© Hindawi Publishing Corp. RIESZ BASIS PROPERTY OF TIMOSHENKO BEAMS WITH BOUNDARY FEEDBACK CONTROL

A Timoshenko beam equation with boundary feedback control is considered. By an abstract result on the Riesz basis generation for the discrete operators in the Hilbert spaces, we show that the closed-loop system is a Riesz system, that is, the sequence of generalized eigenvectors of the closed-loop system forms a Riesz basis in the state Hilbert space.


Introduction.
The boundary feedback stabilization problem of a hybrid system has been studied extensively in the last decade.Many important results have been obtained.Among them, most of studies in the literatures are concerned with Euler-Bernoulli and Rayleigh beams; there are a few results for Timoshenko beams (cf.[3,5,6,7,9]), which are mainly focused on the stability of the closed-loop system.Though it is important to obtain the exponential stability of the system, it is also very interesting to study the rate of the exponential decay of the system.It is well known that if the system satisfies the spectrum-determined growth assumption, then the rate of the exponential decay can be easily estimated via the spectra of the system operator, see [2].Furthermore, if the system is a Riesz one, that is, the set of the generalized eigenvectors of the system operator forms a Riesz basis of the state Hilbert space, then the spectrum-determined growth assumption is satisfied.In [1], the Riesz basis property was used to give some quantitative information of the rate of the exponential decay for a simpler Euler-Bernoulli beam system with no tip mass.For Euler-Bernoulli and Rayleigh beam systems, some further results concerning the Riesz basis property of the systems can be found in [4,8].

2.
The state space and eigenvalue problem.First, we recall the state space and the operator defined in [9].Let where where the superscript T denotes the transposition of a matrix, the inner product in Ᏼ is defined by with the domain (2.4) Then the closed-loop system (1.1) with (1.2) becomes the following evolution equation in Ᏼ: where (2.6) The following lemma can be found in [9].
Lemma 2.1.The operator Ꮽ is dissipative and generates a C 0 -semigroup with exponential decay.
In order to investigate the rate of the exponential decay of the closed-loop system (1.1) with (1.2), we study the Riesz basis property of the generalized eigenvector system of Ꮽ.For this purpose, we need the following lemma.
Proof.It follows from Lemma 2.1 that 0 ∈ ρ(Ꮽ).Then for any from which we obtain where x, x ≤ s ≤ . (2.9) Then it is easy to see the compactness of Ꮽ −1 .Now, we consider the eigenvalue problem of Ꮽ.Let λ ∈ C be an eigenvalue of Ꮽ and Y = [w,z,ϕ,ψ,ξ,η] T ∈ Ᏼ an eigenvector corresponding to λ, then the functions w(x) and ϕ(x) satisfy Denote by µ 1 and µ 2 the two roots of the quadratic equation that is, (2.13) In the case of µ 1 = µ 2 , we define functions w j (λ, x) and ϕ j (λ, x) for j = 3, 4 by where A and B are two constants to be determined.Then w(λ, x), ϕ(λ, x) satisfy (2.16) From (2.10), we obtain (2.17) For an eigenpair (λ, Y ) of Ꮽ, the determinant Γ (λ) of the coefficient matrix of the above linear equation system must be vanishing.Here where (2.19) Therefore, we have the following result.

