Exponential Stability of Non-uniform Timoshenko Beam With Indeﬁnite Damping Under Boundary Control

In this paper we study the stabilization of a non-uniform Timoshenko beam with indeﬁnite damping terms under boundary control and prove how the damping terms can a ﬀ ect the decay rate asymptotically. Using the theory of perturbed problems, we obtain the stability and establish the spectrum determined growth condition for the problem. Moreover, when the two damping terms are indeﬁnite, we provide a condition to obtain the exponential stability


Introduction
We study the following variable coefficients Timoshenko beam with two indefinite damping terms under boundary feedback controls given by: (S ) : = 0, t > 0 EI(1) ∂ϕ ∂x (t, 1) + αϕ(t, 1) + β ∂ϕ ∂t (t, 1) = 0, t > 0 K(1) ϕ (t, 1) − ∂ω ∂x (t, 1) = 0. t > 0 (1) Here α and β are positive feedback constants, ω(t, x) is a lateral displacement and ϕ(t, x) is the bending angle of the beam at position x and time t.The length of the beam is chosen to be unity.The coefficients m(x) > 0 and K(x) > 0 are respectively the mass density and the shear stiffness of a cross section.Furthermore, I m (x) > 0 is the rotatory inertia, EI(x) > 0 is the flexural rigidity, b 1 (x) and b 2 (x) are continuously differentiable damping terms that are allowed to change signs in [0,1].
Due to indefiniteness of functions b 1 (x) and b 2 (x) on interval [0, 1], we impose the following assumptions The same problem was investigated in (Wang, 2004) with α = 0 and β = 0.In (Feng et al, 2001) the authors study a constant coefficients Timoshenko beam system without damping with boundary controls and obtain Riesz basis property.
There are two commonly used approaches to study these perturbed systems.
The first is due to (Huang, 1985), which says that if the resolvent R(λ, A + B) is uniformly bounded along the imaginary axis, then the operator A + B generates an exponentially stable semigroup on the energy space.
The second one is due to (Renardy, 1993) and (Xu & Feng, 2002) which says that the semigroup generated by the operator A + B with bounded linear operator B will satisfy the spectrum determined growth condition if the operator A (not necessarily skew-adjoint) satisfies the following property: There exists N > 0 such that the spectrum σ(A) = {λ n } of A is separated and simple when |λ n | ≥ N and there is a sequence of generalized eigenfunctions of A that forms a Riesz basis in the state Hilbert space.
There is a third approach, namely the Riesz basis approach, which shows that the generalized eigenfunctions of system form a Riesz basis, and then deduces the spectrum determined growth condition and various stability results from the eigenvalue distribution of the system, see (Wang & Yung, 2006), (Wang et al, 2004) and (Wang, 2004).
In this paper, we use the second approach because we need the spectrum determined growth condition of system to prove the exponential stability.Eventually, asymptotic expressions of the eigenvalues of system (S ) are obtained and the spectrum determined growth condition of system is deduced by second approach and stability is established.
The rest of the paper is organized as follows.In section 2, we convert system (S ) into an evolution equation in an appropriate Hilbert space framework, and then prove that the system generates a C 0 −semigroup.
In order to solve the eigenvalue problem, we shall use a space-scaling transformation to derive an equivalent eigenvalue boundary problem and this leads to much simpler asymptotic expansions.In section 3, we shall apply technique in (Wang, 2004) to the fundamental solutions of the eigenvalue boundary problem, and then use results to expand the characteristic determinant in deducing the asymptotic behavior of the eigenvalues.Furthermore, in the last section, we also obtain conditions for the exponential stability of the system for indefinite damping terms.We shall operate under the following conditions: 0 < β < r 2 (1)EI(1) and β > r 2 (1)EI(1).

State Space Setup and Eigenvalue Problem
We start our investigation by formulating the problem on some state Hilbert space.Let where and let H k [0, 1] (k = 1, 2) be the usual Sobolev space of order k.The inner product in H is defined by where , in which the superscript τ denotes transpose of a vector or a matrix, and the notation ' denotes the derivative with respect to x.
In view of system (S ), we define two linear operators A and B in Hilbert space H respectively by and If we let Y t = (ω t , z t , ϕ t , ψ t ) τ , then the system (S ) can be rewritten into an abstract evolution equation in H as following: (11) We have the following results.
For x = 1, we get .
Thus we obtain: Moreover f 1 can also be obtained by substituting the above expression of f 3 (x) into (17) and using the boundary condition f 1 (0) = 0 to yield Theorem 2 Let A and B be given by ( 9) and (10), then A generates a C 0 -semigroup on H. Furthermore, since B is bounded, A + B also generates a C 0 -group on H.
Since the spectrum determined growth condition will later be shown to be true for A + B, the stability of (S ) and ( 11) hinge on the behavior of the eigenvalues of A + B, and we are now in a position to study the eigenvalues problem.
and that ω and ϕ must satisfy the following characteristic equations For convenience, we let which has been stated in (Wang, 2004).Here 1 r 1 (x) and 1 r 2 (x) are usually called wave of the system (S ) and not equal in general.Now (18) becomes And if we let then ( 22) becomes where and A 0 (x), A 1 (x) and A 2 (x) are three matrix functions see (Wang, 2004) .
Under the same procedure, the boundary conditions ( 19) can be written as follows: with We get the following result: Theorem 3 The characteristics equations (18) together with boundary conditions (19) are equivalent to the first order linear system (25) with boundary conditions (27) .

Asymptotic Behavior of Eigenfrequencies of System (S)
We want to find an asymptotic expression for eigenvalues of A + B. It is accomplished by expanding the characteristic determinant with asymptotic expressions of the fundamental matrix solutions of (25).A crucial step is an invertible matrix transformation which is very powerful in the sense that it can be applied to a lot other coupled problems as well, see (Wang & Yung, 2006), (Wang, 2004) and (Xu & Feng, 2002) Theorem 4 For λ ∈ C with |λ| large enough.the first order linear system (25) has fundamental matrix of solutions given by: Φ (., λ) = P (., λ) Ψ (., λ) , where P (., λ) is invertible matrix defined by and Ψ (., λ) can be given by: Proof. for the details of the proof, See (Wang, 2004, Chapter 2, pp. 24-32).
We have the following result on the spectrum of A + B.
Theorem 7 Let A + B given by ( 8)-( 10) .Then generalized eigenfunctions of the operator A + B of system (11) are complete in H.
Theorem 8 The system (11) is Riesz-spectral system (in the sense that eigenfunctions form a Riesz basis in H) and so it satisfies the spectrum determined growth condition.
We now ready to discuss the stability of the Timoshenko beam system (S ).Under assumptions (2) and (3), the Theorems 9.3.3 and 9.4.2 imply that for each ε > 0 there are at most finitely many eigenvalues lying outside of two strips: is satisfied.

Conclusion
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, we may cite the work of Akian (2022), (Feng et al, 2001), (Liping et al, 2019), (Nasser, 2011), (Nasser, 2013), (Wang, 2004, Chapter 9) and (Xu & Feng, 2002), which are We consider the important following result:Theorem 9 Suppose that assumptions (8)-(20) hold.1.If both b 1 (x) and b 2 (x) are non-negative and exist intervals I 1 and I 2 in [0; 1] such that: b 1 (x)| I 1 > 0 and b 2 (x)| I 2 > 0 then the system (11) is exponentially stable.2. If b 1 (x) and b 2 (x) are indefinite, then the system (11) is exponentially stable when conditions: