The influence of damping on the asymptotic behavior of solution for laminated beam

1.   Introduction and relevance of subject

Mathematical modeling is indispensable in engineering, natural science, and applied mathematics to capture the effects of both memory and delay ingrained in the studied actualities. To this end, the inclusion of both of them is often simplified for presentation purposes, as a specific description of basic operations can be intricate for mathematical manipulation. A key question to address is how certain behaviors are related to memory and delays. In this study, we investigate the joint impact of an infinite memory, distributed delay, and micro-temperature effects on the system (1.1).

In the current work, we study the following thermoelastic laminated beam, together with structural damping, infinite memory, distributed delay, and micro-temperatures effects:

where

and the initial and boundary conditions are given by

Here, $\varpi$ denotes the transverse displacement, $\phi$ represents the rotation angle, $\psi$ is relative to the amount of slip occurring along the interface, $\theta$ is the temperature difference and $r$ is the micro-temperature vector. The coefficients $\delta, \; \beta, \; \varrho, \; I_\varrho, \; G,$ and $D, \;$ are positive constants representing the adhesive stiffness, the adhesive damping parameter, the density, the shear stiffness, the flexible rigidity and the mass moment of inertia, respectively. We denote by the positive constants $c, \; k_0, \; k_1, \; k_2, \; k_3, \; d, \; \gamma, \; \alpha, \; m, \;$ the physical parameters describing the coupling between the various constituents of the materials.

Herein, $\varsigma_1, \; \varsigma_2$ are positive numbers such that $0 < \varsigma_1 < \varsigma_2,$ and $\mu_2$ is an $L^{\infty}$ function satisfying the following assumption:

● The function $\mu_2: [\varsigma_1, \varsigma_2]\rightarrow \mathbb{R}$ is bounded and it fulfills

To motivate our work, let us recall some earlier related results. For the problems with the Timoshenko system with/without thermal law, one can see the works ^{[4,6,7,16,23,24]} and for problems related to thermoelasticity, we mention for instance ^{[8,10,11,13,17,18]}.

We start with the laminated beam model, which has become quite popular, and both scientists and engineers are interested in it. This model is a pertinent study topic, because of the wide industry applicability of such materials. Hansen and Spies in ^[12] were the first to introduce the following beam with two layers by developing this mathematical model:

The laminated beam equations have produced some results so far, most of which are focused on the system's stability and existence. Provided that the assumption of equal wave speeds holds, it was demonstrated that system (1.3) is exponentially stable, when linear damping terms are incorporated in two of the three equations. However, if they are included in the three equations, then the system decays exponentially with no restriction on the speeds of wave propagations, see, for instance ^[1,22].

Lately, a renewed focus on investigating the asymptotic behavior of the solutions of several thermoelastic laminated beams has grown. For more details about this topic the reader may consult ^[2,9,20].

The thermoelastic laminated beam problem together with nonlinear weights and time-varying delay was the study topic of Nonato et al. in ^[20], where the authors considered two cases (with and without the structural damping) and proved an exponential decay result for both of them. Distributed delay is one of the main damping factors in our model. It is used to model systems in which there is a delay of uncertain duration. The physical interpretation of this term differs from the delayed differential equation, as it can take several values. For example, in incoming signals, distributed delay shortens the setup and lengthens the hold time. Even moderate distributed delay likely makes setup time negative on those inputs that are directly connected to the register.

The infinite memory is a critical aspect in addressing problems, and it has been explored in various contexts such as the work of Liu and Zhao ^[14], in which they considered a thermoelastic laminated beam model with past history. The authors managed to establish both exponential and polynomial stabilities, depending on the kernel function for the system involving structural damping and with no constraint on the wave speeds. Moreover, concerning the system in the absence of structural damping, they were able to establish both exponential and polynomial stabilities, in case of equal wave speeds and lack of exponential stability in the opposite case.

The time delays problems are one of the most significant and active research areas recently. Numerous studies have demonstrated that delay can lead to instability unless certain conditions are incorporated, and it also can lead to distinct solutions that differ from those found in prior studies. Therefore, the issue of stability for systems that involve delay is highly crucial. To learn more about this term, we refer the reader to the following papers ^[3,5,21].

In ^[19], Nicaise and Pignotti made a study on the following wave equation, together with linear frictional damping and internal distributed delay:

and assuming that

the authors managed to prove that the solution is exponentially stable.

