Loss of Exponential Stability for a Delayed Timoshenko System Symmetrically in Both Viscoelasticity and Fractional Boundary Controls

Abstract

The stability analysis of Timoshenko beam systems that incorporate delays and fractional boundary controls is a complex area of study in the field of viscoelasticity. Our study aims to balance the symmetric influence of internal viscoelastic damping and boundary fractional damping in a structured way. The goal is to establish a system where both effects contribute symmetrically to the overall stability and dynamics. In this paper, we study the stability of certain hyperbolic evolution problems, in particular, a Timoshenko system in viscoelasticity with fractional time delay and fractional boundary controls. We prove, under assumptions on the data, the lack of exponential stability decay rate when $\eta \ge 0$ and polynomial stability decay rate when $\eta >0$ using energy methods.

1. Introduction and Statement of the Problem

The Timoshenko beam theory is a mathematical model that improves on the Euler–Bernoulli beam theory by accounting for shear deformation and rotational inertia effects. It is widely used in structural mechanics to describe the behavior of beams subjected to dynamic loads, vibrations, and wave propagation. It is governed by a system of coupled partial differential equations that describe both transverse displacement $\phi (x,t)$ and rotational angle $\varphi (x,t)$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{({\phi }_x+\varphi )}_x=f\left(\phi \right),\hfill \\ {\rho }_2{\varphi }_{tt}-b{\varphi }_{xx}+k({\phi }_x+\varphi )=f\left(\varphi \right),\hfill \end{array}\right.\end{array}$$

(1)

where $f\left(\phi \right),f\left(\varphi \right)$ are external forces. It is widely studied, and many quantitative/qualitative results have been obtained in the last decades. Timoshenko systems with indefinite damping are considered in [1]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{({\phi }_x+\varphi )}_x=0,\hfill \\ {\rho }_2{\varphi }_{tt}-b{\varphi }_{xx}+k({\phi }_x+\varphi )+a\left(x\right){\varphi }_t=0,\hfill \end{array}\right.\end{array}$$

(2)

A precise description of the decay rate properties has given in the literature (see [2,3,4]). For Timoshenko systems in viscoelasticity, in [5], with nonlinear damping, it is considered as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\phi }_{tt}-{({\phi }_x+\varphi )}_x=0,\hfill \\ {\varphi }_{tt}-{\varphi }_{xx}+({\phi }_x+\varphi )+\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds+{|{\varphi }_t|}^{m\left(x\right)-2}{\varphi }_t=0,\hfill \end{array}\right.\end{array}$$

(3)

The interaction between the two different types of damping terms is based an explicit decay rate of solutions established by the author; previous results generalized that in [6]. The well posedness of a linear Timoshenko beam system with infinite memory is obtained by means of the resolvent family theory in [7]. In addition, exponential stability is proven, where the memory kernel under consideration is assumed to be non-monotonic (see [8,9]). In [10], a rod with an attached oscillator was considered for which the equations of motion took into account the inertia of the rotational motion of the rod. The Timoshenko beam is understood as a rod, the description of which takes into account transverse shear and inertia of section rotation. For the constructed model, the possibility of generalizing the approach to construct the frequency equation developed in many works has been investigated (see [11]). The paper [12] presents a numerical method for non-linear analysis of vibrations of Timoshenko pipes that transport liquid. The limit cycle and the associated vibration amplitude for the flow-induced vibration problem were discussed. An analysis is performed to evaluate the influence of the flow velocity and the liquid/pipe mass ratio on the amplitude of the limit cycle oscillation. The dynamics and stability of short pipes that transport liquid are considered using the Timoshenko beam theory for a pipe [13]. It has been shown that a refined Timoshenko beam theory is necessary for an adequate description of the dynamic behavior of extremely short pipes. The trend in the development of this dynamic type of research is towards more complex mathematical models describing the oscillatory processes with different damping terms. Several theoretical studies are devoted to the mathematical modeling of elastic oscillations. However, modeling of the Timoshenko beam theory taking into account the viscoelastic properties of the material of the structures with delay has been considered in relatively few works. In [14], a mathematical model of viscoelastic pipes transporting liquid was developed. When considering elastic systems, the internal friction of the material is taken into account using the Kelvin–Voigt model. The dimensionless equation of transverse motion and the associated classical and non-classical boundary conditions are obtained using the variational approach. Currently, the problem of a Timoshenko beam in a viscoelastic material is of great theoretical and applied interest. To date, many approaches have been developed to solve such problems, but none of them can adequately reflect the real effects of the interaction between the delay and the fractional boundary controls. Basically, these approaches describe individual stages of the processes that occur in the pipeline. Therefore, the construction of mathematical models that allow the study of dynamic processes in viscoelasticity with delay is of undoubted scientific and practical interest.

From the above review, we can conclude that the development of dynamic models made of composite materials with different damping terms, taking into account the fractional boundary controls, is a rather complex and relevant research task. In this article, we discuss the lack of exponential stability decay rate when $\eta \ge 0$ and polynomial stability decay rate when $\eta >0$ of solution of some hyperbolic systems like uni-dimensional Timoshenko system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{({\phi }_x+\varphi )}_x=0,\hfill \\ {\rho }_2{\varphi }_{tt}-b{\varphi }_{xx}+k({\phi }_x+\varphi )+\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds+\gamma {\partial }_t^{\alpha ,\eta }\varphi (x,t-\tau )=0,\hfill \end{array}\right.\end{array}$$

(4)

where $(x,t)\in (0,L)\times (0,\infty )$ and $\gamma ,{\rho }_1,{\rho }_2,k,b$ are positive constants, $\phi =\phi (x,t)$ denotes vertical displacements, and $\varphi =\varphi (x,t)$ denotes shear angle displacement. To this system, we add the boundary conditions with fractional boundary control

$$\left\{\begin{array}{cc}\phi (0,t)=\phi (L,t)=\varphi (0,t)=\varphi (L,t)=0,\hfill & t\in (0,\infty )\hfill \\ k{\phi }_x(L,t)=-{\gamma }_1{\partial }_t^{\alpha ,\eta }\phi (L,t),\hfill & t\in (0,\infty )\hfill \\ b{\varphi }_x(L,t)=-{\gamma }_2{\partial }_t^{\alpha ,\eta }\varphi (L,t),\hfill & t\in (0,\infty ).\hfill \end{array}\right.$$

(5)

Here, ${\partial }_t^{\alpha ,\eta }$ represents the generalized Caputo’s fractional derivative of order $\alpha$ with respect to t, it is given in [15] as following

$${\partial }_t^{\alpha ,\eta }\omega \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}\underset{0}{\overset{t}{\int }}{(t-s)}^{-\alpha }{e}^{-\eta (t-s)}{\displaystyle \frac{d\omega }{ds}}\left(s\right)ds, \eta \ge 0.$$

In the present work, we investigate two different types of dissipative effects at the boundary where ${\gamma }_1>0$ and ${\gamma }_2>0$, and the initial conditions

$$\left\{\begin{array}{c}\phi (x,0)={\phi }_o, {\phi }_t(x,0)={\phi }_1,\hfill \\ \varphi (x,0)={\varphi }_o, {\varphi }_t(x,0)={\varphi }_1,\hfill \\ {\varphi }_t(x,t-\tau )=f_o(x,t-\tau ).\hfill \end{array}\right.$$

(6)

Here, $({\phi }_o,{\phi }_1,{\varphi }_o,{\varphi }_1)$ is the initial dad; it belongs to a suitable function space, $f_o$ is prescribed, and the time delay is given by ${\varphi }_t(x,t-\tau )$ where $\tau$ is a positive constant. It is well known that although Timoshenko beam systems with certain damping mechanisms can achieve exponential stability, the introduction of fractional boundary controls and delays may alter this behavior, leading to a lack of exponential and polynomial stability. Understanding these dynamics is essential for designing and controlling viscoelastic systems in engineering applications.

