In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet–Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.
In this work, we consider the Timoshenko system given by
with the following initial conditions
and with several types of boundary conditions, we will precise them later. Here the coefficients a, b, and c are positive constants and we would like to understand precisely what is the influence of these coefficients on the indirect boundary stability and exact controllability of (1.1).
In the first part of this paper, we study the stability of the Timoshenko system (1.1)–(1.2) with only one boundary fractional damping, i.e, System (1.1)–(1.2) is subject to the following boundary conditions
Here the coefficients η, γ, and α are non-negative, strictly positive and in respectively. Fractional calculus includes various extensions of the usual definition of derivative from integer to real order, including the Riemann–Liouville derivative, the Caputo derivative, the Riesz derivative, the Weyl derivative, cf. [45]. In this paper, we consider the Caputo’s fractional derivative
In the second part of this paper, we study the exact controllability of the Timoshenko system (1.1)–(1.2) with only one boundary control v is applied on the right boundary of the first equation, the second equation is indirectly controlled by means of the coupling between the equations, i.e, System (1.1)–(1.2) is subject to the following boundary conditions
The Timoshenko system is usually considered as describing the transverse vibration of a beam and ignoring damping effects of any nature. Precisely, we have the following model, which was developed by Timoshenko in 1921 (see in [49]),
where φ is the transverse displacement of the beam, and ψ is the rotation angle of the filament of the beam. The coefficients ρ, , E, I and K are respectively the density (the mass per unit length), the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section and the shear modulus respectively.
The fractional derivatives are nonlocal and involve singular and non-integrable kernels . We refer the readers to [45] and the rich references therein for the mathematical description of the fractional derivative. The fractional order or, in general, of convolution type is not only important from the theoretical point of view but also for applications as they naturally arise in physical, chemical, biological, ecological phenomena see for example [39], and references therein. They are used to describe memory and hereditary properties of various materials and processes. For example, in viscoelasticity, due to the nature of the material microstructure, both elastic solid and viscous fluid like response qualities are involved. Using Boltzmann assumption, we end up with a stress-strain relationship defined by a time convolution. Viscoelastic response occurs in a variety of materials, such as soils, concrete, rubber, cartilage, biological tissue, glasses, and polymers (see in [13,36,50]). In our case, the fractional dissipation describes an active boundary viscoelastic damper designed for the purpose of reducing the vibrations (see in [37]).
The notion of indirect damping mechanisms has been introduced by Russell in [44], and since this time, it retains the attention of many authors, for instance, let us quote the papers of Alabau [3,5] for a general studies on the hyperbolic systems with indirect boundary stabilizations and [4,6] for indirect boundary observability and controllability of weakly coupled hyperbolic systems. Note nevertheless that the above system does not enter in the framework of these papers. Let us now mention some known results related to the stabilization of the Timoshenko beam. Kim and Renardy in [27] considered Timoshenko system (1.6) with two boundary controls they establish an exponential decay result for the system (1.6). Raposo et al. in [42] showed that Timoshenko system (1.6) with homogeneous Dirichlet boundary conditions and two internal distributed dissipation is exponentially stable. Soufyane and Wehbe in [47] showed that Timoshenko system (1.6) with one internal distributed dissipation law is exponentially stable if and only if the wave propagation speeds are equal, otherwise, only the strong stability holds. Indeed, Rivera and Racke in [43] improved the previous results and showed an exponential decay of the solution of the system when the coefficient of the feedback admits an indefinite sign. Muñoz Rivera and Racke in [41] studied nonlinear Timoshenko system of the form with homogeneous boundary conditions, they showed an exponential decay of the solution of the system if and only if the wave propagation speeds are equal, otherwise, only the polynomial stability holds. Alabau-Boussouira [7] extended the results of [41] to the case of nonlinear feedback , instead of , where α is a globally Lipchitz function satisfying some growth conditions at the origin. In fact, if the wave propagation speeds are equal she established a general semi-explicit formula for the decay rate of the energy at infinity. Otherwise, she proved polynomial decay in the case of different speed of propagation for both linear and nonlinear globally Lipschitz feedbacks. Ammar-Khodja et al. in [9] considered a linear Timoshenko system with memory type under homogeneous boundary conditions. They established uniform stability if and only if the wave speeds are equal and g decays uniformly. Also, they proved an exponential decay if g decays at an exponential rate and polynomially if g decays at a polynomial rate. Ammar-Khodja et al. in [10] studied the decay rate of the energy of the nonuniform Timoshenko beam with two boundary controls acting in the rotation-angle equation. In fact, under the equal speed wave propagation condition, they established exponential decay results up to an unknown finite dimensional space of initial data. In addition, they showed that the equal speed wave propagation condition is necessary for the exponential stability. However, in the case of non-equal speeds, no decay rate has been discussed. The result in [10] has been recently improved by Wehbe et al. in [14] where are established nonuniform stability and an optimal polynomial energy decay rate of the Timoshenko system with only one dissipation law on the boundary. In addition to the previously cited papers. The stability of Timoshenko system with a different kind of damping has been also studied [1,8,14,15,22,23,38,48,51]. For the stabilization of the Timoshenko beam with non-linear term, we mention [7,20,25]. In [17], Benaissa and Benazzouz considered the Timoshenko beam system with two dynamic control boundary conditions of fractional derivative type
where and are positive constant. They showed that the System (1.7) is not uniformly stable by spectral analysis. Hence, using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov, they established a polynomial energy decay rate of type .
In the first part of this paper, unlike [17], we study the stability of the Timoshenko system (1.1) with only one fractional derivative (1.3). We show that the energy of the System (1.1)–(1.3) has a polynomial decay rate (see Theorems 2.12, 2.13). Moreover, in some cases, we obtain the optimal order of polynomial stability (see Theorem 2.12). In addition to the previously cited papers, we mention [2,37] for the stability of wave equation with fractional damping.
