RIESZ BASIS PROPERTY AND EXPONENTIAL STABILITY FOR ONE-DIMENSIONAL THERMOELASTIC SYSTEM WITH VARIABLE COEFFICIENTS∗

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case. We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a consequence, the exponential stability of the system is concluded. Mathematics Subject Classification. 35Q79, 37L15, 47B06. Received April 23, 2021. Accepted September 25, 2021.


Introduction
It is known that the derivation of the classical heat equation presumes the conducting body to be rigid and hence ignores the interaction between thermal effects and mechanical effects. Since many engineering processes are controlled by the temperature, the generalization of the heat equation which incorporates the effect of thermo mechanical coupling and visa versa has been studied by many authors in the last a few decades. A typical phenomenon for thermoelasticity is that the heat dissipation alone is used as a control to stabilize exponentially the thermoelastic system through in-domain and boundary weak connections. A simple one-dimensional linear thermoelasticity for a homogeneous and isotropic body was derived ( [3], p. 3) by incorporating the effect of thermo mechanical coupling and the effect of inertia, which is described by u(0, t) = u(1, t) = θ(0, t) = θ(1, t) = 0, t > 0, (1.1) where u(x, t) represents the displacement of the string vibration, θ(x, t) represents the absolute temperature, and k > 0 represents the thermal conductivity. The coupling constant r > 0 which is a measure of the mechanical thermal coupling present in the system is generally much smaller than one. The exponential stability of system (1.1) was first obtained in [17] by frequency domain multiplier method. In [4], the distribution of eigenvalues of system (1.1) was studied, and the Riesz basis property of the system was obtained in [5]. There are some computation errors in [4,5], which were corrected in a recent monograph ( [9], Sect. 3.6). By taking the memory effect into account, the second equation of (1.1) was replaced by θ t (x, t) + ru xt (x, t) − (k * θ xx )(x, t) = 0 in [25], where the sign * represents the convolution product, and the exponential stability was obtained as well. The exponential stability and analyticity of abstract linear thermoelastic system were studied in [16]. An interesting fact presented in [6] is that the first real eigenvalue of system (1.1) associated with heat equation must be greater than the first eigenvalue of the pure heat equation, guessed from physical point of view in [20]. This explains clearly the interaction of the mechanical-thermal coupling. The heat makes coupled system exponentially stable and conversely, the vibration of the string increases the temperature of the object.
In this paper, we consider system (1.1) with variable coefficients described by where a(x) > 0 represents the velocity of the wave subsystem and k(x) > 0 represents the thermal conductivity, both are varying with spatial variable. The total energy of system (1.2) is given by ( [3], p.4): Formally, the derivative of E(t) with respect to time t satisfieṡ which shows that E(t) is non-increasing with time. However, the right-hand side of (1.4) depends only on the gradient of temperature which is considered as controller and does not depend on the elastic subsystem explicitly. Such a weak coupling gives rise to a serious problem for the stability of system (1.2). Although the frequency domain multiplier method developed in [17] might be applicable to obtain the exponential stability of the system, some other profound problems like the spectrum-determined growth condition cannot be answered by the multiplier method. To tackle this problem, we adopt the Riesz basis approach. First, throughout the paper, we always assume that a(x), k(x) ∈ C 2 [0, 1]. (1.5) Our goal of this paper is to show that the variable coefficients thermoelastic system (1.2) is a Riesz spectral system in the energy state space. This follows from the following several facts: (a) the system is well-posed and the system operator has compact resolvent; (b) the spectrum of the system consists of two branches: one branch has an asymptote which is parallel to the imaginary axis from the left side and the other distributes along the real axis like 1-D heat equation; (c) the generalized eigenfunctions of the system form a Riesz basis for the state space and therefore the spectrum determined growth condition holds. The latter implies the exponential stability of the system.
The difficulty in spectral analysis for the current system is caused by the variable coefficients. A classical method of finding the asymptotic eigenvalues was introduced in Section 4 of [18] which consists of two steps. The first step is to transform the "dominant term" of the eigenfunction into a uniform "dominant term" by space scaling and state transformation, where no variable coefficient is involved any more, and the second step is to approximate the eigenfunctions of the system by those of the uniform "dominant equation". This approach has been successfully used in dealing with the Euler-Bernoulli beam equations with variable coefficients in [7,12,24]. However, this approach is not applicable to the thermoelastic system investigated in this paper because the "dominant term" of the eigenfunction cannot be transformed into a uniform form. In [21], the eigenvalue problem of a Timoshemko beam was written as Y = λA(x)Y from which the polynomial operator pencil method can be applied. While, the eigenvalue problem of the thermoelastic system is equivalent to Y = A(λ, x)Y where λ and x in A(λ, x) are inseparable. This results in failure of the method used in [21] to deal with the thermoelastic system. The essential reason behind is that the thermoelastic system is a coupled system consisting of a wave subsystem and heat subsystem, and the eigenvalues of the two subsystems are of different orders. We therefore first transform the eigenvalue problem of (1.2) into a first order matrix differential equation and find out the explicit asymptotic expression of the matrix fundamental solutions by introducing an invertible matrix function and using the asymptotic technique for the first order matrix differential equation, referred to as the matrix operator pencil method (see, e.g., [22,23]). This method has been used to spectral analysis for system of coupled partial differential equations in [13,26]. However, there is a remarkable difference between [13,26] and the present one in the process of investigation, that is, the invertible function matrix defined in [26] either associates with the eigenvalues only or the spatial variables only [13], while in present paper, the invertible matrix function is associated with both eigenvalues and the spatial variable. This is caused by the fact that the eigenvalues in those systems discussed in [13,26] are of the same order, which is not true for thermoelastic systems (1.1) and (1.2).
The remaining part of this paper is organized as follows. In the next section, Section 2, we transform system (1.2) into an evolution equation in the energy Hilbert space and the well-poseness of the system is concluded. Asymptotic estimation of the eigenvalues is presented in Section 3. Based on the asymptotic eigenvalues, Section 4 is devoted to calculating the asymptotic expression of the corresponding eigenfunctions. The Riesz basis generation of the system is developed in Section 5, which leads to the spectrum-determined growth condition. As a consequence, the system is shown to be exponentially stable, which covers the main result of [17] on exponential stability of constant coefficients thermoelastic system as a special case.