This paper is organized as follows. In the first section after the introduction, we present some preliminary remarks. In the third and fourth sections, we prove the lack of exponential stability and polynomial stability of the system using energy methods.

2. Preliminary and Augmented System

In this section, we state some necessary assumptions to present our main results. Throughout this paper, the standard Lebesgue space $L^2(0,L)$ and the Sobolev space $H^1(0,L)$ are used. We introduce the energy space

$$\mathcal{H}={\left(H^1(0,L)\times L^2(0,L)\times L^2(-\infty ,+\infty )\right)}^2\times L^2(0,L),$$

and define the following new variable z by

$$z(x,\rho ,t)={\varphi }_t(x,t-\tau \rho ), x\in (0,L), t>0, \rho \in (0,1).$$

Then

$$\begin{array}{c}\hfill \begin{array}{ccccc}{z}_t(x,\rho ,t)& =& {\displaystyle \frac{\partial {\varphi }_t(x,t-\rho \tau )}{\partial (t-\rho \tau )}}*{\displaystyle \frac{\partial (t-\rho \tau )}{\partial t}}& =& {\displaystyle \frac{\partial {\varphi }_t(x,t-\rho \tau )}{\partial (t-\rho \tau )}},\\ \\ z_{\rho }(x,\rho ,t)& =& {\displaystyle \frac{\partial {\varphi }_t(x,t-\rho \tau )}{\partial (x,t-\rho \tau )}}*{\displaystyle \frac{\partial (t-\rho \tau )}{\partial \rho }}& =& -\tau {\displaystyle \frac{\partial {\varphi }_t(x,t-\rho \tau )}{\partial (t-\rho \tau )}}.\end{array}\end{array}$$

(7)

It is not hard to see

$$\tau z_t(x,\rho ,t)+z_{\rho }(x,\rho ,t)=0, (x,\rho ,t)\in (0,L)\times (0,1)\times (0,\infty ).$$

(8)

Then, system (4) are transformed to

$$\left\{\begin{array}{cc}{\rho }_1{\phi }_{tt}-k{({\phi }_x+\varphi )}_x=0,\hfill & \\ {\rho }_2{\varphi }_{tt}-b{\varphi }_{xx}+k({\phi }_x+\varphi )+\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds+\gamma {\partial }_t^{\alpha ,\eta }\varphi (x,t-\tau )=0,\hfill & \\ \tau z_t(x,\rho ,t)+z_{\rho }(x,\rho ,t)=0,\hfill \end{array}\right.$$

(9)

with $x\in (0,L)$, $\rho \in (0,1)$, and $t\in (0,\infty )$, the initial and boundary conditions are

$$\left\{\begin{array}{cc}\phi (L,t)=\phi (0,t)=0, \varphi (L,t)=\varphi (0,t)=0,\hfill & t>0\hfill \\ \phi (x,0)={\phi }_o,{\phi }_t(x,0)={\phi }_1,\varphi (x,0)={\varphi }_o,\varphi {(x,0)}_t={\varphi }_1,\hfill & x\in (0,L)\hfill \\ z(x,\rho ,0)=f_o(x,-\rho \tau ), z(x,0,t)={\varphi }_t(x,t),\hfill \end{array}\right.$$

We reformulate the model (4) into an augmented system; to this end, we need the following theorem.

Theorem 1

([16]). Define the function ϑ as follows:

$$\begin{array}{cc}\vartheta \left(\nu \right)={\left|\nu \right|}^{(2\alpha -1)/2}& -\infty <\nu <+\infty ,0<\alpha <1.\end{array}$$

Then, the ‘input’ U and the ‘output’ O of the system

$${\partial }_t\phi (\nu ,t)+({\nu }^2+k)\phi (\nu ,t)-U\left(t\right)\vartheta \left(\nu \right)=0,$$

$$\phi (\nu ,0)=0,$$

$$O\left(t\right)={\left(\pi \right)}^{-1}sin\left(\alpha \pi \right)\underset{-\infty }{\overset{+\infty }{\int }}\vartheta \left(\nu \right)\phi (\nu ,t)d\nu ,$$

have a relationship that is given by

$$O=I^{1-\alpha ,\eta }U=D^{\alpha ,\eta }U,$$

where

$$\left[I^{\alpha ,\eta }f\right]\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}{e}^{-\eta (t-s)}f\left(s\right)ds.$$

The following lemma is given by [17], and it is important for our reformulation.

Lemma 1.

Let

$$D_{\eta }=\{\lambda \in \mathbb{C}:Re\lambda +\eta >0\}\cup \{\lambda \in \mathbb{C}:Im\lambda \ne 0\}$$

If $\lambda \in D_{\eta }$, then

$$\underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{{\vartheta }^2\left(\nu \right)}{\lambda +\eta +{\nu }^2}}d\nu ={\displaystyle \frac{\pi }{sin\left(\alpha \pi \right)}}{(\lambda +\eta )}^{\alpha -1}.$$

Putting $\zeta =\gamma {\pi }^{-1}sin\left(\alpha \pi \right)$ and ${\zeta }_i={\gamma }_i{\pi }^{-1}sin\left(\alpha \pi \right);i=1,2$, the system (9) may be recast as the augmented model

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{tt}-k{({\phi }_x+\varphi )}_x=0\hfill \\ {\partial }_t{\chi }_1(\xi ,t)+({\xi }^2+\eta ){\chi }_1(\xi ,t)-{\phi }_t(L,t)\mu \left(\xi \right)=0\hfill \\ {\rho }_2{\varphi }_{tt}-b{\varphi }_{xx}+k({\phi }_x+\varphi )+\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds+\zeta \underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(\xi ,t-\tau )d\xi =0\hfill \\ {\partial }_t{\chi }_2(\xi ,t)+({\xi }^2+\eta ){\chi }_2(\xi ,t)-{\varphi }_t(L,t)\mu \left(\xi \right)=0\hfill \\ \tau z_t(x,\rho ,t)+z_{\rho }(x,\rho ,t)=0\hfill \\ \\ {\varphi }_t(x,t-\tau )=f_o(x,t-\tau ), \rho \in [0,1]\hfill \\ {\chi }_1(\xi ,0)={\chi }_{o1}\left(\xi \right)=0, {\chi }_2(\xi ,0)={\chi }_{o2}\left(\xi \right)=0\hfill \\ z(x,\rho ,0)=f_o(x,-\tau ), z(x,0,t)=f_o(x,t)\hfill \\ \phi \left(0\right)=\phi \left(L\right)=\varphi \left(0\right)=\varphi \left(L\right)=0\hfill \\ \phi (x,0)={\phi }_o, {\phi }_t(x,0)={\phi }_1, \varphi (x,0)={\varphi }_o, {\varphi }_t(x,0)={\varphi }_1\hfill \\ \\ k{\phi }_x(L,t)=-{\zeta }_1\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_1(\xi ,t)d\xi \hfill \\ b{\varphi }_x(L,t)=-{\zeta }_2\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(\xi ,t)d\xi .\hfill \end{array}\right.\end{array}$$

(10)

We introduce two new dependent variables $u={\phi }_t$ and $v={\varphi }_t$ and let

$$U={(\phi ,u,{\chi }_1,\varphi ,v,{\chi }_2,z)}^T,$$

be a solution of system (10). Concerning the relaxation function $\varpi$, we assume that $\varpi :R^{+}\to R^{+}$ is a nondecreasing differentiable function satisfying

$$\begin{array}{c}\hfill \varpi \left(0\right)>0, b-{\int }_0^{\infty }\varpi \left(s\right)ds\ge \ell >0,\end{array}$$

(11)

and

$$\begin{array}{c}\hfill \varpi \left(t\right)\le -\xi {\varpi }^{\prime }\left(t\right), \forall t\ge 0, \xi >0.\end{array}$$

(12)

The following notation will be used

$$(v\circ w)\left(t\right)={\int }_0^{\infty }v(t-s){\parallel w\left(t\right)-w\left(s\right)\parallel }^{2}ds.$$

Definition 1.