We now turn to the second set of results of the paper, which addresses controllability issues of the Timoshenko system with different types of control. For the boundary control, Zhang and Hu in [52] studied the exact controllability of a Timoshenko beam with dynamic boundary controls. Since the controlled Timoshenko system connects with a rigid antenna at one end, the authors introduced two new variables in order to describe their actions. The obtained system was described by two partial and two ordinary differential equations. By using the HUM method, the exact controllability of the system is proved in the energy space. In [11], Araruna and Zuazua considered the dynamical one-dimensional Mindlin–Timoshenko system for beams. They analyzed how its controllability properties depend on the modulus k of elasticity in shear. In particular, under some assumptions on the initial conditions, they proved that the exact boundary controllability property of the Kirchhoff system is obtained as a singular limit, as . For internal control, we mention [24,29]. For the controllability of non-homogeneous Timoshenko beam we mention [46].
In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system (1.1) with the boundary conditions (1.5) while waves propagate with equal or different speeds. We use the Hilbert Uniqueness Method introduced by Lions (see [30,31] for example). To this aim, by using Ingham’s theorem, we first establish the inverse and the direct observability inequalities for the homogeneous Timoshenko system. Next, we use the Hilbert Uniqueness Method, to get the exact controllability for the Timoshenko system (1.1) with the boundary conditions (1.5) inappropriate functional spaces of terminal data.
This paper is organized as follows: In Section 2, we study the stability of the Timoshenko with only one fractional derivative. In Section 2.1, we prove the well-posedness of System (1.1) with the boundary conditions (1.3). In Section 2.2, we prove the strong stability of the system in the lack of the compactness of the resolvent of the generator. In Section 2.3, we prove that the Timoshenko system (1.1) considered with the boundary conditions (1.3) is non-uniformly stable when the speeds of the propagation of the waves are either equal or different. More precisely, we show that an infinite number of eigenvalues approach the imaginary axis. In Section 2.4, we prove the polynomial stability of the system, with a faster polynomial decay rate if the waves propagate with equal speed: the energy of System (1.1)–(1.3) has a polynomial decay rate (see Theorem 2.13). In Section 3, we study the exact controllability of the Timoshenko system (1.1) with the boundary conditions (1.5). In Section 3.1, we set the framework of the homogeneous Timoshenko system (1.1) and we establish the characteristic equation satisfied by the eigenvalues of the operator . Next, in Section 3.2, we prove the exact controllability of the System (1.1) with the boundary conditions (1.5) while waves propagate with the same speed, i.e., . Depending on number theoretical properties of the constants a, b, and c, we deduce the corresponding observability spaces. In Section 3.3, we consider the case where the waves propagate with different speeds and show the exact controllability of the System (1.1) with the boundary conditions (1.5) and the corresponding observability spaces depending on the parameter a, b, and c.
Stability of Timoshenko system with fractional derivative
In this section, we study the stability of the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.3). Here a, b, c, γ are strictly positive constants, and .
Augmented model and well-posedness
In this part, using a semigroup approach, we establish well-posedness result for the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.3). First, from Theorem 2 in [37] (see also Theorem A.4), Timoshenko system (1.1)–(1.2) with the boundary conditions (1.3) can be rewritten as the following augmented model
with the following boundary conditions
and the following initial conditions
where and be the function defined almost everywhere on . Since , one has that . The energy of System (2.1)–(2.6) is given by
Let be a regular solution of (2.1)–(2.6). Multiplying (2.1), (2.2) and (2.3) by , , and respectively, then using the boundary conditions in (2.4)–(2.5), we get
Thus, System (2.1)–(2.6) is dissipative in the sense that its energy is non increasing with respect to the time t. Let us define the energy space by
such that and . It is easy to check that the spaces and are Hilbert spaces over equipped respectively with the norms: and , where denotes the usual norm of . The energy space is equipped with the inner product defined by
for all and in . We use to denote the corresponding norm. We define the linear unbounded operator in by
and
The condition is imposed to insure the existence of in (2.5).
If is the state of (2.1)–(2.6), then the Timoshenko system is transformed into the first order evolution equation on the Hilbert space given by
For the well-posedness of Problem (2.1)–(2.6), according to Lumer–Phillips theorem (see [35,40]), we need to prove that the operator is m-dissipative. Hence, we prove the following proposition.
The unbounded linear operatoris m-dissipative in the energy space.
For , one has
which implies that is dissipative. Here Re is used to denote the real part of a complex number. We next prove the maximality of . For , we show the existence of , unique solution of the equation
Equivalently, one must consider the system given by
with the following boundary conditions
Let . Multiplying Equations (2.9) and (2.10) by and respectively, integrating in and taking the sum, then using by parts integration and the boundary conditions in (2.11), we get
where
such that
By using the fact that , , and , we can easily check that and are well defined. Thanks to (2.13), (2.14) and using the fact that , we have that a is a bilinear continuous coercive form on , and L is a linear continuous form on . Then, using Lax–Milgram theorem, we deduce that there exists unique solution of the variational Problem (2.12). From (2.8) and applying the classical elliptic regularity we deduce that , completing the proof of the proposition. □
Thanks to Lumer–Phillips theorem (see [35,40]), we deduce that generates a -semigroup of contractions in and therefore Problem (2.1)–(2.6) is well-posed. Then, we have the following result.
For any, the Problem (
2.1
)–(
2.6
) admits a unique weak solution. Moreover, if, then.
Strong stability
Our main result in this part is the following theorem.
Assume that, then the semigroup of contractionsis strongly stable onin the sense thatfor allif
For the proof of Theorem 2.4, according to Theorem A.2 of Arendt and Batty, we need to prove that the operator has no pure imaginary eigenvalues and is countable, where denotes the spectrum of . The argument for Theorem 2.4 relies on the subsequent lemmas.
Assume thatand conditionholds. Then, one has
Let and , such that
Equivalently, we have
First, a straightforward computation gives
consequently, we deduce that
Inserting the above equation in (2.16) and using the fact that , we get
If , by elementary computations, one deduces that and consequently . If , combining Equations (2.17)–(2.19), we get the following system
If condition is holds true, then using the boundary conditions in (2.21), we can easily check that is the unique solution of (2.20). Consequently and the conclusion follows (For more details about the proof see Appendix B). □
Assume that. Then, the operatoris not invertible and consequently.
Let , and assume that there exists such that: , it follows that
Hence, we deduce that , which contradicts the fact that . Consequently, the operator is not invertible, as claimed. □
The following lemma is a technical result to be used in the proof of Lemma 2.9 given below.
Assume that conditionholds and assume that eitherorand. Then, for any, the following systemadmits a unique strong solution, where
Since , under the assumptions of the above lemma, it is easy to check that
We distinguish two cases.
