kβδ coincides with the asymptotic distribution of theeigenvalues of the operator iLβδ which, in turn, coincides with the asymptotic distribution of the aeroelasticmodes given by (4.14) and (4.15). The set of generalized eigenvectors of the operator Kβδ forms a Riesz basisin H .

It can be readily seen from theorems 4.3 and 4.4 that the leading terms in the asymptoticrepresentations for the aeroelastic modes and for the eigenvalues of Kβδ do not depend on the airstream speed u. However, the remainder terms in these asymptotic representations may dependon u. Theorem 4.5 deals with the dependence of the remainder terms upon u. The result has not beenproved in author’s previous works. However, it is crucial for the description of the stability region(see §6).

Theorem 4.5. Let the second condition from (3.13) and (4.13) be satisfied. Then, the asymptoticapproximations (4.14) and (4.15) as well as the asymptotic approximations for the eigenvalues of Kβδ

are uniform with respect to the speed of an air stream, u.

The proof of this statement is a generalization of the proof of the spectral asymptotics fromreferences [11,13]. Because it is quite lengthy and technical, we have presented the proof in theelectronic supplementary material (see theorem 2 of the electronic supplementary material).



10


...................................................

Remark 4.6. As shown in reference [14], the set of the aeroelastic modes is asymptotically closeto the set of the eigenvalues of the matrix differential operator (iLβδ) defined in (4.3) and (4.4).An important property of Lβδ is that it is a dissipative operator in H . Let {λβn }n∈Z′

⋃{λδn}n∈Z′ be thenotation for the two-branch spectrum of iLβδ . An immediate consequence of the dissipativityis that the spectrum of the operator (iLβδ) is located in the closed left half-plane of thecomplex plane. We recall (see [11,18]) that the set of the normalized generalized eigenvectors{ϕβn }n∈Z′

⋃{ϕδn}n∈Z′ of the operator Lβδ forms a Riesz basis in H . The biorthogonal Riesz basis{ψβn }n∈Z′

⋃{ψδn}n∈Z′ is constructed from the generalized eigenvectors of the adjoint operator L ∗βδ ,

and the following relations hold

(ϕβn ,ψβm)H = (ϕδn,ψδm)H = δnm, (ϕβn ,ψδm)H = (ϕδn,ψβm)H = 0.

This means that any solution of the evolution problem (2.5) can be given in the form of anexpansion with respect to the generalized eigenvectors of Lβδ

Ψ (x, t) =∑n∈Z′

(Ψ0,ψβn )H eλβn tϕ

βn (x) +

∑n∈Z′

(Ψ0,ψδn)H eλδntϕδn(x). (4.16)

Using (4.16) and the fact that �λβn ≤ 0 and �λδn ≤ 0, one obtains the following estimate for thesolution:

‖Ψ (x, t)‖2H ≤ C e−2γ t‖Ψ0‖2

H , where γ = inf{|�λβn |, |�λδn|, n ∈ Z′}. (4.17)

with some absolute constant C. Estimate (4.17) shows that the dissipativity of the operator iLβδ

implies that the ground-vibration dynamics (2.5), whose generator is iLβδ , is exponentially stable.Even though the set of the aeroelastic modes is asymptotically close to the eigenvalues of theoperator iLβδ , the fact that the set of the aeroelastic modes is confined to the closed left half-plane of the complex plane is not valid, in general. So for the expansion with respect to the modeshapes the estimate similar to (4.17) is not valid any more owing to the existence of unstableaeroelastic modes.

5. Non-local energy functional for the reduced modelFor the rest of the paper, we study the evolution equation (2.8), for which the operator Kβδ is thedynamics generator, i.e.

Ψ (x, t) = (KβδΨ )(x, t), Ψ (x, 0) =Ψ0(x), (5.1)

whereΨ (x, t) = (h(x, t), h(x, t),α(x, t), α(x, t)) and the dynamical variables h,α defined in (3.1) satisfythe boundary conditions (3.9)–(3.11), i.e. the operator Kβδ has the same domain as Lβδ (see (4.4)).