Riesz basis property of generalized eigenvector system of Ꮽ.
In this section, we study the Riesz basis property of generalized eigenvector system of Ꮽ.We recall that the basis {ϕ n | n ≥ 1} of a Hilbert space Ᏼ is said to be a Riesz basis if it is equivalent to some orthonormal basis {e n | n ≥ 1} of Ᏼ, that is, there is a bounded invertible linear operator T on Ᏼ such that T ϕ n = e n for all n ≥ 1.For the linear system (2.5), if the set of the generalized eigenvectors of the operator Ꮽ forms a Riesz basis of the state Hilbert space Ᏼ, then the linear system (2.5) is called a Riesz system.
In the sequel, we prove that (2.5) is indeed a Riesz system.The following lemma can be found in [4].Lemma 3.1 [4].Let Ꮽ be a densely defined discrete operator in a Hilbert space Ᏼ and {φ n | n ≥ 1} a Riesz basis of Ᏼ. Assume that there are an integer N 1 ≥ 0 and a sequence of generalized eigenvectors Then the following assertions hold.
(1) There are an integer N 2 > N 1 and generalized eigenvectors where each σ n is counted according to its algebraic multiplicity.
(3) If there is an integer N 3 > 0 such that σ n = σ m for all n, m > N 3 , then there is an integer N 4 > N 3 such that any σ n for n > N 4 is algebraically simple.
From Lemma 3.1, it follows that in order to obtain the Riesz basis property of generalized eigenvector system of Ꮽ, we need to know some eigenvalues with corresponding eigenvectors of Ꮽ and their asymptotic behavior.Now, we discuss the asymptotic behavior of eigenvalues of Ꮽ.A lengthy computation shows that when J − ρ 2 ν = 0 and M − ρ Thus, for λ ∈ σ (Ꮽ), it is sufficient to consider λ lying in some vertical zone of complex plane, parallel to the imaginary axis.For λ in this zone with |λ| large enough, we have In the case of J − ρ 2 ν = 0 and M − ρ 1 γ = 0, for n ∈ Z, the set of all integers denote Then M sinh ω (1)  n + γρ 1 cosh ω (1)  n = 0, J sinh ω (2)  n + νρ 2 cosh ω (2)  n = 0.
Corollary 3.3.Let Ᏼ and Ꮽ be as before.Let T (t) be the C 0 -semigroup generated by Ꮽ.If J −ρ 2 ν = 0 and M −ρ 1 γ = 0, then the solution of system (2.5) with initial data F ∈ Ᏼ can be expressed as e µ j t P s j ,i (F , t)Y j,i + |n|≥N e λ (1)  n t F,Y * λ (1)   n Y λ (1)   n + |n|≥N e λ (2)  n t F,Y * λ (2)   n Y λ (2)   n . (3.37) 4. Exponential decay of the closed-loop system.In Section 3, we discussed the Riesz basis property of system (2.5) in Ᏼ and gave the solution expression of the system.In this section, we discuss the exponential decay of the closed-loop system.Since (2.5) is a Riesz system, according to [2], we have the following theorem.Theorem 4.1.Let Ᏼ and Ꮽ be as before.If J − ρ 2 ν = 0 and M − ρ 1 γ = 0, then system (2.5) satisfies the spectrum-determined growth assumption in the state space Ᏼ.
Let T (t) be the C 0 -semigroup generated by Ꮽ.The rate of exponential decay of T (t) is defined by and the bound s(Ꮽ) of the spectrum of Ꮽ is given by Theorem 4.1 implies that ω(T ) = s(Ꮽ).In order to estimate s(Ꮽ), denote From the discussion in Section 3, we know that the lines Re λ = ω 1 and Re λ = ω 2 are two asymptotic lines of the spectrum of Ꮽ. Noticing that ω 2 < ω 1 < 0, obviously for any ε > 0 with ω 1 +ε < 0, there are only finitely many eigenvalues of Ꮽ outside the zone If there is an eigenvalue λ of Ꮽ such that Re λ > ω 1 , then there exists at least one eigenvalue λ 0 of Ꮽ such that Re λ 0 = s(Ꮽ).Since Ꮽ is a dissipative operator and σ (Ꮽ) ∩ iR = ∅, we have s(Ꮽ) < 0. Summarizing the above discussion, we have the following result.
Theorem 4.2.Let Ꮽ and Ᏼ be as before and T (t) the C 0 -semigroup of contractions generated by Ꮽ.Then it holds that and hence the closed-loop system is exponentially stable.
We have proven that system (2.5) is a Riesz system.However, from the previous discussion we also can see that the asymptotic behavior of the eigenvalues of the closed-loop system operator Ꮽ is dependent only upon the feedback parameters γ and ν.So, if we take α = 0 and β = 0 in the feedback controls, then it is not difficult to prove that the corresponding closed-loop system is also a Riesz system.In this case, it is natural to ask whether the corresponding closed-loop system decays still exponentially.The answer is positive.In fact, if we denote by Ꮽ 1 the operator Ꮽ with α = 0 and β = 0, then Ꮽ 1 is also dissipative.In order to prove the exponential stability of the C 0 -semigroup generated by Ꮽ 1 , similar to above, it is enough to show that there is no eigenvalue of Ꮽ 1 on imaginary axis.It is easy to see that 0 from which it follows that w ( ) − ϕ( ) = 0 and ϕ ( ) = 0. Then w(x) and ϕ(x) satisfy Thus, w( ) = w ( ) = 0 and ϕ( ) = ϕ ( ) = 0, and hence according to the general theory of ordinary differential equations, it follows that w(x) = ϕ(x) = 0 for all x ∈ [0, ], that is, Y = 0.This shows that there is no eigenvalue of Ꮽ 1 on the imaginary axis.Notice the fact that when J − ρ 2 ν = 0 and M − ρ 1 γ = 0, the lines Re λ = ω 1 and Re λ = ω 2 are two asymptotic lines of σ (Ꮽ 1 ) and the system associated with Ꮽ 1 is a dissipative Riesz system; we can assert that the closed-loop system decays still exponentially and the rate of the exponential decay of the system is just equal to the supremum of real parts of the spectra of Ꮽ 1 .Let T 1 (t) be the contraction semigroup generated by Ꮽ 1 .Similar to above, we have the following result.
Theorem 4.3.Let Ꮽ 1 , Ᏼ, and ω 1 be defined as before.Let T 1 (t) be the contraction semigroup generated by Ꮽ 1 .Then it holds that Notice that the exponential decay rates of the systems associated with Ꮽ and Ꮽ 1 are different.In fact, we can show that ω(T ) ≤ ω(T 1 ).It is very interesting to clarify the dependence of the exponential decay rate on the parameters α and β.

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n
.29) Using Lemma 3.1, the required result follows.According to the proof of Theorem 3.2, there exists an integer N > 0 such that λ for |n| ≥ N are simple eigenvalues of Ꮽ, and Y (λ