Case 1. and : System (2.22) becomes
Let . Multiplying the first and the second equations of (2.23) by and respectively, integrating in and taking the sum, then using by parts integration and the boundary conditions in (2.23), we get
The left hand side of (2.24) is a bilinear continuous coercive form on , and the right hand side of (2.24) is a linear continuous form on . Using Lax–Milgram theorem, we deduce that there exists a unique solution of the variational Problem (2.24). Hence, by applying the classical elliptic regularity we deduce that System (2.23) has a unique strong solution .
Case 2. and : we first define the linear unbounded operator by
For any , let us consider the following system
Let . Multiplying the first and the second equations of (2.25) by and respectively, integrating in and taking the sum, then using by parts integration, the boundary conditions in (2.25) and the fact that (), we get
for all . Thanks to (2.26) and using Remark 2.8, we have that the left hand side of (2.26) is a bilinear continuous coercive form on , and the right hand side of (2.26) is a linear continuous form on . Then, using Lax–Milgram theorem, we deduce that there exists unique solution of the variational Problem (2.26) and deduce that System (2.25) has a unique strong solution . In addition, we have
where . It follows, from the above inequality and the compactness of the embeddings into , that the inverse operator is compact in . Then, applying to (2.22), we get
Consequently, by Fredholm’s alternative, proving the existence of solution of (2.27) reduces to proving . Indeed, if , then . It follows that
Multiplying the first and the second equations of (2.28) by and respectively, integrating in and taking the sum, then using by parts integration and the boundary conditions in (2.28), we get
Hence, take the imaginary part of the above equation, to get
where Im stands for the imaginary part of a complex number. Since , we get . Then, System (2.28) becomes
It is now easy to see that if is a solution of System (2.29), then the vector defined by: belongs to , and . Therefore, . Using Lemma 2.5, we get . This implies that System (2.27) admits a unique solution due to Fredholm’s alternative, hence System (2.22) admits a unique solution . Thus, the proof of the lemma is complete. □
We use the previous lemma to deduce the following one.
Assume that eitherorand. Thenis surjective.
Let , we look for solution of
Equivalently, we consider the following system
with the following initial conditions
where is defined in Lemma 2.7 and
Since and , under the hypotheses of the lemma, it is easy to check that
Let such that
Setting and in (2.32)–(2.34), we obtain
Using Lemma 2.7, we get that System (2.35) has a unique solution . Therefore, System (2.31)–(2.34) admits a solution . Thus, we define , , and
we conclude that the Equation admits a solution , hence the conclusion. □
We are now in a position to conclude the proof of Theorem 2.4.
Using Lemma 2.5, we have that has no pure imaginary eigenvalues. According to Lemmas 2.5, 2.6, 2.9 and with the help of the closed graph theorem of Banach, we deduce that if and if . Thus, we get the conclusion by applying Theorem A.2 of Arendt and Batty. The proof of the theorem is complete. □
Lack of exponential stability
In this part, we use the classical method developed by Littman and Markus in [32] (see also [21]), to show that the Timoshenko system (2.1)–(2.6) is not exponentially stable. Our main result in this part is the following theorem.
The semigroup generated by the operatoris not exponentially stable in the energy space.
For the proof of Theorem 2.10, we recall the following definitions: the growth order and the spectral bound of are defined respectively as
From the Hille–Yoside theorem (see also Theorem 2.1.6 and Lemma 2.1.11 in [21]), we see that: . By the previous results, one clearly has that and the theorem would follow if equality holds in the previous inequality. It therefore amounts to show the existence of a sequence of eigenvalues of whose real parts tend to zero. For this aim, we need to study the asymptotic behavior of the spectrum of . We prove the following proposition.
There existsufficiently large and two sequencesandof simple eigenvalues ofsatisfying the following asymptotic behavior:
Case 1. If, then
Case 2. Ifand,, then
Case 3. Ifand,, then
Case 4. Ifand,, then
Since is dissipative, we fix small enough and we study the asymptotic behavior of the eigenvalues λ of in the strip . First, we determine the characteristic equation satisfied by the eigenvalues of . For this aim, let be an eigenvalue of and let be an associated eigenvector such that . Then, the corresponding eigenvalue problem is given by
Equivalently, we have
The characteristic polynomial associated with System (2.43) is given by
In order to proceed, we set the following notation. Here and below, in the case where z is a non zero non-real number, we define (and denote) by the square root of z, i.e., the unique complex number with positive real part whose square is equal to z.
Our aim is to study the asymptotic behavior of the large eigenvalues λ of in S. A careful examination shows that Q admits four distinct roots if . In case of equal wave propagation speed (i.e. ), this is automatically true and, in case of different wave propagation speeds, this again holds true by taking λ large enough. Hence, the four distinct roots of Q are given by: , , , , such that
Here and below, for simplicity we denote by . Then the general solution of (2.43) is given by
From the boundary conditions in (2.43) at , for λ large enough, we get . Moreover, the boundary conditions in (2.43) at can be expressed by
where
and
Denoting the determinant of a matrix M by , one gets that
Equation (2.43) admits a non trivial solution if and only if . Next, using the asymptotic expansion in , we get that the largest roots of are given as in (2.36) if and given as in (2.37)–(2.42) if (For more details about the proof see Appendix C). □
From Proposition 2.11 the operator has two branches of eigenvalues with eigenvalues admitting real parts tending to zero. Hence, the energy corresponding to the first and second branch of eigenvalues has no exponential decay. Therefore the total energy of the Timoshenko system (2.1)–(2.6) has no exponential decay both in the equal speed case, i.e., or in the different speed case, i.e., when . □
Polynomial stability
In the case where is not exponentially stable, we look for a polynomial decay rate. Our results are gathered in the following two theorems.
Assume that,and conditionholds. Then there existssuch that, for every, the energy of the System (
2.1
)–(
2.6
) has the optimal polynomial decay rate of type, i.e.,where
Assume that,and conditionholds. Then, for almost all real number, there existssuch that for every, we have
Since , for the proof of Theorem 2.12 and Theorem 2.13, according to Theorem A.3 of Borichev and Tomilov, we need to prove that
where if and if .