The goal of this section is to introduce the energy functional naturally associated with equation(5.1) (see formula (5.11)) and to prove that this functional is positively definite (lemma 5.2). Theenergy functional we present here plays a crucial role in the proof of our main result in §6.The functional is non-local in the time variable and can be called a memory-type energy. Thesystem governed by equation (5.1) is not conservative and, accordingly, the energy we introduceis not conserved on the solutions of equation (5.1). The formula (5.11), which defines our energyfunctional, is quite complicated and is not intuitively obvious. Thus, instead of just giving anunmotivated definition of the energy, we present below a heuristic derivation of the formula andgive a physical interpretation of each term in it.

Let us apply the matrix

M =

⎡⎢⎢⎢⎣

1 0 0 00 m 0 S0 0 1 00 S 0 I

⎤⎥⎥⎥⎦



11


...................................................

to both sides of equation (5.1). Taking into account formulae (4.3) and (4.6) by direct calculations,we obtain the following equation

⎡⎢⎢⎢⎢⎣

h(x, t)

mh(x, t) + Sα(x, t)

α(x, t)

Sα(x, t) + Iα(x, t)

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

−Ed4

dx4 0 0 −πρu

0 0 0 1

0 πρu Gd2

dx2 + πρu2 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

h(x, t)

h(x, t)

α(x, t)

α(x, t)

⎤⎥⎥⎥⎥⎦

− πρu

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0

0 1 u(

12

− a)

0 0 0 0

0(

12

− a)

u(

12

− a) (

12

− a)2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

h(x, t)

h(x, t)

α(x, t)

α(x, t)

⎤⎥⎥⎥⎥⎦ . (5.2)

This system written component-wise yields

mh(x, t) + Sα(x, t) + Eh′′′′(x, t) + πρuh(x, t)

+ πρu( 32 − a)α(x, t) + πρu2α(x, t) = 0

and Sh(x, t) + Iα(x, t) − Gα′′(x, t) − πρu( 12 + a)h(x, t)

+ πρu( 12 − a)2α(x, t) − πρu2( 1

2 + a)α(x, t) = 0.

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(5.3)

Note system (5.3) contains all terms from the left-hand side of system (3.20) combined with theentries of the matrix M from (4.6). We deal with the solutions of system (5.3) satisfying boundary

conditions (3.9)–(3.11). Let us take the first equation of system (5.3) and multiply it by ˙h, then takethe equation complex conjugated to the first equation from (5.3) and multiply it by h. Summingup these two equations we obtain

mddt

|h(x, t)|2 + S[α ˙h + ¨αh](x, t) + E[h′′′′ ˙h + h′′′′h](x, t)

+ 2πρu|h(x, t)|2 + πρu( 32 − a)[α ˙h + αh](x, t) + πρu2[α ˙h + αh](x, t) = 0. (5.4)

Now, we take the second equation of system (5.3) and multiply it by ˙α, then we take the equation,which is complex conjugate to the second equation of (5.3), and multiply it by ˙α. Summing upthese two equations, we obtain

S[h ˙α + ¨hα](x, t) + Iddt

|α(x, t)|2 − G[α′′ ˙α + α′′α](x, t) − πρu(

12

+ a)

× [h ˙α + ˙hα](x, t) + 2πρu(

12

− a)2

|α(x, t)|2 − πρu2(

12

+ a)

ddt

|α(x, t)|2 = 0. (5.5)

Combining equations (5.4) and (5.5), we obtain the following relation

ddt

{m|h(x, t)|2 + I|α(x, t)|2 − πρu2

(12

+ a)

|α(x, t)|2 + S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)]}

+ E[h′′′′(x, t) ˙h(x, t) + h′′′′(x, t)h(x, t)] − G[α′′(x, t) ˙α(x, t) + α′′(x, t)α(x, t)]

+ 2πρuJ(x, t) = 0, (5.6)

where

J(x, t) = b[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)] + |h(x, t)|2 + b2|α(x, t)|2

+ u2

[α(x, t) ˙h(x, t) + α(x, t)h(x, t)], b = 12

− a. (5.7)



12


...................................................

It can be easily seen that J can be written in the form

J(x, t) =∣∣∣h(x, t) + bα(x, t) + u

2α(x, t)

∣∣∣2 +[

b2|α(x, t)|2 −∣∣∣bα(x, t) + u

2α(x, t)

∣∣∣2] . (5.8)

Indeed,

J(x, t) =∣∣∣h + bα + u

2α

∣∣∣2 (x, t) + b2|α(x, t)|2 −∣∣∣bα + u

2α

∣∣∣2 (x, t)

=[ ˙h (bα + u

2α)

+ h(

b ˙α + u2α)]

(x, t) + |h(x, t)|2 + b2|α(x, t)|2

= |h(x, t)|2 + b[ ˙hα + h ˙α](x, t) + b2|α(x, t)|2 + u2

[ ˙hα + hα](x, t).

Owing to representation (5.8), we can rewrite equation (5.6) in the form

ddt

{m|h(x, t)|2 + I|α(x, t)|2 − πρu2

(12

+ a)

|α(x, t)|2 + S[α ˙h + ˙αh](x, t)}

+ E[h′′′′ ˙h + h′′′′h](x, t) − G[α′′ ˙α + α′′α](x, t) + 2πρu∣∣∣(h + bα + u

2α)

(x, t)∣∣∣2

= −2πρu[

b2|α|2 −∣∣∣bα + u

2α

∣∣∣2] (x, t). (5.9)

Now, we use a heuristic consideration (quite traditional) to derive a non-local energy functionalassociated with this equation. Let us discard the right-hand side of equation (5.9) and assumethat we consider a cantilever beam model, i.e. the right end is free: h′′(L, t) = h′′′(L, t) = α′(L, t) = 0.Integrating both sides of equation (5.9) and using the clamped-free boundary conditions, weobtain equation (5.10)

ddt

{12

∫L

0

{E|h′′(x, t)|2 + G|α′(x, t)|2 + m|h(x, t)|2 + I|α(x, t)|2

+ S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)] − πρu2(

12

+ a)

|α(x, t)|2

+ 2πρu∫ t

0

∣∣∣h(x, τ ) + bα(x, τ ) + u2α(x, τ )

∣∣∣2 dτ}

dx}

= 0. (5.10)

This equation means that the following functional:

E (t) = 12

∫L

0

{E|h′′(x, t)|2 + G|α′(x, t)|2 + m|h(x, t)|2 + I|α(x, t)|2

+ S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)] − πρu2(

12

+ a)

|α(x, t)|2

+ 2πρu∫ t

0

∣∣∣h(x, τ ) + bα(x, τ ) + u2α(x, τ )

∣∣∣2 dτ}

dx (5.11)

is not changing in time, being evaluated on the functions h,α satisfying equation (5.9) with theright-hand side replaced with zero and subjected to the clamped-free boundary conditions.

Definition 5.1. The functional defined in (5.11) is called the non-local energy functional associatedwith the problem given by operator equation (5.1) or, equivalently, by system (5.2) on thefunctions satisfying the boundary conditions (3.9)–(3.11).

The following interpretation of the different terms of equation (5.11) can be given. Namely{ 1

2∫L

0 E|h′′(x, t)|2 dx} represents the potential energy of a structure owing to the bending

displacement; { 12

∫L0 G|α′(x, t)|2} dx—the potential energy owing to the torsional displacement;

{ 12

∫L0 m|h(x, t)|2 dx}—the kinetic energy owing to the bending motion; { 1

2∫L

0 I|α(x, t)|2 dx}—the

kinetic energy owing to the torsion motion; { 12

∫L0 S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)] dx}—the kinetic



13


...................................................

energy owing to coupling between bending and torsion; {πρu∫L

0 dx∫t

0 |h(x, τ )( 12 − a)α(x, τ ) +

(u/2)α(x, τ )|2 dτ }—the energy of vibrations owing to flow–structure interaction.

Lemma 5.2. Let conditions (3.13) be satisfied. Then, for each t ≥ 0, the functional E (t), defined by (5.11)on smooth functions h and α satisfying conditions (3.11), is positively definite, i.e. E (t) ≥ 0 and E (t) = 0 ifand only if h(x, t) = α(x, t) = 0, x ∈ (0, L), t ≥ 0.

Proof. To prove that E (t)> 0 for non-trivial (h, α), it suffices to show that for (h,α) = (0, 0)

(i) E|h′′(x, t)|2 + m|h(x, t)|2 + I|α(x, t)|2 + S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)] ≥ 0

and (ii)∫L

0[G|α′(x, t)|2 − πρu2

(12

+ a)

|α(x, t)|2] dx> 0.