The energy $\mathcal{E}\left(t\right)$ associated with the solution of (10) is defined by

$$\begin{array}{cc}\mathcal{E}\left(t\right)\hfill & ={\displaystyle \frac{{\rho }_1}{2}}\parallel {\phi }_t{\parallel }_2^2+{\displaystyle \frac{{\rho }_2}{2}}\parallel {\varphi }_t{\parallel }_2^2+{\displaystyle \frac{k}{2}}\parallel {\phi }_x{+\varphi \parallel }_2^2+{\displaystyle \sum _{i=1}^{i=2}}{\displaystyle \frac{{\zeta }_i}{2}}\parallel {\chi }_i{\parallel }_2^2+{\displaystyle \frac{\tau }{2}}\underset{0}{\overset{1}{\int }}{\parallel z\parallel }_2^{2}d\rho \hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left(b-\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right)ds\right){\parallel {\varphi }_x\parallel }_2^2+{\displaystyle \frac{1}{2}}(\varpi \circ {\varphi }_x).\hfill \end{array}$$

(13)

We suppose that $\varpi$ is a positive function and that ${\varphi }_t^2$ does not increase with respect to t.

Lemma 2.

Let ${(\phi ,{\chi }_1,\varphi ,{\chi }_2,z)}^T$ be a solution of (10); then, we have

$$\begin{array}{c}{\mathcal{E}}^{\prime }\left(t\right)=-{\displaystyle \sum _{i=1}^2}{\zeta }_i\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta ){\chi }_i^2(\xi ,t)d\xi -\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(x,t-\tau )v(x,t)d\xi dx\hfill \\ -{\displaystyle \frac{1}{2}}\underset{0}{\overset{L}{\int }}\left(z^2(x,1,t)-z^2(x,0,t)\right)dx-\underset{0}{\overset{+\infty }{\int }}\varpi (t-\tau ){\varphi }_x(L,\tau )v\left(L\right)d\tau .\hfill \end{array}$$

(14)

Remark 1.

Since

$${\mathcal{E}}^{\prime }\left(t\right)\le 0.$$

Then, system (10) is considered a dissipative system. We will discuss the effect of the dissipative terms on the problem.

3. Lack of Exponential Stability When $\eta \ge 0$

The absence of exponential stability implies that the energy decay over time does not adhere to these rates. This characteristic is crucial for applications requiring precise control over vibration damping and stability, as it affects the predictability and efficiency of the system’s response to disturbances. Here, we show that the $C^o$-semi-group $e^{\mathcal{A}t}$ associated with the system (10) is not exponentially stable.

Theorem 2.

The semigroup generated by the operator $\mathcal{A}$ is not exponentially stable.

Proof. 

We consider the system (10). The abstract formulation of (10) is $\mathcal{A}U=U^{\prime }$

$$\mathcal{A}\left(\begin{array}{c}\phi \hfill \\ u\hfill \\ {\chi }_1\hfill \\ \varphi \hfill \\ v\hfill \\ {\chi }_2\hfill \\ z_t\hfill \end{array}\right)=\left(\begin{array}{c}u\hfill \\ {\displaystyle \frac{k}{{\rho }_1}}{({\phi }_x+\varphi )}_x\hfill \\ -({\xi }^2+\eta ){\chi }_1+u\left(L\right)\mu \left(\xi \right)\hfill \\ v\hfill \\ {\displaystyle \frac{b}{{\rho }_2}}{\varphi }_{xx}-{\displaystyle \frac{k}{{\rho }_2}}({\phi }_x+\varphi )-{\displaystyle \frac{1}{{\rho }_2}}\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds-{\displaystyle \frac{\zeta }{{\rho }_2}}\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(\xi ,t-\tau )d\xi \hfill \\ -({\xi }^2+\eta ){\chi }_2+v\left(L\right)\mu \left(\xi \right)\hfill \\ -{\displaystyle \frac{1}{\tau }}{z}_{\rho }(x,\rho ,t),\hfill \end{array}\right)$$

(15)

the domain $\mathcal{D}\left(A\right)$ is defined by

$$\mathcal{D}\left(\mathcal{A}\right)=\left\{\begin{array}{c}U\in \mathcal{H}:\phi ,\varphi \in H^2(0,L)\cap H_L^1(0,L),u,v\in H_L^1(0,L),z\in L^2(0,L)\hfill \\ -({\xi }^2+\eta ){\chi }_1+u\left(L\right)\mu \left(\xi \right)\in L^2\left(\mathbb{R}\right),-({\xi }^2+\eta ){\chi }_2+v\left(L\right)\mu \left(\xi \right)\in L^2\left(\mathbb{R}\right),\hfill \\ k{\phi }_x(L,t)+{\zeta }_1\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_1(\xi ,t)d\xi =0,\hfill \\ b{\varphi }_x(L,t)+{\zeta }_2\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(\xi ,t)d\xi =0,\hfill \\ |\xi |{\chi }_i\in L^2\left(\mathbb{R}\right),i=1,2.\hfill \end{array}\right\}$$

Let ${\mathcal{H}}_1$ be the subspace of $\mathcal{H}$ defined by

$${\mathcal{H}}_1=\left\{U\in \mathcal{H}:U=(\phi ,u,{\chi }_1,0,0,0,0)\right.\}$$

and ${\mathcal{A}}_1=\mathcal{A}/{\mathcal{H}}_1$. Observe that the generator $\mathcal{A}$ becomes the operator ${\mathcal{A}}_1$ defined by

$$\mathcal{D}\left({\mathcal{A}}_1\right)=\left\{\begin{array}{c}U=(\phi ,u,{\chi }_1,0,0,0,0)\in {\mathcal{H}}_1:\phi \in H^2(0,L)\cap H_L^1(0,L),u\in H_L^1(0,L),\hfill \\ -({\xi }^2+\eta ){\chi }_1+u\left(L\right)\mu \left(\xi \right)\in L^2\left(\mathbb{R}\right),\hfill \\ k{\phi }_x(L,t)+{\zeta }_1\underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_1(\xi ,t)d\xi =0,\hfill \\ |\xi |{\chi }_1\in L^2\left(\mathbb{R}\right).\hfill \\ \phi \left(L\right)=0,\hfill \end{array}\right\}$$

and

$${\mathcal{A}}_1{U}_1={\displaystyle \frac{d}{dt}}{U}_1,$$

(16)

with $U_1={(\phi ,u,{\chi }_1,0,0,0,0)}^T\in \mathcal{D}\left({\mathcal{A}}_1\right)$, $U_1^{\prime }={(u,{\phi }_{tt},{\chi }_{1t},0,0,0,0)}^T$ then

$${\mathcal{A}}_1\left(\begin{array}{c}\phi \hfill \\ u\hfill \\ {\chi }_1\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \end{array}\right)=\left(\begin{array}{c}u\hfill \\ {\displaystyle \frac{k}{{\rho }_1}}{\phi }_{xx}\hfill \\ -({\xi }^2+\eta ){\chi }_1+u\left(L\right)\mu \left(\xi \right)\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \end{array}\right).$$

(17)

Let $\lambda$ be an eigenvalue of ${\mathcal{A}}_1$ with associated eigenvector U; then,

$${\mathcal{A}}_{1}U=\lambda U,$$

is equivalent to

$$\left\{\begin{array}{c}\lambda \phi -u=0,\hfill \\ \lambda u-{\displaystyle \frac{k}{{\rho }_1}}{\phi }_{xx}=0\hfill \\ (\lambda +{\xi }^2+\eta ){\chi }_1\left(x\right)-u\left(L\right)\mu \left(\xi \right)=0.\hfill \end{array}\right.$$