We will argue by contradiction. Suppose there exists , with and
such that
Equivalently, we have
such that
In the following, we will prove that, under Condition (H1), and (2.47), one also gets that , hence reaching the desired contradiction. For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. From (2.49) and (2.51), we remark that
On the other hand, for all , taking the inner product of (2.48) with U in , then using the fact that U is uniformly bounded in , we get
Let, we have
First, from the boundary condition, we have
using Cauchy–Shariwz inequality, we get
Since for all , then inserting (2.55) in the above inequality, we obtain the first asymptotic estimate of (2.56). Next, from (2.53), we get
Multiplying the above inequality by , then integrating over with respect to the variable ξ and applying Cauchy–Schwarz inequality, we obtain
such that
It is easy to check that
Moreover, may be simplified by defining a new variable . Substituting ξ by in Equation (2.58) (i.e., in ), then using the fact that , we get
where is a positive constant number. Inserting (2.58) and (2.59) in (2.57), then using (2.48), (2.55) and the fact that , we deduce that
Since and , we have , consequently, from the above inequality, we get the second asymptotic estimate of (2.56). Thus, the proof of the lemma is complete. □
Let, we have
First, multiplying Equation (2.50) by in , then using by parts integration, we get
Using the fact that in , in , and , are bounded in , we get
Inserting (2.62) in (2.61), we get
Similarly, multiplying Equation (2.52) by in , then using by parts integration, the fact that , in , in , , are bounded in , and (2.54), we get
Finally, adding (2.63) and (2.64), we get (2.60), which concludes the proof of the lemma. □
Let. We have the following two cases:
Case 1. If, then
Case 2. If, let, then for almost all real number, we have
Let , then Equations (2.50) and (2.52) can be written as
such that
By the variation of constant formula, the solution of (2.67) is: . Therefore,
On the other hand, from (2.56), we have
The eigenvalues μ of the matrix B are the roots of the following characteristic polynomial
Since , then the above characteristic polynomial has four distinct pure imaginary roots: , , , , where and are given by
Since the eigenvalues of B are simple, then B is a diagonalizable matrix and for all , we have
such that
and
Now, using (2.71), (2.48), asymptotic expansion and by parts integration, we can easily check that
Next, inserting Equations (2.69), (2.71) and (2.72) in (2.68), then using asymptotic expansion and the fact that , we get
For more details about how we proved Equations (2.71)–(2.74) see Appendix D. We now distinguish two cases.
Case 1. Assume that , using asymptotic expansion in (2.70), we get
Inserting (2.75) in (2.73)–(2.74), then using asymptotic expansion, we get
such that
Case 1.1. If there exist no integers such that , then and , therefore from (2.76), we get
Hence, from (2.77), we get (2.65).
Case 1.2. Assume that , we divide the proof into two cases: Case 1.2.1, if and Case 1.2.2, if . Since the argument of two cases is entirely similar, we will only provide one of them.
Assume that , then and , consequently, from (2.76), we get
Adding (2.78) and (2.79), we get
Hence, from (2.79) and (2.80), we get (2.65).
Case 2. Assume that , using asymptotic expansion in (2.70), we get
Inserting (2.81) in (2.73)–(2.74), then using asymptotic expansion, we obtain
Adding (2.82) and (2.83), we get
Let , from (2.47), (2.56) and (2.60), we get . Our aim is to show that , suppose that there exist two positive constant numbers such that , then from (2.83) and (2.84), we get
It follows from Equation (2.85), there exists such that
Subtracting the second equation of (2.86) from the first equation of (2.86), then using the fact that , we get
From Theorem 1.10 in [19], we have for almost all real numbers ζ there exists infinitely many integers n, m such that
Let , then from (2.87) and (2.88) there exist infinitely many integers n, m such that
Since for a positive constant, then the estimate (2.89) can be written as . Consequently, , which is impossible. Therefore, , we conclude the proof. □
Step 1. The energy decay estimation. We distinguish two cases:
Case 1. If there exist no integers such that , let , then from (2.56) and (2.65), we get
Inserting (2.90) in (2.60), we get which contradicts (2.47). Thus, (H1) holds true with . The conclusion then follows by applying Theorem A.3.
Case 2. If there exists such that , let , then from (2.56) and (2.65), we get
Inserting (2.91) in (2.60), we get which contradicts (2.47). Now applying Theorem A.3, we conclude the proof.
Step 2. The optimality. For the optimality of (2.46), let and for , set
where and are the simple eigenvalues of . Moreover, let be the normalized eigenfunction corresponding to . We introduce the following sequence
Therefore, we have , . Hence, from Proposition 2.11 (Cases 2, 3, 4), we get
where , , are non zero real numbers. Hence, for all there exists such that
Thus, we deduce that . Finally, thanks to Theorem A.3, we cannot expect the energy decay rate . Therefore, estimate (2.46) is optimal. □
For almost all real numbers , let , then from (2.56) and (2.66), we have
Inserting (2.92) in (2.60), we get which contradicts (2.47). The result follows from Theorem A.3. □
Exact controllability of the Timoshenko system
In this section, we study the indirect boundary exact controllability of the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5). For a given and initial data belonging to a suitable space, the aim of this section is to find a suitable control v such that the solution of the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5), given by , is driven to zero in time T, i.e.,
Spectral compensation for homogeneous Timoshenko system
The aim of this section is to compute the eigenvalues and the eigenvectors associated to the homogeneous Timoshenko system. For this aim, we consider the homogeneous Timoshenko system
with the following initial conditions
here a, b and c are strictly positive constants. The energy of System (3.1)–(3.2) is given by
a direct computation gives . Thus, the energy of the solution is conserved. Let us define the energy space by
with the inner product defined by
for all , . We use to denote the corresponding norm. We define the linear unbounded operator in by
and
Therefore, we can write the System (3.1)–(3.2) as an evolution equation on the Hilbert space :
One clearly that is a maximal dissipative operator on , then by Lumer Philips’s Theorem (see Theorem 4.3 in [40]), is the infinitesimal generator of a -semigroup of contractions on . Therefore, the Problem (3.1)–(3.2) is well-posed and we have the following result.
For any, System (
3.3
) admits a unique weak solution, such thatMoreover, if, then the Problem (
3.3
) admits a unique strong solution.
Since is a closed operator with a compact resolvent, its spectrum consists entirely of isolated eigenvalues with finite multiplicities (see Theorem 6.29 in [26]). Moreover, it is easy to check that .