⎫⎪⎪⎬⎪⎪⎭ (5.12)

It can be readily estimated that

E|h′′(x, t)|2 + m|h(x, t)|2 + I|α(x, t)|2 + S[α(x, t) ˙h(x, t) + ˙α(x, t)h(x, t)]

≥ E|h′′(x, t)|2 + (√

m|h(x, t)| −√

I|α(x, t)|)2 + 2(√

mI|α| − S)|α(x, t)||h(x, t)|. (5.13)

Using formulae (3.18) for m, I and S, we have that �≡ mI − S2 >mI − S2 > 0, and therefore

(√

mI − S)|α(x, t)| |h(x, t)| = �√mI + S

|α(x, t)| |h(x, t)| ≥ 0. (5.14)

Substitution (5.14) into (5.13) and yields estimate (5.12) (i). Now we prove estimate (5.12) (ii).Owing to the boundary condition α(0, t) = 0, we obtain that ‖α‖2

L2(0,L) ≤ L2/2‖α′‖2L2(0,L) by one-

dimensional Poincare inequality. Now, we estimate the integral from (5.12) (ii) as follows∫L

0

[G|α′(x, t)|2 − πρu2

(12

+ a)

|α(x, t)|2]

dx ≥ 2L2 G‖α‖2

L2(0,L)

− πρu2(

12

+ a)

‖α‖2L2(0,L) =

(2L2 G − πρu2

(12

+ a))

‖α‖2L2(0,L). (5.15)

For −1 ≤ a ≤ 1, we have (2/L2)G − πρu2( 12 + a) ≥ (2/L2)G − 3

2πρu2. Because u satisfies the secondestimate from (3.13), the positivity of the difference (2/L2)G − πρu2( 1

2 + a) is obvious.Finally, we prove that E (t) = 0 implies that h(x, t) = α(x, t) = 0, x ∈ (0, L), t ≥ 0. Using (5.13), we

obtain that the equation E (t) = 0 yields

(i) h′′(x, t) = 0, (ii)√

m|h(x, t)| −√

I|α(x, t)| = 0 and (iii) h(x, t)α(x, t) = 0. (5.16)

Combining (ii) and (iii) from (5.16), we obtain that h(x, t) = 0 and α(x, t) = 0, which means thath(x, t) = h(x), α(x, t) = α(x). From (i) of (5.16), we obtain that h(x) = ax + b and to satisfy theboundary conditions at x = 0, h(0) = h′(0) = 0, we obtain h(x) = 0.

Evaluating E (t) from (5.11) on h = 0 and α(x, t) = α(x), we obtain that

E (t) = 12

∫L

0

{G|α′(x)|2 − πρu2

(12

+ a)

|α(x)|2 + 2πρut(u

2

)2|α(x)|2

}dx = 0. (5.17)

It is clear that E (t) from (5.17) is a linear function of t and it can be equal to zero only if‖α‖2

L2(0,L) = 0, which yields α = 0.The lemma is completely shown. �

6. Proof of main result: circle of instabilityHere, we prove the main result of the paper. Our main object of interest is the dynamics generatorKβδ of the evolution problem (5.1). Let us denote the spectrum of Kβδ by {λn}n∈Z′ . Accordingto theorems 4.3–4.5, this spectrum splits asymptotically into two branches, the β-branch and theδ-branch. For this reason, it will be convenient at some point below to use an alternative notationand denote the spectrum by {λβn }n∈Z′

⋃{λδn}n∈Z′ . Now, we formulate the main result.



14


...................................................

Theorem 6.1. For each value u of the air stream speed satisfying the second condition of (3.13), thereexists R(u)> 0 such that the following statement holds. If an eigenmode λn satisfies |λn|>R(u), then thiseigenmode is stable, i.e. �λn < 0. Formula (6.1) provides an estimate on R(u) :

R(u) = C√

ρ

G�δ u3/2, (6.1)

where C> 0 is an absolute constant.

Corollary 6.2. For each u, all the unstable eigenmodes are located inside the ‘circle of instability’ |λ| =R(u). The number of these eigenmodes is always finite.