Spectral analysis
In this section, we consider the eigenvalue problem for system operator A. For any (f, g, h) ∈ D(A) and Reλ < 0, let A(f, g, h) = λ(f, g, h) . Then, g(x) = λf (x), and f (·) and h(·) satisfy: A direct calculation transforms (3.1) into a coupled system of second order ordinary differential equations: In order to solve these equations, we shall use matrix operator pencil method which is a standard technique for the asymptotic expression of eigenpairs.
Then, (3.2) becomes a first order matrix differential equation: and In the sequel, we diagonalize the leading term for ρ in (3.4). Define a matrix P (x, ρ) by (3.6) It is easy to check that for Reλ < 0 and λ = ρ 2 , the matrix P (x, ρ) is invertible when λ = − γ 2 k(x) . A direct computation gives that Taking the derivative of (3.8) with respect to x and substituting Φ (x) given by the first equation of (3.3), we obtain: . By using power series expansion, we find that M (x, ρ) and have the following asymptotic expression as |ρ| → ∞: and (3.14) and Thus, On the basis of these transformations, we are now in a position to find a asymptotic expression for the fundamental matrix solution of system (3.9).

21)
and all entries ofΨ i (x) are uniformly bounded in [0, 1], i = 0, 1, 2, · · · . Here, Proof. We first assume that λ = ρ 2 = − γ 2 k(x) for any x ∈ [0, 1]. Substituting (3.12) into (3.10), we obtian which is a diagonal matrix. It follows that E(·, ρ) given by (3.17) is a fundamental matrix solution to in which the right-hand side involves the higher order terms of ρ from the right side of (3.19). Now we look for a fundamental matrix solution of (3.19) given by (3.20). Taking the derivative of both sides of (3.20) with respect to x leads to Comparing it with the right-hand side of (3.19): and letting each coefficient with same power of ρ of both hands be equal, we arrive at . . .
Note that in the process of proving Theorem 3.1,Ψ 1 (·) is of the following form:  is a fundamental matrix solution for the system (3.3).
Theorem 3.4. Let A be defined by (2.1). Then, the eigenvalues of A contain at least two families: where λ 1n and λ 2n have the following asymptotic expansions: Proof. Since all eigenvalues are symmetric to the real axis, we only need to consider those λ which lie in the second quadrant of the complex plane: It is easy to check that for any ρ ∈ S, Since the positive functions a(·) and k(·) are bounded, a direct computation shows that p i (i = 1, 2, . . . , 6) and Q 1 , Q 2 are constants and |F 1 |, |F 4 | are not larger than 1, according to the expressions of p i (i = 1, 2, . . . , 6), Q j (j = 1, 2) and F k (k = 1, 2, . . . , 4) given by (3.33) and (3.34).
By multiplying some factors, we make each entry of the determinant ∆(ρ) be bounded as |ρ| → ∞: It is easily seen that From this, we see that ∆(ρ) = 0 if and only if which is equivalent to Since Q 1 , Q 2 and F 4 are bounded, there holds and hence Similarly, when ρ ∈ S 2 , it is easy to verify that Re(ρ 2 ) = |ρ| 2 cos(arg(ρ 2 )) ≤ |ρ| 2 cos(3π/4) < 0, Thus, there exists a positive constant C 2 , such that

Hence ∆(ρ) = 0 if and only if
which is equivalent to Therefore, F 1 satisfies Therefore, the second branch of eigenvalues λ 2n = ρ 2 2n has the asymptotic expression given by (3.36). The proof is complete.
Notice that the asymptotic expressions (3.36) are exactly the same as those obtained in Section 3.6 of [9], for the eigenvalues of the thermoelastic system with constant coefficients, which given by However, we do not know whether λ 2n are real until Corollary 5.6 in Section 5.