We will now study the spectrum of the System (3.1)–(3.2). Let be an eigenvalue of the operator and a corresponding eigenvector. Using the fact that , we get that with . Then, the corresponding eigenvalue problem is given by
For some constants , let
be a solution of (3.4). It follows that
which has a non-trivial solution if and only if
The solution of (3.4) is given by , such that
System (3.8) admits a non zero solution if and only if
Solving the above equation, we get
From the boundary conditions in (3.4), the System (3.4) has a non-trivial solution if and only if and/or . Taking
we get
In this case, the solution of (3.4), , is uniquely written as defined in (3.5).
Observability and exact controllability under equal speeds wave propagation condition
In this part, assume that the waves propagate with the same speeds, i.e., . In this case, we study exact controllability of a one dimensional Timoshenko system (1.1)–(1.2) with the boundary conditions in (1.5). For this aim, first, we prove the following Observability theorem.
Assume that,and conditionholds. Then, for all solutionthat solve the Problem (
3.1
)–(
3.2
) there exists a positive constantdepending only on a, b, c such that the following direct inequality holds:Moreover, there exists a positive constantdepending only on a, b, c such that the following inverse observability inequalities hold:
Case 1. If there exist no integerssuch that, then
Case 2. If there existssuch that, then there exists Hilbert space, defined byequipped with the following norm, such that the inverse observability holds:where,are the eigenfunctions of the operator(see the eigenfunctions in Proposition
3.4
and Remark
3.5
).
For the proof of Theorem 3.3, we use the spectrum method. For this aim, we need to study the asymptotic behavior of the spectrum of . We prove the following proposition.
Assume thatand conditionholds. Then, the eigenvalues ofhas the following asymptotic behaviorwith the corresponding eigenfunctions
Assume that , by solving (3.7), we get
Using the asymptotic expansion in (3.16), we get (3.13). Next, for , setting
in (3.6), we get the corresponding eigenfunctions (3.14). Similarly for , setting
in (3.6), we get the corresponding eigenfunctions (3.15). □
If , then from Equation (3.16), we can easily check that the eigenvalues , are simple and different from zero. Then, we set the eigenfunctions of the operator as
From the asymptotic expansions (3.13) and (3.14)–(3.15), we can easily prove that form a Riesz basis in the energy space . We distinguish different types of observability inequalities, while depending on the constants a, b, c. In fact, we are going to see in Proposition 3.6 that if there exist no integers such that , then the eigenvalues satisfy a uniform gap condition. In this case, we will apply the usual Ingham’s theorem (see Theorems 4.3, 9.2 in [28]) in order to get observability inequalities hold in the energy space . In the case where there exists such that , then the eigenvalues of the same branch satisfy a uniform gap condition, while on different branches they can be asymptotically close at a rate of order (see Proposition 3.6). Thus, the usual Ingham’s Theorem used in the case is no longer valid, and therefore we will use a general Ingham-type theorem, which tolerates asymptotically close eigenvalues (see Theorem 9.4 in [28]).
Assume thatand conditionholds, then there exist two constants,depending only on the constants a, b, c such thatMoreover, we have the following two cases:
Case 1. If there exist no integerssuch that, then there exists a constantdepending only on the constants a, b, c such that the two branches of eigenvalues ofsatisfy a uniform gap condition
Case 2. If there existssuch that, then there exist constantsdepending only on the constants a, b, c such that for all, forlarge enough, we haveand there exist infinitely many integers m, n such that
First, from (3.16) and the fact that all the eigenvalues , are simple, it follows that (3.17). We now divide the proof into two cases:
Case 1. There exists no integer such that . First, from the asymptotic expansions (3.13), we have
Since there exists no integer such that , then there exists , such that
Therefore, from (3.21), we get (3.18).
Case 2. There exists such that . Again from the asymptotic expansions (3.13), we have
We distinguish two cases:
If , then there exists such that , therefore from (3.22), it follows that (3.19).
If , then from (3.22), we obtain . Consequently, we get (3.19).
Moreover, if , then from the previous inequality there exists such that (3.20) holds, which concludes the proof of the proposition. □
Assume thatand conditionholds. If there existssuch that, then we adjust the branches of eigenvalues into one sequencesuch thatis strictly increasing. IfthenWe say that,is a chain of close exponents relative to γ of length 2.
When there exists such that , since the eigenvalues of the same branch satisfy a uniform gap condition, then from (3.23), we deduce that , belong to different branches. If , belong to the same branch of eigenvalues, then from (3.17), we get . Thus, we obtain the first assertion of (3.24). Otherwise, if , belong to different branches, then using the fact that , belong to different branches, then , belong to the same branch of eigenvalues. In this case, from (3.17), it follows that
From the above inequality and (3.23), we get
Therefore, we obtain the first assertion of (3.24). The same argument verifies the second assertion of (3.24). Thus, the proof of the proposition is complete. □
From Proposition 3.6, it follows that
On the other hand, if there exist no integer such that , due to the fact that the eigenvalues , satisfy a uniform gap condition, then the inverse observability inequality is true in the energy space . Otherwise, if there exists integer such that , due to the fact that the eigenvalues can be asymptotically close, then the inverse observability inequality is not true in the energy space . For this reason, we defined the weighted spectral space (see in (3.11)). Since the System is a Riesz basis in the energy space , the space is obviously a Hilbert space equipped with the norm . We are now ready to prove our observability inequalities results.
We divide the proof into two main steps. Our first aim is to prove the direct inequality (3.9). Given any initial data , such as , then the solution of (3.3) can be written as
Therefore, we have
where
Since the eigenvalues of the same branch satisfy a uniform gap condition, applying the usual Ingham’s theorem (see Theorem 9.2 in [28]), we get
On the other hand, from (3.26), we get
Inserting (3.27) in (3.28) and using (3.25), we get the inequality (3.9).
Our next aim is to prove the inverse observability inequalities. We divide the proof into two cases:
Case 1. There exists no integer such that . Given any initial data , such as: , then the solution of (3.3) can be written as
Therefore, we have
From (3.25), we can rewrite the above equation as
Following a generalization of Ingham’s theorem (see Theorem 9.2 in [28]), the sequence forms a Riesz basis in provided that , where is the upper density of the sequence , defined as , where denotes the largest number of terms of the sequence contained in an interval of length r. To be more precise, . Therefore,
Hence, we get (3.10).