To prove theorem 6.1, we have to complete some preliminary steps.For each eigenvalue λn, equation (5.1) has a solution given explicitly in the form

Ψn(x, t) = eλnt(hn(x), λnhn(x),αn(x), λnαn(x))T. (6.2)

Obviously, (hn(x), λnhn(x),αn(x), λnαn(x))T is an eigenvector of Kβδ . It can be easily seen that thepair of functions (hn(x),αn(x))T satisfies the system of two coupled ordinary differential equations

mλ2h(x) + Sλ2α(x) + Eh′′′′

(x) + πρuλh(x)

+ πρu(

32

− a)λα(x) + πρu2α(x) = 0

and Sλ2h(x) + Iλ2α(x) − Gh′′(x) − πρu(

12

+ a)λh(x)

+ πρub2λα(x) − πρu2(

12

+ a)α(x) = 0,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(6.3)

and the boundary conditions

h(0) = h′(0) = α(0) = 0, h′′′(L) = 0

and Eh′′(L) + βλh′(L) = 0, Gα′(L) + δλα(L) = 0,

}(6.4)

Let Φn(x) = (hn(x),αn(x))T. Formulae (6.3) and (6.4) mean that λn and Φn are an eigenvalue andthe corresponding eigenvector of the following non-selfadjoint quadratic operator pencil [25]:

λ2

[m SS I

]Φ(x) + λπρu

⎡⎢⎢⎢⎣

1(

32

− a)

−(

12

+ a) (

12

− a)2

⎤⎥⎥⎥⎦Φ(x)

+

⎡⎢⎢⎣

Ed4

dx4 πρu2

0 −Gd2

dx2 + πρu2(

12

+ a)⎤⎥⎥⎦Φ(x) = 0,

⎡⎢⎢⎣E

d2

dx2 0

0 Gddx

⎤⎥⎥⎦Φ

∣∣∣∣∣∣∣∣x=L

+ λ

⎡⎣β d

dx0

0 δ

⎤⎦Φ

∣∣∣∣∣∣x=L

= 0

and

⎡⎢⎣ d3

dx3 0

0 0

⎤⎥⎦Φ

∣∣∣∣∣∣∣x=L

= 0,

⎡⎣ d

dx0

0 0

⎤⎦Φ

∣∣∣∣∣∣x=0

= 0,

[1 00 1

]Φ

∣∣∣∣∣x=0

= 0.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(6.5)

Polynomial pencil (6.5) has been studied in reference [18]. Its spectrum coincides withthe spectrum of the operator Kβδ . At this point, we switch to an alternative notation{λβn }n∈Z′

⋃{λδn}n∈Z′ for the spectrum, as was explained above, and denote by {Φβn }n∈Z′⋃{Φδn}n∈Z′

the corresponding eigenvectors of the pencil. In reference [18], we have obtained the asymptotic



15


...................................................

approximation for the components of the vectorsΦβn andΦδn as |n| → ∞. In what follows, we needasymptotic approximation as |n| → ∞ only for the α− component of Φβn and Φδn, i.e. for αβn andαδn. We reproduce the necessary results from reference [18].

Theorem 6.3 ([18]). (i) For n> 0 and x ∈ [0, L], the asymptotic representation for the α− componentof the δ− branch eigenfunction can be given by formula (6.6):

αδn(x) = sin

{(πn + i

2lnδ +

√GI

δ −√

GI

)xL

}+ O

(1√n

), n → ∞. (6.6)

For n< 0 and x ∈ (0, L), the following relation holds: αδn(x) = αδ|n|(x).(ii) For n> 0 and x ∈ [0, L], the asymptotic representation for the α− component of the β− branch

eigenfunction can be given by formula (6.7):

αβn (x) = sin

(dm2x

L

)+ Vβ [−eiπmx/L + i e−iπmx/L + (1 − i) e−πmx/L] + O

(1n

), n → ∞, (6.7)

where m = n − 14 , d = π2L−1(mI − S2)−1 IE1/2G−1/2, and

Vβ = − 14 [K+ + K−e−2idm2

] eim(π+dm), K± = δ−1√

GI ± 1. (6.8)

For n< 0 and x ∈ [0, L], the following relation holds: αβn (x) = αβ|n|(x). All estimates O(·) in formulae (6.6)

and (6.7) are uniform with respect to x ∈ [0, L].

Having the necessary information on the spectrum and the eigenfunctions of the operator Kβδ ,we are in a position to introduce definition (6.4).