(18)

With $u\left(L\right)=\lambda \phi \left(L\right)=0$, we obtain

$$\left\{\begin{array}{c}{\lambda }^2\phi -{\displaystyle \frac{k}{{\rho }_1}}{\phi }_{xx}=0\hfill \\ (\lambda +{\xi }^2+\eta ){\chi }_1\left(x\right)=0\hfill \end{array}\right.\iff \left\{\begin{array}{c}{\lambda }^2\phi -{\displaystyle \frac{k}{{\rho }_1}}{\phi }_{xx}=0\hfill \\ {\chi }_1\left(x\right)=0\Rightarrow {\phi }_x\left(L\right)=0.\hfill \end{array}\right.$$

(19)

with the following conditions

$$\left\{\begin{array}{c}{\phi }_x\left(L\right)=0\hfill \\ \phi \left(L\right)=0.\hfill \end{array}\right.$$

(20)

We set $X={(\phi ,{\phi }_x)}^T$, and we have the following system

$${\displaystyle \frac{d}{dx}}X=\tilde{B}X.$$

(21)

It is equivalent to

$$\left(\begin{array}{c}{\phi }_x\\ {\phi }_{xx}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ {\displaystyle \frac{{\lambda }^2{\rho }_1}{k}}& 0\end{array}\right)\left(\begin{array}{c}\phi \\ {\phi }_x\end{array}\right).$$

(22)

We remark that if $|\tilde{B}|={\displaystyle \frac{-{\lambda }^2{\rho }_1}{k}}\ne 0$, then $\tilde{B}$ is not singular. The characteristic polynomial of $\tilde{B}$ is

$$|\tilde{B}-SI|=0\iff \left|\left(\begin{array}{cc}-S& 1\\ {\displaystyle \frac{{\lambda }^2{\rho }_1}{k}}& -S\end{array}\right)\right|=S^2-{\displaystyle \frac{{\lambda }^2{\rho }_1}{k}}=0\iff S=\pm \lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}.$$

(23)

We set $t_1\left(\lambda \right)=\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}$ and $t_2\left(\lambda \right)=-t_1\left(\lambda \right)$ The solution $\phi$ is given by

$$\phi \left(x\right)=\sum _{i=1}^2{C}_i{e}^{t_{i}x}.$$

(24)

We have ${\phi }_x\left(x\right)={\displaystyle \sum _{i=1}^2}{t}_i\left(\lambda \right)C_i{e}^{t_i\left(\lambda \right)x}$, ${\phi }_x\left(L\right)=0$, and $\phi \left(L\right)=0$; then, we obtain

$$\left(\begin{array}{cc}1& 1\\ t_1\left(\lambda \right)e^{t_1\left(\lambda \right)L}& t_2\left(\lambda \right)e^{t_2\left(\lambda \right)L}\end{array}\right)\left(\begin{array}{c}{C}_1\\ C_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

We calculate the determinant of the last system, and we obtain

$$f\left(\lambda \right)=-\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}{e}^{-L\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}}\left(1+e^{2L\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}}\right)=0,$$

(25)

is equivalent to

$$\left\{\begin{array}{cc}1+e^{2L\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}}=0,\hfill & \\ \\ \lambda =0,\hfill \end{array}\right.$$

⇔

$$\left\{\begin{array}{cc}{e}^{2L\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}}=-1,\hfill & \\ \\ \lambda =0,\hfill \end{array}\right.$$

gives

$$\left\{\begin{array}{cc}2L\lambda \sqrt{{\displaystyle \frac{{\rho }_1}{k}}}=i(2K+1)\pi ,\hfill & K\in Z\hfill \\ \\ \lambda =0,\hfill \end{array}\right.$$

⇔

$$\left\{\begin{array}{cc}{\lambda }_K={\displaystyle \frac{i(2K+1)\pi \sqrt{k}}{2L\sqrt{{\rho }_1}}},\hfill & K\in Z\hfill \\ \\ \lambda =0,\hfill \end{array}\right.$$

So, if $Re\left({\lambda }_K\right)=0$, then the operator ${\mathcal{A}}_1$ has non-exponential decaying branches of eigenvalues. As a result, the operator $\mathcal{A}$ is not exponentially stable. □

4. Polynomial Stability When $\eta >0$

Theorem 3.

$\eta >0$, the semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ is polynomially stable and

$$\parallel S_{\mathcal{A}}\left(t\right)U_o{\parallel }_{\mathcal{H}}\le {\displaystyle \frac{1}{t^{\frac{1}{2-2\alpha }}}}{\parallel U_o\parallel }_{\mathcal{D}\left(\mathcal{A}\right)}.$$

Proof. 

We will need to study the resolvent equation

$$(i\lambda -A)U=F, \forall \lambda \in \mathbb{R}.$$

$$\left\{\begin{array}{cc}\hfill & i\lambda \phi -u=f_1,\hfill \\ \hfill & i\lambda {\rho }_{1}u-k{({\phi }_x+\varphi )}_x={\rho }_1{f}_2\hfill \\ \hfill & (i\lambda +{\xi }^2+\eta ){\chi }_1-u\left(L\right)\mu \left(\xi \right)=f_3\hfill \\ \hfill & i\lambda \varphi -v=f_4,\hfill \\ \hfill & i\lambda {\rho }_{2}v-b{\varphi }_{xx}+({\phi }_x+\varphi )+\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)ds+\zeta \underset{-\infty }{\overset{+\infty }{\int }}\mu \left(\xi \right){\chi }_2(\xi ,t-\tau )d\xi ={\rho }_2{f}_5,\hfill \\ \hfill & (i\lambda +{\xi }^2+\eta ){\chi }_2-v\left(L\right)\mu \left(\xi \right)=f_6\hfill \\ \hfill & i\tau \lambda z+z_{\rho }(x,\rho ,t)=\tau f_7.\hfill \end{array}\right.$$

(26)

The inner product in $\mathcal{H}$ is

$$\begin{array}{ccc}\hfill {\langle U,\tilde{U}\rangle }_{\mathcal{H}}& =& -\underset{0}{\overset{L}{\int }}\underset{0}{\overset{+\infty }{\int }}\varpi (t-s){\varphi }_x\left(s\right){\tilde{\varphi }}_x\left(t\right)dsdx+\sum _{i=1}^2{\zeta }_i\underset{-\infty }{\overset{+\infty }{\int }}{\chi }_i(\xi ,t)\widetilde{{\chi }_i}(\xi ,t)d\xi \hfill \\ & +& \tau \underset{0}{\overset{L}{\int }}\underset{0}{\overset{1}{\int }}z(x,\rho ,t)\tilde{z}(x,\rho ,t)d\rho dx\hfill \\ & +& \underset{0}{\overset{L}{\int }}({\rho }_{1}u\tilde{u}+{\rho }_{2}v\tilde{v}+k({\phi }_x+\varphi )({\tilde{\phi }}_x+\tilde{\varphi })+b{\varphi }_x{\tilde{\varphi }}_x\hfill \\ & +& \underset{0}{\overset{+\infty }{\int }}\varpi (t-s)({\varphi }_x\left(t\right)-{\varphi }_x\left(s\right))({\tilde{\varphi }}_x\left(t\right)-{\tilde{\varphi }}_x\left(s\right))ds)dx.\hfill \end{array}$$

We have

$$|\mathcal{R}e\langle AU,U\rangle |=|\mathcal{R}e\langle (i\lambda -A)U,U\rangle |\le {\parallel U\parallel }_{\mathcal{H}}.{\parallel F\parallel }_{\mathcal{H}}.$$

From (26)₃, we obtain

$$u\left(L\right)\mu \left(\xi \right)=(i\lambda +{\xi }^2+\eta ){\chi }_1-f_3.$$

Let us multiplying this last equation by ${(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)$, we obtain

$$\begin{array}{c}\hfill {(i\lambda +{\xi }^2+\eta )}^{-1}u\left(L\right){\mu }^2\left(\xi \right)=\mu \left(\xi \right){\chi }_1-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_3\left(\xi \right),\end{array}$$

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{+\infty }{\int }}|{(i\lambda +{\xi }^2+\eta )}^{-1}u\left(L\right){\mu }^2\left(\xi \right)|d\xi =\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_1-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_3\left(\xi \right)|d\xi ,\end{array}$$

$$\begin{array}{c}\hfill |u\left(L\right)|\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}|\mu \left(\xi \right){|}^{2}d\xi =\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_1-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_3\left(\xi \right)|d\xi ,\end{array}$$

$$\begin{array}{c}\hfill |u\left(L\right)|S=\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_1-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_3\left(\xi \right)|d\xi ,\end{array}$$

with

$$S=\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}{|\mu \left(\xi \right)|}^{2}d\xi$$

$$\begin{array}{cc}|u(L)|S\hfill & \le \underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right)||{\chi }_1|d\xi +\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}|\mu \left(\xi \right)||f_3\left(\xi \right)|d\xi \hfill \\ \hfill & \le \underset{-\infty }{\overset{+\infty }{\int }}{({\xi }^2+\eta )}^{\pm {\displaystyle \frac{1}{2}}}|\mu \left(\xi \right)||{\chi }_1|d\xi +\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}|\mu \left(\xi \right)||f_3\left(\xi \right)|d\xi .\hfill \end{array}$$

Using the Cauchy–Schwartz’s inequality, we obtain

$$\begin{array}{cc}|u(L)|S\hfill & \le \sqrt{\underset{-\infty }{\overset{+\infty }{\int }}{({\xi }^2+\eta )}^{-1}{|\mu \left(\xi \right)|}^{2}d\xi }\sqrt{\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta ){|{\chi }_1|}^{2}d\xi }\hfill \\ \hfill & +\sqrt{\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-2}{|\mu \left(\xi \right)|}^{2}d\xi }\sqrt{\underset{-\infty }{\overset{+\infty }{\int }}{|f_3\left(\xi \right)|}^{2}d\xi }.\hfill \end{array}$$

(27)

We obtain

$$\begin{array}{c}|u\left(L\right)|S\le \mathcal{U}(\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_1{|}^2{d\xi )}^{{\displaystyle \frac{1}{2}}}+\mathcal{V}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){{|}^{2}d\xi )}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

with

$$\mathcal{U}=(\underset{-\infty }{\overset{+\infty }{\int }}{({\xi }^2+\eta )}^{-1}|\mu \left(\xi \right){|}^2{d\xi )}^{{\displaystyle \frac{1}{2}}} \mathrm{and} \mathcal{V}=(\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-2}|\mu \left(\xi \right){{|}^{2}d\xi )}^{{\displaystyle \frac{1}{2}}}.$$

We have

$$\begin{array}{ccc}\hfill {|u\left(L\right)|}^2{S}^2& \le & {\mathcal{U}}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta ){|{\chi }_1|}^{2}d\xi \hfill \\ & +& {\mathcal{V}}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_3{\left(\xi \right)|}^{2}d\xi )+2\mathcal{U}\mathcal{V}(\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_1{|}^2{d\xi )}^{{\displaystyle \frac{1}{2}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){{|}^{2}d\xi )}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Using $2PQ\le P^2+Q^2; if P\ge 0, Q\ge 0$, we have

$$\begin{array}{c}{|u\left(L\right)|}^2{S}^2\le 2{\mathcal{U}}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_1{|}^{2}d\xi +2{\mathcal{V}}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^{2}d\xi .\hfill \end{array}$$

(28)

We obtain

$$\begin{array}{c}\hfill {|u\left(L\right)|}^2\le 2{\left({\displaystyle \frac{\mathcal{U}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_1{|}^{2}d\xi +2{\left({\displaystyle \frac{\mathcal{V}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^{2}d\xi \end{array}$$

Using the same method as in the sixth equation of (26), we have

$$v\left(L\right)\mu \left(\xi \right)=(i\lambda +{\xi }^2+\eta ){\chi }_2-f_6.$$

Let us multiplying this last equation by ${(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)$, we obtain

$$\begin{array}{c}\hfill {(i\lambda +{\xi }^2+\eta )}^{-1}v\left(L\right){\mu }^2\left(\xi \right)=\mu \left(\xi \right){\chi }_2-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_6\left(\xi \right),\end{array}$$

$$\begin{array}{c}\hfill \underset{-\infty }{\overset{+\infty }{\int }}|{(i\lambda +{\xi }^2+\eta )}^{-1}v\left(L\right){\mu }^2\left(\xi \right)|d\xi =\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_2-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_6\left(\xi \right)|d\xi ,\end{array}$$

$$\begin{array}{c}\hfill |v\left(L\right)|\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}|\mu \left(\xi \right){|}^{2}d\xi =\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_2-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_6\left(\xi \right)|d\xi ,\end{array}$$

$$\begin{array}{c}\hfill |v\left(L\right)|S=\underset{-\infty }{\overset{+\infty }{\int }}|\mu \left(\xi \right){\chi }_2-{(i\lambda +{\xi }^2+\eta )}^{-1}\mu \left(\xi \right)f_6\left(\xi \right)|d\xi ,\end{array}$$

with

$$S=\underset{-\infty }{\overset{+\infty }{\int }}|(i\lambda +{\xi }^2+\eta ){|}^{-1}{|\mu \left(\xi \right)|}^{2}d\xi .$$

We obtain the same result:

$$\begin{array}{c}{|v\left(L\right)|}^2\le 2{\left({\displaystyle \frac{\mathcal{U}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_2{|}^{2}d\xi +2{\left({\displaystyle \frac{\mathcal{V}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_6\left(\xi \right){|}^{2}d\xi \hfill \end{array}.$$

Finally

$$\begin{array}{c}{|u\left(L\right)|}^2\le 2{\left({\displaystyle \frac{\mathcal{U}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_1{|}^{2}d\xi +2{\left({\displaystyle \frac{\mathcal{V}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^{2}d\xi \hfill \\ {|v\left(L\right)|}^2\le 2{\left({\displaystyle \frac{\mathcal{U}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_2{|}^{2}d\xi +2{\left({\displaystyle \frac{\mathcal{V}}{S}}\right)}^2\underset{-\infty }{\overset{+\infty }{\int }}|f_6\left(\xi \right){|}^{2}d\xi .\hfill \end{array}$$

(29)

We have

$$\begin{array}{ccccc}S& =& \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{\mu {\left(\xi \right)}^2}{(|\lambda |+{\xi }^2+\eta )}}d\xi & =& {\displaystyle \frac{\pi }{sin\alpha \pi }}{(|\lambda |+\eta )}^{\alpha -1}\\ {\mathcal{U}}^2& =& \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{\mu {\left(\xi \right)}^2}{({\xi }^2+\eta )}}d\xi & =& {\displaystyle \frac{\pi }{sin\alpha \pi }}{\eta }^{\alpha -1}\\ {\mathcal{V}}^2& =& \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{\mu {\left(\xi \right)}^2}{(|\lambda |+{\xi }^2{+\eta )}^2}}& \le & \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{\mu {\left(\xi \right)}^2}{(|\lambda |+{\xi }^2+\eta )}}={\displaystyle \frac{\pi }{sin\alpha \pi }}{(|\lambda |+\eta )}^{\alpha -1}.\end{array}$$

We have

$${|u\left(L\right)|}^2+{|v\left(L\right)|}^2\le 2{\left({\displaystyle \frac{\mathcal{U}}{S}}\right)}^2\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_i{|}^{2}d\xi +2{\left({\displaystyle \frac{\mathcal{V}}{S}}\right)}^2(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^2+|f_6\left(\xi \right){|}^2)d\xi ,$$

$$\begin{array}{ccc}\hfill {|u\left(L\right)|}^2+{|v\left(L\right)|}^2& \le & {\displaystyle \frac{2{\eta }^{\alpha -1}sin\alpha \pi }{\pi {(|\lambda |+\eta )}^{2\alpha -2}}}\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta ){|{\chi }_i|}^{2}d\xi \hfill \\ & & +{\displaystyle \frac{2sin\alpha \pi }{\pi {(|\lambda |+\eta )}^{\alpha -1}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3{\left(\xi \right)|}^2+|f_6\left(\xi \right){|}^2)d\xi ,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {|u\left(L\right)|}^2+{|v\left(L\right)|}^2& \le & {\displaystyle \frac{2{\eta }^{\alpha -1}}{\pi {(|\lambda |+\eta )}^{2\alpha -2}}}\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta ){|{\chi }_i|}^{2}d\xi \hfill \\ & & +{\displaystyle \frac{2}{\pi {(|\lambda |+\eta )}^{\alpha -1}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3{\left(\xi \right)|}^2+|f_6\left(\xi \right){|}^2)d\xi ,\hfill \end{array}$$

or $\eta >0$,

$${|u\left(L\right)|}^2+{|v\left(L\right)|}^2\le {\displaystyle \frac{2{\eta }^{\alpha -1}}{{\pi |\lambda |}^{2\alpha -2}}}\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_i{|}^{2}d\xi +{\displaystyle \frac{2}{{\pi |\lambda |}^{\alpha -1}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^2+|f_6\left(\xi \right){|}^2)d\xi .$$

Then

$${|u\left(L\right)|}^2+{|v\left(L\right)|}^2\le {\displaystyle \frac{2{\eta }^{\alpha -1}}{{\pi |\lambda |}^{2\alpha -2}}}\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_i{|}^{2}d\xi +{\displaystyle \frac{2{\eta }^{\alpha -1}}{{\pi |\lambda |}^{\alpha -1}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^2+|f_6\left(\xi \right){|}^2)d\xi ,$$

$${|u\left(L\right)|}^2+{|v\left(L\right)|}^2\le {\displaystyle \frac{C}{{|\lambda |}^{2\alpha -2}}}\sum _{i=1}^2\underset{-\infty }{\overset{+\infty }{\int }}({\xi }^2+\eta )|{\chi }_i{|}^{2}d\xi +{\displaystyle \frac{C}{{|\lambda |}^{\alpha -1}}}(\underset{-\infty }{\overset{+\infty }{\int }}|f_3\left(\xi \right){|}^2+|f_6\left(\xi \right){|}^2)d\xi .$$

We deduce that

$${|u\left(L\right)|}^2+{|v\left(L\right)|}^2\le {C|\lambda |}^{2-2\alpha }{\parallel U\parallel }_H\parallel F\parallel +{C\parallel F\parallel }_H^2.$$

It follows that

$$|{\phi }_x{\left(L\right)|}^2+|{\varphi }_x{\left(L\right)|}^2\le C\left({\displaystyle \frac{1}{{|\lambda |}^{2\alpha }}}+1\right){\parallel U\parallel }_H\parallel F\parallel +C{\displaystyle \frac{1}{{|\lambda |}^2}}{\parallel F\parallel }_H^2.$$

Using the following Lemma

Lemma 3.

Let $q\in H^1(0,L)$, we have

$${\mathcal{E}}_{\phi }\left(L\right)={\left[q{\tau }_{\phi }\right]}_0^L+2k\mathcal{R}e\underset{0}{\overset{L}{\int }}q{\varphi }_x\overline{{\phi }_x}dx+R_1,$$

(30)

and

$${\mathcal{E}}_{\varphi }\left(L\right)={[q{\tau }_{\psi }]}_0^L-{k[q|\varphi |}^2{]}_0^L-2k\mathcal{R}e\underset{0}{\overset{L}{\int }}q{\phi }_x\overline{{\varphi }_x}dx+k\underset{0}{\overset{L}{\int }}{q}^{\prime }\left(x\right){|\varphi |}^{2}dx+R_2,$$

(31)

where $R_i$ satisfies

$$|R_i{|\le C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}},i=1,2,$$

for a positive constant C.

With the notation

$${\tau }_{\phi }\left(\alpha \right)={\rho }_1{|u\left(\alpha \right)|}^2+k{|{\phi }_x\left(\alpha \right)|}^2,$$

$${\tau }_{\varphi }\left(\alpha \right)={\rho }_2{|v\left(\alpha \right)|}^2+b{|{\varphi }_x\left(\alpha \right)|}^2,$$

$${\mathcal{E}}_{\phi }\left(L\right)=\underset{0}{\overset{L}{\int }}{\tau }_{\phi }\left(s\right)ds,$$

$${\mathcal{E}}_{\varphi }\left(L\right)=\underset{0}{\overset{L}{\int }}{\tau }_{\varphi }\left(s\right)ds.$$

Let us multiplying the second equation of (26) by $q{\overline{\phi }}_x$,

$$\underset{0}{\overset{L}{\int }}i\lambda {\rho }_{1}q{\overline{\phi }}_{x}u-\underset{0}{\overset{L}{\int }}k{\phi }_{xx}q{\overline{\phi }}_{x}dx-\underset{0}{\overset{L}{\int }}k{\psi }_{x}q{\overline{\phi }}_{x}dx=\underset{0}{\overset{L}{\int }}{\rho }_1{f}_{2}q{\overline{\phi }}_{x}dx,$$

⇔

$$-{\rho }_1\underset{0}{\overset{L}{\int }}\overline{i\lambda {\phi }_x}qu-k\underset{0}{\overset{L}{\int }}{\phi }_{xx}q{\overline{\phi }}_{x}dx-k\underset{0}{\overset{L}{\int }}q{\psi }_x{\overline{\phi }}_{x}dx=\underset{0}{\overset{L}{\int }}{\rho }_1{f}_{2}q{\overline{\phi }}_{x}dx,$$

or $i\lambda {\phi }_x=f_{1x}+u_x$

$$-{\rho }_1\underset{0}{\overset{L}{\int }}q{u}_{x}udx-k\underset{0}{\overset{L}{\int }}q{\phi }_{xx}{\overline{\phi }}_{x}dx-k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx={\rho }_1\underset{0}{\overset{L}{\int }}q{f}_2{\overline{\phi }}_{x}dx+{\rho }_1\underset{0}{\overset{L}{\int }}q{f}_{1x}udx$$

⇔

$$-{\displaystyle \frac{{\rho }_1}{2}}\underset{0}{\overset{L}{\int }}q{\displaystyle \frac{{d|u\left(x\right)|}^2}{dx}}dx-{\displaystyle \frac{k}{2}}\underset{0}{\overset{L}{\int }}q{\displaystyle \frac{d|{\phi }_x{|}^2}{dx}}dx-k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx={\rho }_1\underset{0}{\overset{L}{\int }}q{f}_2{\overline{\phi }}_{x}dx+{\rho }_1\underset{0}{\overset{L}{\int }}q{f}_{1x}udx$$

⇔ $\begin{array}{c}-{\rho }_1\left[{q|u\left(x\right)|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|u\left(x\right)|}^{2}dx\right]-k\left[q|{\phi }_x{\left(x\right)|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|{\phi }_x\left(x\right)|}^{2}dx\right]-2k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx=\\ 2{\rho }_1\underset{0}{\overset{L}{\int }}q\left(f_2{\overline{\phi }}_x+f_{1x}u\right)dx\end{array}$⇔

$$\begin{array}{ccc}& & \underset{0}{\overset{L}{\int }}{q}^{\prime }\left({\rho }_1{|u\left(x\right)|}^2+k{|{\phi }_x\left(x\right)|}^2\right)dx\hfill \\ & =& q\left({\rho }_1{|u\left(x\right)|}^2{|}_0^L+k|{\phi }_x\left(x\right){{|}^2|}_0^L\right)+2k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx+2{\rho }_1\underset{0}{\overset{L}{\int }}q\left(f_2{\overline{\phi }}_x+f_{1x}u\right)dx.\hfill \end{array}$$