Case 2. There exists such that . Given any initial data , such as: , consequently, the solution of (3.3) can be written as
Therefore, we have
We now arrange the two branches of eigenvalues , into one sequence such that the sequence is strictly increasing. From Proposition 3.7, all the chain , of close exponents relative to γ is of length 2. Moreover, we denote by the coefficient before or in (3.29). Let A and B be defined as and . Then, we can rewrite (3.29) as
where denotes the divided difference of the chain of close exponents , relative to γ
It follows from Theorem 9.4 in [28], that the sequence: , forms a Riesz sequence in provided that . Thus, we have
On the other hand, from (3.20), we get
Inserting the above inequality into (3.30) and returning to the previous notations, we get
Therefore, from the above inequality and Equation (3.25), we get
consequently, we obtain the inequality (3.12). Thus, the proof of the theorem is complete. □
In the case where there exist no integers such that , the inverse observability inequality is true in the energy space . Otherwise, in the case where there exists such that , the inverse observability inequality holds in weighted spectral space . The aim of the next part is to get the observability or exact controllability in usual functional spaces.
Observability inequality in usual spaces. Using the asymptotic expansions (3.14)–(3.15), we have
For any , we define the space
According to Theorem 3.1 stated in [34] (see also Theorem A.5), we can state the following result.
Assume that,() and conditionholds. Then, we have the following identifications
We see that and are Riesz basis in . On the other hand, we have and are Bessel sequences in . Then, the result obtained directly from Theorem A.5. □
Furthermore, for any , we define
Thus, with the pivot space , we have . Then, it follows that
Consequently, we have the following observability result.
Assume that,() and conditionholds. Let, then there exists a constantsuch that the following direct inequality holds:for all solutionthat solve the homogeneous Cauchy Problem (
3.1
)–(
3.2
). Moreover, there exists a constant, such that the following inverse observability inequality holds:
Assume that , if (), then the two branches of eigenvalues are close in the order . Due to the closeness of the eigenvalues, the observability space losses two derivatives. Consequently, the observability holds in the space of type: . Moreover, the control space are of type: with suitable boundary conditions.
Exact controllability in usual spaces. In this part, using HUM method, we establish exact controllability result for the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5). It is interesting to notice that the observability of the System (3.1)–(3.2) suggests the exact controllability of the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5) (see [28,30,31] for example). Then, from Theorem 3.9, we get the following result (For more details about the proof of Theorem 3.11 see Appendix E).
Assume that,, and conditionholds:
If there exists no integersuch that, let
If there existssuch that, let,
then there existssuch that the solution of the Timoshenko system (
1.1
)–(
1.2
) with the boundary conditions (
1.5
) satisfies the null final conditions
Observability and exact controllability when the speeds of propagation are different
In this part, we study the exact controllability of a one dimensional Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5) in the case when the speeds of propagation are different, i.e, . Similar to Section 3.2, we use the spectrum method. For this aim, we need to study the asymptotic behavior of the spectrum of . We prove the following proposition.
Assume thatand conditionholds. Then, the eigenvalues ofasymptotic behaviorwith the corresponding eigenfunctions
First, by solving (3.7) and using the fact that , we get
Using the asymptotic expansion in (3.34), we get (3.31). Next, for , setting
in (3.6), we get the corresponding eigenfunctions (3.32). Similarly for , setting
in (3.6), we get the corresponding eigenfunctions (3.33). □
If , then from Equation (3.34), we can easily check that the eigenvalues , are simple and different from zero. Then, we set the eigenfunctions of the operator as
In fact, using the asymptotic expansions (3.31) and (3.32)–(3.33), we can easily prove that form a Riesz basis in the energy space . Here the eigenvalues of the same branch satisfy a uniform gap condition, but the eigenvalues of different branches can be asymptotically close at rate depends on the parameters a, b, c (see Proposition 3.14). Again in this subsection we will use a general Ingham-sort theorem.
Assume thatand conditionholds. Then, there exists a constantdepending only on a, b, c such thatMoreover, there exist constantsdepending only on a, b and c such that
1. Ifis a rational number different fromfor all integers p, q, then for all, forlarge enough, we haveand there exist infinitely many integers m, n such that2. Iffor some integers,, then for all, forlarge enough, we haveand there exist infinitely many integers m, n such that3. For almost all positive irrational numberand all, forlarge enough, we haveand there exist infinitely many integers m, n such that
The assertion (3.35) follows directly from the asymptotic expansions (3.34) and the fact that all the eigenvalues are geometrically simple. Using the asymptotic expansions (3.31), we have
If , then the estimates (3.36), (3.38) and (3.40) are trivial. Otherwise, if , then and (3.42) becomes
Therefore, it is sufficient to consider the leading term in (3.43).
1. Let be a reduced rational number. Then, is a root of the integer polynomial of second degree. Since for all integers p, q, then the integer polynomial is irreducible. This means that is a quadratic algebraic number. Thanks to the Liouville’s theorem on the approximation of algebraic numbers (see Theorem 1.2 in [19]), there exists a constant , depending only on , such that for all , we have
On the other hand, since is an irrational number, using the Dirichlet’s classic theorem on number theory (see Theorem 1.1 in [19]), there exist infinitely many integers m, n such that
Therefore, we get the estimates (3.36)–(3.37).
2. Let , be a reduced rational number. We return to (3.42), we get
If or for all , then from the above inequality, we get , hence we get (3.38). Otherwise, if and , then from (3.31), we deduce that and therefore we get (3.38).
On the other hand, by taking and , , and using the asymptotic expansions (3.31), we easily get that
consequently, we get the estimate (3.39).
3. Let . Firstly, from Khintchine’s theorem on Diophantine approximation (see Theorem 1.10 in [19]), for almost all irrational number , there exist only finitely many integers m, n such that
It follows from (3.43), that for almost all irrational number , there exists a constant and , large enough, such that, for all , , we have
This gives the estimate (3.40). Secondly, from Khintchine’s theorem on Diophantine approximation (see Theorem 1.10 in [19]) for almost all irrational real number , there exist infinitely many integers such that
Therefore, we get the estimate (3.41), which concludes the proof of the proposition. □
Similar to Proposition 3.7, we can prove the following proposition.