Definition 6.4. An eigenvalue λn of the operator Kβδ is called a strongly stable mode if theenergy (5.11) evaluated on the corresponding solution (6.2) of the evolution problem (5.1) isa strictly decreasing function of t : En(t)< 0. Here, En(t) is the notation for the energy (5.11)evaluated on Ψn(x, t) from (6.2).

Proposition 6.5. If λn = rn + iωn is a strongly stable mode, then it is stable in the usual sense, i.e.rn = �λn < 0. The inverse statement is not true in general.

Proof. Substitute (6.2) into (5.11) to obtain

En(t) = 12

e2rnt∫L

0

{E|h′′

n(x)|2 + G|α′n(x)|2 − πρu2

(12

+ a)

|αn(x)|2

+|λn|2[m|h′′n(x)|2 + I|α′′

n(x)|2 + S(αn(x)hn(x) + αn(x)hn(x))]}

dx

+ 2πρu∫L

0

∣∣∣∣λnhn(x) + bλnαn(x) + u2αn(x)

∣∣∣∣2 dx∫ t

0e2rnτ dτ . (6.9)

From (6.9), it follows that for any n ∈ Z′, we have

En(t) = An e2rnt + Bne2rnt − 1

2rn, An > 0, Bn > 0, if rn = 0

and En(t) = An + Bnt, if rn = 0.

⎫⎪⎬⎪⎭ (6.10)

Obviously, both functions of (6.10) increase for rn ≥ 0. Therefore, because En(t) decreases, we havern < 0.

The proposition is shown. �

We need one technical result given below. In the sequel, if two positive sequences {an}n∈Z′ and{bn}n∈Z′ are connected by relations C1an ≤ bn ≤ C2an with two absolute constants C1 and C2, thenwe write an � bn.



16


...................................................

Lemma 6.6.

(i) For the set of the α-components of the δ-branch eigenfunctions the following relation holds

‖αδn‖L2(0,L) � |αδn(L)|, n ∈ Z′. (6.11)

(ii) For the set of the α-components of the β-branch eigenfunctions the following relations hold

|αβn (L)| � 1, ‖αβn ‖L2(0,L) ≤ C|αβn (L)|, n ∈ Z′, (6.12)

with C being some absolute constant.

Proof. Let n> 0. Based on (5.6), we obtain the following estimates for |αδn(L) :

|αδn(L)| =∣∣∣∣∣sin

(πn + i

2lnδ +

√GI

δ −√

GI

)∣∣∣∣∣+ O(

1√n

)=∣∣∣∣∣sinh

(12

lnδ +

√GI

δ −√

GI

)∣∣∣∣∣+ O(

1√n

). (6.13)

If δ = δR + iδI with δR > 0, then ln((δ +√

GI)/(δ −√

GI)) = A + iB, where

A = 12

ln

∣∣∣∣∣ δR +√

GI + iδI

δR −√

GI + iδI

∣∣∣∣∣2

= 12

ln|δ|2 + 2δR

√GI + GI

|δ|2 − 2δR

√GI + GI

, (6.14)

and

B = Argδ +

√GI

δ −√

GI= Arg

(δR +√

GI + iδI)(δR −√

GI − iδI)

|δR −√

GI + iδI|2= − tan−1 2δI

√GI

|δ|2 − GI. (6.15)

It can be easily seen from (6.14) and (6.15) that when |δ| = GI, both A and B are well defined and|A|> 0. It means that | sinh(A/2 + i(B/2))|> 0. Thus, (5.7) implies that there exist two absoluteconstants, D1 and D2, such that

0<D1 ≤ |αδn(L)| ≤ D2 <∞. (6.16)

It remains to be shown that the sequence {αδn(x)}n∈Z′ is almost normalized in L2(0, L), i.e. thereexist two absolute constants, D3 and D4, such that

0<D3 ≤ ‖αδn‖L2(0,L) ≤ D4 <∞. (6.17)

Combining (6.16) and (6.17) yields (6.11). To prove (6.17), we use (5.6) and estimate ‖αδn‖L2(0,L) asfollows

‖αδn‖L2(0,L) =∫L

0

∣∣∣∣∣sin

{(πn + i

2lnδ +

√GI

δ −√

GI

)τ

L

}∣∣∣∣∣2

dτ + O(

1√n

)