We obtain

$${\mathcal{E}}_{\phi }\left(L\right)=q{\tau }_{\phi }{\left(x\right)|}_0^L+2k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx+2{\rho }_1\underset{0}{\overset{L}{\int }}q\left(f_2{\overline{\phi }}_x+f_{1x}u\right)dx,$$

then, we have

$${\mathcal{E}}_{\phi }\left(L\right)=q{\tau }_{\phi }{\left(x\right)|}_0^L+2k\underset{0}{\overset{L}{\int }}q{\varphi }_x{\overline{\phi }}_{x}dx+R_1,$$

where

$$R_1=2{\rho }_1\underset{0}{\overset{L}{\int }}q\left(f_2{\overline{\phi }}_x+f_{1x}u\right)dx,$$

with

$$|R_1{|\le C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}.$$

Similarly, let us multiply the fifth equation of (26) by $q{\overline{\varphi }}_x$,

$$\begin{array}{c}\underset{0}{\overset{L}{\int }}i\lambda q{\overline{\varphi }}_x{\rho }_{2}vdx-b\underset{0}{\overset{L}{\int }}q{\overline{\varphi }}_x{\varphi }_{xx}dx+k\underset{0}{\overset{L}{\int }}q{\phi }_x{\overline{\varphi }}_{x}dx+k\underset{0}{\overset{L}{\int }}q\varphi {\overline{\varphi }}_{x}dx+\underset{0}{\overset{L}{\int }}\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)q{\overline{\varphi }}_{x}dsdx\hfill \\ +\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}q\mu \left(\xi \right){\chi }_2(\xi ,t-\tau ){\overline{\varphi }}_{x}d\xi dx=\underset{0}{\overset{L}{\int }}{\rho }_{2}q{f}_5{\overline{\varphi }}_{x}dx\hfill \end{array}$$

Or $\overline{i\lambda {\varphi }_x}=f_{4x}+v_x$

$$\begin{array}{c}-{\rho }_2\underset{0}{\overset{L}{\int }}q{v}_{x}vdx-b\underset{0}{\overset{L}{\int }}q{\overline{\varphi }}_x{\varphi }_{xx}dx+k\underset{0}{\overset{L}{\int }}q\varphi {\overline{\varphi }}_{x}dx+k\underset{0}{\overset{L}{\int }}q{\phi }_x{\overline{\varphi }}_{x}dx\underset{0}{\overset{L}{\int }}\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right){\varphi }_{xx}(x,t-s)q{\overline{\varphi }}_{x}dsdx\hfill \\ +\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}q\mu \left(\xi \right){\chi }_2(\xi ,t-\tau ){\overline{\varphi }}_{x}d\xi dx=\underset{0}{\overset{L}{\int }}{\rho }_{2}q\left(f_5{\overline{\varphi }}_x+f_{4x}v\right)dx\hfill \end{array}$$

⇔

$$\begin{array}{c}-{\displaystyle \frac{{\rho }_2}{2}}\underset{0}{\overset{L}{\int }}q{\displaystyle \frac{d}{dx}}{|v\left(x\right)|}^{2}dx-{\displaystyle \frac{b}{2}}\underset{0}{\overset{L}{\int }}q{\displaystyle \frac{d}{dx}}|{\varphi }_x{|}^{2}dx+{\displaystyle \frac{k}{2}}\underset{0}{\overset{L}{\int }}q{\displaystyle \frac{d}{dx}}{|\varphi |}^2+k\underset{0}{\overset{L}{\int }}q{\phi }_x{\overline{\varphi }}_{x}dx+{\displaystyle \frac{1}{2}}\underset{0}{\overset{L}{\int }}\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right)q{\displaystyle \frac{d}{dx}}{|{\varphi }_x|}^{2}dsdx\hfill \\ +\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}q\mu \left(\xi \right){\chi }_2(\xi ,t-\tau ){\overline{\varphi }}_{x}d\xi dx=\underset{0}{\overset{L}{\int }}{\rho }_{2}q\left(f_5{\overline{\varphi }}_x+f_{4x}v\right)dx\hfill \end{array}$$

⇔

$$\begin{array}{ccc}& & -{\rho }_2\left[{q|v\left(x\right)|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|v\left(x\right)|}^{2}dx\right]-b\left[q|{\varphi }_x{|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|{\varphi }_x|}^{2}dx\right]\hfill \\ & =& -2k\underset{0}{\overset{L}{\int }}q{\phi }_x{\overline{\varphi }}_{x}dx-k\left[{q|\varphi |}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|\varphi |}^{2}dx\right]\hfill \\ & & -\underset{0}{\overset{+\infty }{\int }}\varpi \left(s\right)\left[q|{\varphi }_x{|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }{|{\varphi }_x|}^{2}dx\right]ds-2\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}q\mu \left(\xi \right){\chi }_2(\xi ,t-\tau ){\overline{\varphi }}_{x}d\xi dx\hfill \\ & & +\underset{0}{\overset{L}{\int }}{\rho }_{2}q\left(f_5{\overline{\varphi }}_x+f_{4x}v\right)dx.\hfill \end{array}$$

We obtain

$$\begin{array}{c}{\mathcal{E}}_{\varphi }\left(L\right)=q{\tau }_{\varphi }{\left(x\right)|}_0^L-2k\underset{0}{\overset{L}{\int }}q{\phi }_x{\overline{\varphi }}_{x}dx-{kq|\varphi |}^2{|}_0^L+k\underset{0}{\overset{L}{\int }}{q}^{\prime }{|\varphi |}^{2}dx+R_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill R_2& =& \underset{0}{\overset{L}{\int }}{\rho }_{2}q\left(f_5{\overline{\varphi }}_x+f_{4x}v\right)dx-\underset{0}{\overset{+\infty }{\int }}g\left(s\right)(q|{\varphi }_x{|}^2{|}_0^L-\underset{0}{\overset{L}{\int }}{q}^{\prime }|{\varphi }_x{|}^{2}dx)ds\hfill \\ & & -2\zeta \underset{0}{\overset{L}{\int }}\underset{-\infty }{\overset{+\infty }{\int }}q\mu \left(\xi \right){\chi }_2(\xi ,t-\tau ){\overline{\varphi }}_{x}d\xi dx,\hfill \end{array}$$

With

$$|R_2{|\le C\parallel U\parallel }_H{\parallel F\parallel }_{\mathcal{H}}.$$

If we take $q\left(x\right)=x$ we have

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)=x{\tau }_{\phi }{\left(x\right)|}_0^L+2k\underset{0}{\overset{L}{\int }}x{\varphi }_x{\overline{\phi }}_{x}dx+R_1\hfill \\ {\mathcal{E}}_{\varphi }\left(L\right)=x{\tau }_{\varphi }{\left(x\right)|}_0^L-2k\underset{0}{\overset{L}{\int }}x{\phi }_x{\overline{\varphi }}_{x}dx-{kx|\varphi |}^2{|}_0^L+k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx+R_2,\hfill \end{array}$$

which become

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)=L{\tau }_{\phi }\left(L\right)+2k\underset{0}{\overset{L}{\int }}x{\varphi }_x{\overline{\phi }}_{x}dx+R_1\hfill \\ {\mathcal{E}}_{\varphi }\left(L\right)=L{\tau }_{\varphi }\left(L\right)-2k\underset{0}{\overset{L}{\int }}x{\phi }_x{\overline{\varphi }}_{x}dx-{kL|\varphi \left(L\right)|}^2+k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx+R_2.\hfill \end{array}$$