Assume thatand conditionholds. We rearrange the two branches of eigenvalues into one sequencesuch thatis strictly increasing. Ifthenand.
From Proposition 3.12, it follows that
We now define the following weighted spectral spaces
and
The factor in will be omitted for . Since the System is a Riesz basis in the energy space , we get that the spaces and are obviously a Hilbert spaces equipped respectively with the norms
In fact, to get the observability we need to use a weaker norm for the second equation in order that has the same order as . For this reason we multiplied the eigenvector by n in the spaces and . We are now ready to prove our observability inequalities results.
Assume thatand conditionholds, let. Then, for all solutionthat solve the Problem (
3.1
)–(
3.2
) there exists a constantsuch that the following direct inequality holds:Moreover, there exists a constantdepending only on a, b and c such that the following inverse observability inequalities hold:
Case 1. Ifis a rational, then
Case 2. For almost all irrational number, we have
Similar to Section 3.2, we can prove the direct inequality (3.46). Our next aim is to prove the inverse observability inequalities.
Case 1. Let be a rational. Given any initial data such as: , using the Riesz property the solution of (3.1)–(3.2) can be written as
Hence, we have
We now rearrange the two branches of eigenvalues , into one sequence such that the sequence is strictly increasing. From Proposition 3.15, it follows that all chain , of close exponents relative to γ is of length 2. Then, let A denotes the set of integers such that the condition (3.44) holds true and let . We denote by the coefficient before or in (3.49). We can rewrite it into
such that . From Theorem 9.4 in [28], the sequence , forms a Riesz sequence in provided that . Thus, it follows that
The assertions (3.36) and (3.38) of Proposition 3.14, imply that
Inserting the above inequality into (3.50) and returning to the previous notations, we get
Therefore, from the above inequality and Equation (3.45), we get
therefore, we get the inequality (3.47).
Case 2. For almost all irrational number . Given any initial data , such as:
, then the solution of (3.1)–(3.2) can be written as
Therefore, we have
Similar to case 1, we get
where denoted the coefficient before or in (3.51). Using (3.40) of Proposition 3.14, we get
Inserting the above inequality into (3.52), we get
by inserting (3.45) into the above inequality, we get the inequality (3.47). Thus, the proof of the theorem is complete. □
The weighted spectral spaces and are defined by means of the eigenvectors and with weights. Our aim is to get the observability or exact controllability in usual functional spaces. For this aim, let
For any , we define the spaces
Assume thatand conditionholds. Then, we have
We see that and are Riesz basis in , respectively and are Bessel sequences in . Then, the first assertion of (3.53) follows directly from Theorem A.5. The second assertion of (3.53) can be obtained in the same way. □
Furthermore, for any , we define the spaces
Thus, with the pivot space , we have and . Then, it follows that
Assume that and condition holds. In the case 1, since the two branches of eigenvalues are close in the order of , then the observability space of the first equation losses one derivative because of the closeness of eigenvalues, while that of the second equation losses two derivatives due to the closeness of eigenvalues and the transmission of the modes between the two equations. Therefore, the observability holds in the space of type: . Moreover, the control space are of type with suitable boundary conditions.
Similar to Section 3.2, from Theorem 3.16, we get the following result.
Assume that,and conditionholds.
Ifbe a rational, let,
For almost all irrational number, let,
then there existssuch that the solution of Timoshenko system (
1.1
)–(
1.2
) with the boundary conditions (
1.5
) satisfies the null final conditions
Open problems. 1. If condition is not true (i.e., if (B.3) holds), then (where λ is given in (B.3)) is an eigenvalue of . In this case has a pure imaginary eigenvalues. So, we cannot apply Theorem A.2 (i.e., Arendt and Batty theorem). Consequently, if condition is not true, then strong stability of the Problem (2.1)–(2.6) is still an open problem.
2. In the case that . We proved that the Problem (2.1)–(2.6) is strongly stable (see Theorem 2.4). Moreover, from Lemma 2.6, we see that . In this case, we cannot apply Theorem A.3. Consequently, in the case , polynomial or exponential stability of System (2.1)–(2.6) is still an open problem.
3. It is very important to ask the question about the optimality of the observability inequalities (3.12) and (3.47). In our opinion:
From inequity (3.20), we may find a dense subspace of such that
From inequalities (3.37) and (3.39), we may find a dense subspace of such that
Footnotes
Acknowledgements
The authors would like to thank the referees for their valuable comments and useful suggestions which helped to improve the manuscript.
Notions of stability and theorems used
We introduce here the notions of stability that we encounter in this work.
We now look for necessary conditions to show the strong stability of the -semigroup . We will rely on the following result obtained by Arendt and Batty in [12].
For necessary conditions to show the polynomial stability of the -semigroup . We will rely on the frequency domain approach method has been obtained by Batty in [16], Borichev and Tomilov in [18], Liu and Rao in [33].
For well-posedness of our System (1.1)–(1.2) with the boundary conditions (1.3), we recall Theorem 2 stated in [37].
For the observability inequality, we recall Theorem 3.1 stated in [34]. (Theorem 3.1 in [<a class="bibr" data-ref-id="ref034" href="#ref034" id="a-524">34</a>]).
Letandbe Riesz basis of Hilbert spaces X and Y respectively, andandbe Bessel sequences of X and Y with suitably small bounds respectively. DefineThen, we have.
About Lemma 2.5
In this part, under condition , we prove that the System (2.20)–(2.21) has a unique solution . First, the characteristic polynomial of System (2.20) is
Setting
The polynomial has two distinct real roots and given by:
It is clear that and the sign of depends on the value of with respect to . We hence distinguish the three cases: , , and .
Case 1.: then and set
Then P has four distinct roots , , , and the general solution of (2.20) is given by
where , . Using the boundary conditions in (2.21) at and the fact that , we get . Moreover, from the boundary conditions in (2.21) at and the fact that , we get
yielding that . Therefore, System (2.20)–(2.21) admits only the zero and the proof of the lemma is complete.
Case 2.: in this case , and one gets that
Then P has two simple roots , , and 0 as a double root. Hence the general solution of (2.20) is
where , . From the boundary conditions in (2.21) at , we get . Moreover, from boundary conditions in (2.21) at , we get
Assume first that . It follows that
Therefore, after choosing , one gets that
which contradicts . Hence . It implies that and . Consequently and one gets the conclusion.