=∫L

0

∣∣∣∣sin{(πn − B

2+ i

A2

)τ

L

}∣∣∣∣2 dτ + O(

1√n

). (6.18)

Taking into account that | sin(x + iy)|2 = 12 [cosh 2y − cos 2x], we evaluate the integral (6.18) to have

‖αδn‖2L2(0,L) = −L

2

∫ 1

0cos(2πn − B)τ dτ + L

2

∫ 1

0cosh(Aτ ) dτ + O

(1√n

)

= L sinh A2A

+ O(

1√n

)= 2L sinh((1/2) ln((|δ|2 + 2δR

√GI + GI)/(|δ|2 − 2δR

√GI + GI)))

ln((|δ|2 + 2δR

√GI + GI)/(|δ|2 − 2δR

√GI + GI))

+ O(

1√n

), (6.19)

which obviously implies (6.17).



17


...................................................

Note expression (6.19) makes sense as δR → ∞. Indeed,

limδR→∞

2L sinh((1/2) ln((|δ|2 + 2δR

√GI + GI)/(|δ|2 − 2δR

√GI + GI))

ln((|δ|2 + 2δR

√GI + GI)/(|δ|2 − 2δR

√GI + GI))

= limδR→∞

δR

√GI

|δ|2(4δR

√GI/(|δ|2 − 2δR

√GI + GI))

= limδR→∞

|δ|2 − 2δR

√GI + GI

4|δ|2 = 14

.

Statement (i) of the lemma is shown.Now, we turn to the β-branch and show that |αβn (L)| � 1, i.e. this sequence is bounded above

and below uniformly with respect to n. The fact that this sequence is bounded above followsimmediately from formulae (6.7) and (6.8). Let us show the estimate from below: |αβn (L)| ≥ C. Wehave for n> 0

αβn (L) = sin(dm2) + Vβ [−eiπm + i e−iπm] + O(n−1)

= sin(dm2) − ( 14 )[K+ + K−e−2idm2

] eid m2[−e2iπm + i] + O(n−1)

= sin(dm2) − ( 14 )[K+eidm + +K−e−idm2

]2i + O(n−1)

= 2 sin(dm2) + iδ−1√

GI cos(dm2) + O(n−1). (6.20)

Equation (6.20) obviously means that |αβn (L)|2 � 4 sin2(dm2) + δ−2GI cos2(dm2). Becausesin2(dm2) + δ−2GI cos2(dm2) � 1, we obtain the desired estimate from below: |αβn (L)| � 1, whichyields the estimates of statement (ii).

The lemma is shown. �

Now, we are in a position to prove the main result of the paper.

Proof of theorem 6.1. We start with the formula for E (t) that follows from (5.11)

E (t) = 12

∫L

0

{E[h′′h′′ + h′′ ˙h′′](x, t) + G[α′α′ + ˙α′α′](x, t) + m[h ˙h + h ¨h](x, t)

+ I[α ˙α + α ¨α](x, t) + S[α ˙h + α¨h + ¨αh + ˙αh](x, t)

+ 2πρu∣∣∣h(x, t) + bα(x, t) + u

2α(x, t)

∣∣∣2 − πρu2(

12

+ a)

ddt

|α(x, t)|2}

dx. (6.21)

Integrating by parts twice in the first two integrals of (6.21) and using the boundary conditions(3.9)–(3.11), we obtain

∫L

0E[h′′h′′ + h′′ ˙h′′](x, t) dx = E[h′′(L, t)h′(L, t) + h′′(L, t) ˙h′(L, t)] + E

∫L

0[h′′′′h + h′′′′ ˙h](x, t) dx

= −2E�β|h′(L, t)|2 + E∫L

0[h′′′′h + h′′′′ ˙h](x, t) dx. (6.22)

Similarly, we obtain∫L

0G[α′α′ + ˙α′α′](x, t) dx = −2G�δ|α(L, t)|2 − G

∫L

0[α′′α + ˙αα′′](x, t) dx. (6.23)

With (6.22) and (6.23), formula (6.21) can be modified to

E (t) = 12

∫L

0

{E[h′′′′ ˙h + h′′′′h](x, t) − G[α′′ ˙α + α′′α](x, t) + m

ddt

|h(x, t)|2 + Iddt

|α|2(x, t)

+ Sddt

[α ˙h + ˙αh](x, t) − πρu2(

12

+ a)

ddt

|α|2(x, t) + 2πρu∣∣∣h + bα + u

2α

∣∣∣2 (x, t)}

dx

− {E�β|h′(L, t)|2 + G�δ|α(L, t)|2}. (6.24)



18


...................................................