Asymptotic behavior of eigenfunctions
In this section, we shall consider the asymptotic expression of eigenfunctions which are solutions of (3.2) with respect to eigenvalues given in Theorem 3.4.
Proof it is sufficient to calculate f (x), f (x) and h(x), namely the first three components of Φ(x). From (3.30), we obtain the fundamental matrix solution of system (3.9) as k(x) , and other functions are defined by (3.18) and (3.22) respectively. In the sequal, we calculate the eigenfunctions associated with eigenvalues given by (3.36). Replacing the second row of T RΦ (x, ρ) by the first row ofΦ(x, ρ), we obtain

Riesz basis property and exponential stability
In this section, we shall use the asymptotic expressions of eigenpairs obtained in Section 3 and Section 4 to prove that there exists a sequence of generalized eigenfunctions of operator A, which forms a Riesz basis for H. Furthermore, the exponential stability of the system can be determined by its spectrum distribution.
Before continuing, we recall some notations. For a closed operator A in a Hilbert space H, a nonzero X ∈ H is called a generalized eigenvector of A, corresponding to an eigenvalue λ of A, if there is an integer m > 0 such that (λI − A) m X = 0. If m = 1, X is just an eigenvector. A sequence {X n } ∞ n=1 ⊂ H is called a Riesz basis for H if there exists an orthonormal basis {e n } ∞ n=1 in H and a linear bounded invertible operator T such that T X n = e n , n = 1, 2, · · · .
To prove the Riesz basis property of our system, we recall two associated results. The following Lemma 5.1 comes from [2].
Lemma 5.2. Let A be a densely defined closed linear operator in a Hilbert space H with isolated eigenvalues be a Riesz basis for H. Suppose that there is an integer N ≥ 1 and a sequence of generalized eigenvectors forms a Riesz basis for H.
To prove the Riesz basis property for the operator A defined by (2.1), we define a linear subspace of L 2 (0, 1) by With the inner product in L 2 (0, 1) defined by there holds an orthogonal decomposition of L 2 (0, 1): Then, for any f ∈ L 2 (0, 1), a simple computation shows that there exists a unique direct sum decomposition where b(x) is given in (5.2).
The following Lemma 5.3 is trivial from decomposition (5.3).
Then, {g n (x)} forms a Riesz basis for L 2 0 (0, 1). Proof. The conclusion is trivial by (5.3) and we only need to notice that Lemma 5.4. Let λ n satisfy the conditions of Lemma 5.1 and C n be defined in Lemma 5.3. Then, the vector family {Y n = (cos λ n x − C n , i sin λ n x) } ∞ n=1 forms a Riesz basis for the space L 2 0 (0, 1) × L 2 (0, 1). Proof. Based on Lemma 5.1 and Lemma 5.3, it is easy to check that constitutes a Riesz basis for L 2 0 (0, 1) × L 2 (0, 1). Hence, the sequence cos λ n x − C n sin λ n x , cos λ n x − C n − sin λ n x n∈N also forms a Riesz basis for L 2 0 (0, 1) × L 2 (0, 1). We introduce an invertible bounded operator T 1 = 1 0 0 −i which produces cos λ n x − C n −i sin λ n x = T 1 cos λ n x − C n sin λ n x , cos λ n x − C n i sin λ n x = T 1 cos λ n x − C n − sin λ n x .
The conclusion then follows straightforwardly.
Theorem 5.5. Let A be defined by (2.1), and let X 1n , X 2n be the two families of eigenfunctions given by (4.1) and (4.3) respectively. Then, {X 1n , X 2n , n ∈ Z + } forms a Riesz basis for H the state space of the system (1.2).
It is noted that since the eigenvalues are symmetric about the real axis, theoretically, {λ 2n } are also eigenvalues of A. However, since the two families of asymptotic eigenfunctions of operator A, which correspond to the two branches of eigenvalues {λ 1n , λ 1n , n ∈ N} ∪ {λ 2n , n ∈ N} given by Theorem 3.4, are used in forming a Riesz basis for H only, all λ 2n with sufficiently large n must be real. We therefore obtain the following Corollary 5.6. where λ 1n and λ 2n are given by (3.36) in Theorem 3.4, and λ 2n are real for all sufficiently large n.
As a consequence, we obtain the following Theorem 5.7 which covers main result of [17] on exponential stability as a special case.
Theorem 5.7. Let A be defined by (2.1). Then, the spectrum-determined growth condition holds for e At : ω(A) = S(A) where ω(A) = inf{ω| there exists an M such that e At ≤ M e ωt } is the growth bound of the C 0semigroup, and S(A) = sup{Re(λ)|λ ∈ σ(A)} is the spectral bound of A. Furthermore, the C 0 -semigroup e At is exponentially stable: for some M, ω > 0.
Proof. The spectrum-determined growth condition is a direct consequence of Theorem 5.5. By the expressions of λ 1n and λ 2n given by (3. The exponential stability then follows from the spectrum-determined growth condition, which is similar to [8,10,11].