Case 3.: then and set
Then P has again four distinct roots , , , . The general solution of (2.20) is given by
where , . Using boundary conditions in (2.21) at and the fact that , we get . Assume that and . It follows that
From (B.1) and (B.2), we get
therefore, we get
which contradicts . Hence, or . Using boundary conditions in (2.21) at , we can easily check that . Consequently, and the conclusion follows.
About Proposition 2.11
In this part, we use asymptotic expansion in , to prove (2.36) if and (2.37)–(2.42) if . First, we recall Lemma 2.1 stated in [17].
Our first aim is to show (
2.36
). For this aim, let , then using the asymptotic expansion in (2.44), we get
From (C.1), we get
Next, inserting (C.1) in (2.45), we get
On the other hand, we have
From Lemma C.1 and Equation (C.4), we get
Therefore, from (C.2), (C.3) and (C.5), we get
Let λ be a large eigenvalue of , then from (C.6), λ is a large root of the following asymptotic equation
where
Note that and remains bounded in the strip . The roots of are given by
Finally, with the help of Rouché’s theorem, there exists large enough, such that (), the large roots of h, denoted by , , are close to those of , that is
Consequently, we get (2.36).
Our next aim is to show (
2.37
)–(
2.42
). For this aim, let , then from (2.44), (2.45), and Lemma C.1, we get
and
Hence, we have
We divide the proof into four steps:
Step 1. In this step, we prove that the eigenvalues of , are roots of the following function
First, using the asymptotic expansion in (C.7), we get
From (C.10), we get
Next, using the asymptotic expansion, we get
Consequently, we get
Inserting (C.11) and (C.12) in (C.8), then using the fact that
we get
Equation (2.43) admits a non trivial solution if and only if , i.e., if and only if the eigenvalues of are roots of the function H. Hence, we get (C.9).
Step 2. We look at the roots of . First, from (C.10), we have
From (C.13) and using the fact that is bounded, we get
Next, substituting (C.14) in (C.9), we get
Indeed, using Rouché’s theorem, and the asymptotic Equation (C.15), it is easy to see that the large roots of (denoted by and ) are simple and close to those of , i.e., there exists , such that for all integers , we have
Step 3. We seek to determine . Inserting (C.16) in (C.15), we get
On the other hand, since , we have the asymptotic expansion
Inserting (C.19) in (C.18), then using the fact that , we get
We distinguish two cases:
Case 1. There exists no integer such that . Then, , therefore, from (C.20), we get
substituting the above equation in (C.16), we get the estimates (2.37) and (2.41).
Case 2. If there exists such that . Then, and , therefore, from (C.20), we get
Inserting the above equation in (C.16), we get
since in this case the real part of still does not appear, we need to increase the order of the finite expansion. So, in order to complete the proof of (2.39), we need to show that
For this aim, inserting (C.21) in (C.7), then using the asymptotic expansion, we get
Using the fact that is bounded, we get
Inserting (C.21) and (C.22) in (C.9), then using the asymptotic expansion, we get
Consequently, we obtain
Substituting the above equation in (C.21), we get
again the real part of still does not appear, so we need to increase the order of the finite expansion. For this aim, inserting (C.23) in (C.7) and using the asymptotic expansion, we get
Therefore, we have
Inserting (C.23) and (C.24) in (C.9), then using the asymptotic expansion, we get
Consequently, we obtain
Inserting the above equation in (C.23), we get the estimate (2.39).
Step 4. We seek to determine . Inserting (C.17) in (C.15), we get
On the other hand, since , using the asymptotic expansion in (C.25), we get
We distinguish two cases:
Case 1. There exists no integer such that . Then, , therefore, from (C.26), we get
Inserting the above equation in (C.17), we get the estimates (2.38) and (2.40).
Case 2. If there exists such that . Then, and , therefore, from (C.26), we get
Substituting the above equation into (C.17), we get
since in this case the real part of still does not appear, we need to increase the order of the finite expansion. Inserting (C.27) in (C.7) and using the asymptotic expansion, we get
Therefore, we have
Inserting (C.27) and (C.28) in (C.9), then using the asymptotic expansion, we get
Consequently, we get
Inserting the above equation in (C.27), we get
again the real part of still does not appear, so we need to increase the order of the finite expansion. For this aim, inserting (C.29) in (C.7) and using the asymptotic expansion, we get
Therefore, we have
Inserting (C.29) and (C.30) in (C.9), then using the asymptotic expansion, we get
Consequently, we get
Finally, inserting the above equation into (C.29), we get (2.42).
About Lemma 2.16
Our first aim is to prove Equation (
2.71
). Since the eigenvalues of B are simple, then B is a diagonalizable matrix; i.e., B can be written as
such that
and
where and are defined in (2.70). Therefore, for all , we have , hence we get (2.71).
Our second aim is to prove Equation (
2.72
). First, from (2.71), we have
We distinguish two cases:
Case 1. If , using asymptotic expansion in (2.70), we get
Inserting (D.2) in (D.1), then using asymptotic expansion, (2.48) and the fact that , we get
On the other hand, since , , using by parts integration, we get
where and . From (D.4), (2.48) and (D.2), we get
Substituting (D.5) in (D.3), we get (2.72).
Case 2. If , using asymptotic expansion in (2.70), we get
Inserting (D.6) in (D.1), then using asymptotic expansion, (2.48) and the fact that , we get
Similar to case 1, from (D.4), (2.48) and (D.6), we get (D.5). Finally, substituting (D.5) in (D.7), we get (2.72).
Our third aim is to prove Equations (
2.73
) and (
2.74
). Inserting Equations (2.69), (2.71) and (2.72) in (2.68), we get
We have
On the other hand, from (D.2) and (D.6), we get
Inserting, (D.9) and (D.10) in (D.8), then using the fact that , we get (2.73) and (2.74).
Proof of Theorem 3.19
For the proof of Theorem 3.19, first, we will prove that the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5) admits a unique solution. For this aim, let be a solution of (3.1)–(3.2) and let . After multiplying first and second equation of (1.1) by and , respectively, and integrating their sum over (where ), we get
We introduce the linear form by
From (E.1), we obtain a weak formulation of the Timoshenko system (1.1)–(1.2) with the boundary conditions (1.5)
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