Because h and α satisfy equation (5.6), the representation (6.24) for E (t) can be simplified asfollows

− E (t) = 2πρu∫L

0

[b2|α(x, t)|2 −

∣∣∣bα(x, t) + u2α(x, t)

∣∣∣2]dx + E�β|h′(L, t)|2 + G�δ|α′(L, t)|2. (6.25)

Now, we evaluate E (t) on h(x, t) = eλnthn(x) and α(x, t) = eλntαn(x). Recall that according todefinition 6.4 the result is denoted by En(t). Let Qn(t) be

Qn(t) = |λn|2 e2rnt[E�β|h′n(L)|2 + G�δ|αn(L)|2], where rn = �λn. (6.26)

Rewriting (6.25) for the above h and α and using (6.26), we obtain

En(t) = −2πρu e2rnt|λn|2∫L

0

[b2|αn(x, t)|2 −

∣∣∣∣bαn(x) + u2λn

αn(x)∣∣∣∣2]

dx − Qn(t)

= 2πρu e2rnt|λn|2(

u2

4|λn|2 + burn

|λn|2

) ∫L

0|αn(x)|2 dx − Qn(t)

= e2rnt{πρu2

(u2

+ 2brn

)‖αn‖2

L2(0,L) − |λn|2[E�β|h′n(L)|2 + G�δ|αn(L)|2]

}. (6.27)

Because �β > 0, equation (6.27) yields

En(t) ≤ e2rnt{πρu2

(u2

+ 2brn

)‖αn‖2

L2(0,L) − G�δ|λn|2|αn(L)|2}

. (6.28)

Now, we use lemma 6.6. Note, to prove this lemma, we have to consider the β-branch and theδ-branch of the spectrum separately. Now, when the result is proven, we can return to the originalnotation for the spectrum: {λn}n∈Z′ . Owing to (6.11) and (6.12), estimate (6.28) can be modified to

En(t)|λn|2 ≤ e2rnt

{C1πρu2

(u

2|λn|2 + |2brk||λn|2

)− G�δ

}|αn(L)|2, (6.29)

with C1 being some absolute constant. Obviously, for a given u, one can find a positive integer Nsuch that for all n with |n| ≥ N, the expression in the curly brackets in (6.29) is negative, because�δ > 0. It means that for such n, the energy decreases with time.

Using the spectral asymptotics (4.14) and (4.15), and the second inequality from (3.13), one cansee that there exists N independent on u such that for |n|>N, the following estimate holds

C1πρu2 2brn

|λn|2 − G�δ ≤ −G�δ2

, (6.30)

with C1 being independent on u (lemma 6.6). It is clear that for such n, we have

C1πρu2(

u2|λn|2 + 2brk

|λn|2)

− G�δ ≤ C1πρu3

2|λn|2 − G�δ2

. (6.31)

Therefore, if |λn|>R(u), where R(u) is defined in (6.1) with C = √C1π , then En(t)< 0. It means

that the corresponding mode is strongly stable.The proof is complete. �

Funding statement. Partial support by the National Science Foundation grant no. DMS-1211156 is highlyappreciated by the author.

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http://dx.doi.org/doi:10.1093/imamat/66.4.319

http://dx.doi.org/doi:10.1098/rspa.2005.1579

http://dx.doi.org/doi:10.1002/mma.456

http://dx.doi.org/doi:10.1098/rspa.2003.1217

http://dx.doi.org/doi:10.1155/2010/879519

http://dx.doi.org/doi:10.1006/jmaa.2000.7453

http://dx.doi.org/doi:10.1016/S0016-0032(00)00076-4

http://dx.doi.org/doi:10.1016/0022-460X(89)90637-8

http://dx.doi.org/doi:10.1006/jsvi.1994.1377

http://dx.doi.org/doi:10.1121/1.401260

http://dx.doi.org/doi:10.1121/1.401260


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