If we add these two equations, we obtain

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)+{\mathcal{E}}_{\varphi }\left(L\right)=L{\tau }_{\phi }\left(L\right)+L{\tau }_{\varphi }\left(L\right)-{kL|\varphi \left(L\right)|}^2+k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx+R_1+R_2.\hfill \end{array}$$

using Young’s inequality, we obtain

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)+{\mathcal{E}}_{\varphi }\left(L\right)\le L{\tau }_{\phi }\left(L\right)+L{\tau }_{\varphi }\left(L\right)+{kL|\varphi \left(L\right)|}^2+2k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx+{C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}},\hfill \end{array}$$

for a positive constant C. Alternatively,

$$\begin{array}{ccc}& & L{\tau }_{\phi }\left(L\right)+L{\tau }_{\varphi }\left(L\right)\hfill \\ & =& L\left[{\rho }_1{|u\left(L\right)|}^2+k|{\phi }_x{\left(L\right)|}^2+{\rho }_2{|v\left(L\right)|}^2+b{|{\varphi }_x\left(L\right)|}^2\right]\hfill \\ \\ & \le & C\left({|u\left(L\right)|}^2+{|v\left(L\right)|}^2+|{\phi }_x{\left(L\right)|}^2+{|{\varphi }_x\left(L\right)|}^2\right)\hfill \\ \\ & \le & {C|\lambda |}^{2-2\alpha }{\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+{C\parallel F\parallel }_{\mathcal{H}}^2+C(1+{\displaystyle \frac{1}{{|\lambda |}^{2\alpha }}}){\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+C{\displaystyle \frac{1}{{|\lambda |}^2}}{\parallel F\parallel }_{\mathcal{H}}^2.\hfill \end{array}$$

So, we have

$$\begin{array}{ccc}& & {\mathcal{E}}_{\phi }\left(L\right)+{\mathcal{E}}_{\varphi }\left(L\right)\hfill \\ & \le & {C|\lambda |}^{2-2\alpha }{\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+{C\parallel F\parallel }_{\mathcal{H}}^2+C(1+{\displaystyle \frac{1}{{|\lambda |}^{2\alpha }}}){\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+\hfill \\ & & C{\displaystyle \frac{1}{{|\lambda |}^2}}{\parallel F\parallel }_{\mathcal{H}}^2+{kL|\varphi \left(L\right)|}^2+2k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx+{C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}\hfill \\ & \le & C\left[{|\lambda |}^{2-2\alpha }+{\displaystyle \frac{1}{{|\lambda |}^{2\alpha }}}+1\right]{\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+C\left(1+{\displaystyle \frac{1}{{|\lambda |}^2}}\right){\parallel F\parallel }_{\mathcal{H}}^2+{C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+\hfill \\ & & {kL|\varphi \left(L\right)|}^2+2k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx.\hfill \end{array}$$

If $|\lambda |>1$ and $\alpha \in [0,1]$, we obtain

$${|\lambda |}^{2-2\alpha }+{\displaystyle \frac{1}{{|\lambda |}^{2\alpha }}}+1={|\lambda |}^{2-2\alpha }\left(1+{\displaystyle \frac{1}{{|\lambda |}^2}}+{\displaystyle \frac{1}{{|\lambda |}^{2\alpha -2}}}\right)\le 3{|\lambda |}^{2-2\alpha },$$

with $V=0$; then,

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)+{\mathcal{E}}_{\varphi }\left(L\right)\le {C|\lambda |}^{2-2\alpha }{\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+{C\parallel F\parallel }_{\mathcal{H}}^2+{C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}++2k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx.\hfill \end{array}$$

Alternatively,

$${|\varphi \left(x\right)|}^2={|{\displaystyle \frac{v+f_4}{i\lambda }}|}^2\le {\left({\displaystyle \frac{|v|+|f_4|}{|\lambda |}}\right)}^2=\left({\displaystyle \frac{{|v|}^2+|f_4{|}^2+2|v||f_4|}{{|\lambda |}^2}}\right)\le {\displaystyle \frac{{2|v|}^2+2{|f_4|}^2}{{|\lambda |}^2}},$$

then

$$2k\underset{0}{\overset{L}{\int }}{|\varphi |}^{2}dx\le {\displaystyle \frac{4k}{{|\lambda |}^2}}\underset{0}{\overset{L}{\int }}{|v|}^{2}dx+{\displaystyle \frac{4k}{{|\lambda |}^2}}\underset{0}{\overset{L}{\int }}|f_4{|}^{2}dx\le {\displaystyle \frac{C}{{|\lambda |}^2}}{\parallel U\parallel }_{\mathcal{H}}^2+{\displaystyle \frac{C}{{|\lambda |}^2}}{\parallel F\parallel }_{\mathcal{H}}^2,$$

then

$$\begin{array}{c}{\mathcal{E}}_{\phi }\left(L\right)+{\mathcal{E}}_{\varphi }\left(L\right)\le {C|\lambda |}^{2-2\alpha }{\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}+{C\parallel F\parallel }_{\mathcal{H}}^2+{C\parallel U\parallel }_{\mathcal{H}}{\parallel F\parallel }_{\mathcal{H}}++{\displaystyle \frac{C}{{|\lambda |}^2}}{\parallel U\parallel }_{\mathcal{H}}^2+{\displaystyle \frac{C}{{|\lambda |}^2}}{\parallel F\parallel }_{\mathcal{H}}^2.\hfill \end{array}$$

We have

$${\parallel U\parallel }_{\mathcal{H}}^2\le {C|\lambda |}^{4-4\alpha }{\parallel F\parallel }_{\mathcal{H}}^2,$$

so,

$${\displaystyle \frac{1}{{|\lambda |}^{4-4\alpha }}}{\displaystyle \frac{{\parallel U\parallel }_{\mathcal{H}}^2}{{\parallel F\parallel }_{\mathcal{H}}^2}}\le C$$

$${\displaystyle \frac{1}{{|\lambda |}^{4-4\alpha }}}{\displaystyle \frac{{\parallel U\parallel }_{\mathcal{H}}^2}{{\parallel (i\lambda -A)U\parallel }_{\mathcal{H}}^2}}\le C$$

$${\displaystyle \frac{1}{{|\lambda |}^{2-2\alpha }}}{\parallel {(i\lambda I-A)}^{-1}\parallel }_{\mathcal{H}}\le C.$$

Then, using Borichev–Tomilov’s theorem, there exists a C such that

$$\parallel S_{\mathcal{A}}\left(t\right)U_o{\parallel }_{\mathcal{H}}\le {\displaystyle \frac{C}{t^{\frac{1}{2-2\alpha }}}}{\parallel U_o\parallel }_{\mathcal{D}\left(\mathcal{A}\right)}.$$

We obtain the polynomial stability for $\eta >0$. □

5. Conclusions

Unusually, our model is taken with fractional time delay and fractional boundary conditions to see the impact of their weights on system stability with the effects of viscoelastic term. This last term ensures that the system is damped. The two controls with $\gamma \ne {\gamma }_1\ne {\gamma }_2\ne 0$ are considered to be of the same mathematical type. Studies have shown that Timoshenko beam systems with certain types of damping and delay can not achieve exponential stability. For instance, a system with internal fractional feedback and a delay term has been proven to be polynomial stable under specific conditions. However, when considering fractional boundary controls, the stability behavior can differ. Our research indicates that the presence of fractional boundary controls and delays can lead to a lack of exponential stability and ensure the polynomial stability; see [18].

Our contributions are as follows. Under an appropriate condition on the parameters $\gamma ,{\gamma }_1,{\gamma }_2$, we have the following.

• When $\eta \ge 0$, we found a lack of exponential stability decay rate;
• When $\eta >0$, we found a polynomial decay rate.