Problem (1.1) is considered as a delayed system, and it is also called hereditary systems, posteffect systems, and deviating argument. Distributed delay is a physical phenomenon which is found in a multitude of applications: Many real systems whose temporal evolution is not defined from a simple vector of state (expressed in the present tense) but depends irreducibly on the history of the system. This situation is encountered in the cases-numerous-where a transport of matter, energy or information generates a "dead time" in the reaction: in information and communication technologies (high-speed communication networks, control of networked systems, quality of service in Moving Picture Experts Group (MPEG) video transmissions, tele-operated systems, parallel computing, realtime computing in robotics), in population dynamics and epidemiology (gestation or incubation time), and in mechanics (viscoelasticity). Even if the process does not intrinsically contain a post-effect, its control chain can introduce distributed delays (for example, if the sensors require a significant acquisition/transmission time). For these reasons, it seems reasonable to consider distributed delay as a universal characteristic of the interaction between man and nature (hence, of sciences for engineers). The aim of our study then concerns the interaction between the different damping terms which intervene in the qualitative properties of the energy associated to the system. Before this analysis, we must ensure the existence of unique solution and then we can pass to see the asymptotic behavior of the solution with respect to damping terms. We used classical semigroup theory to find nontrivial results regarding the well-posedness of solutions. Then, under minimal restrictions on the kernel, we found qualitative properties of the solution by contracting an appropriate Lyapnov functional. The main goal is to present fundamental and new techniques for modern models applying science and technology that can stimulate research interest for exploration of mathematical applications in real life sciences.

The rest of the current paper is structured this way: In Section 2, we provide some resources required for our research, then highlight our major results. In Section 3, we establish the well-posedness of the system. In Section 4, we introduce some fundamental lemmas required in the proof later. In Section 5, we demonstrate our general decay result.

2.   Preliminaries and main results

In this section, we provide some materials required in the proof later, then state our major results.

● (A$_1$) Let $g: \mathbb{R}_+\rightarrow \mathbb{R}_+$ be a $C^1$ function which satisfies

● (A$_2$) There exists a strictly increasing convex function ${G}: \mathbb{R}_+\rightarrow \mathbb{R}_+$ of class $C^1(\mathbb{R}_+)\cap C^2(]0, +\infty[)$ which satisfies

such that

Now, we present the following useful inequalities.

Lemma 2.1. The following inequalities are valid,

where $c_1, \; c_2 > 0,$ and

Let us start by introducing (see ^[19])

where

Then, the variables $\eta^t$ and $\mathcal{S}$ surely satisfy

Hence, system (1.1) can be rewritten as

Certainly, system (2.8) is depending on the initial and boundary conditions below

Now, let

Then, system (2.8) is equivalent to

Taking advantage of (2.6), we can rewrite the second equation of (2.10) as

At this step, let us introduce the vector function $U = (\varpi, u, \zeta, \nu, \psi, y, \theta, r, \eta^t, \mathcal{S})^T,$ with

then, system (2.10) becomes

here, $\mathcal{A}: D(\mathcal{A})\subset \mathcal{H}: \rightarrow \mathcal{H}$ stands for a linear operator indicated by

Now, we shall consider the ensuing energy space

where

and

For the space $\mathbb{L}_g,$ we take the following inner product

Furthermore, we consider the following domain

Then, we introduce

We deduce that $\mathcal{H}$ together with (2.12) is a Hilbert space, once we do that, we define $D(\mathcal{A})$ by

Obviously, $D(\mathcal{A})$ is dense in $\mathcal{H}.$

Now, we are ready to state our results.

Theorem 2.1. Let $U_0 \in D(\mathcal{A}),$ then problem (2.9)-(2.10) admits a unique solution

In addition, if $U_0\in \mathcal{H},$ then

We give the energy of the solution of problem (2.8)-(2.9) by

Then, we have the following stability result.

Theorem 2.2. Let $(\varpi, \phi, \psi, \theta, r, \eta^t, \mathcal{S})$ be the solution of (2.8)-(2.9), suppose that ($\boldsymbol{T}$), ($\boldsymbol{A}_1$) and ($\boldsymbol{A}_2$) hold. Then, for any initial data $U_0\in D(\mathcal{A})$ satisfying, for some $p_0\geq0,$

there exist positive constants $\alpha_1, \alpha_2,$ and $\alpha_3,$ such that

where

3.   Existence and uniqueness

In this part, we utilize the semigroup approach to prove our well-posedness result.

Proof of Theorem 2.1. Let's us establish the dissipativity of $\mathcal{A}.$ By (2.12) and for any $U\in D(\mathcal{A}),$ we have

One can notice that

Applying Youg's inequality, we obtain

therefore, by (T) and given $\mathcal{S}(x, 0) = y,$ we end up with

Thereby, $\mathcal{A}$ is dissipative.

Thereafter, we establish the surjectivity of ($I-\mathcal{A}$), that is, we show that

Now, we have

Solving (3.3)$_{10}$ and using $\mathcal{S}(x, 0, \varsigma, t) = y(x, t),$ we find

Hence,

Now, we solve Eq (3.3)$_9$, and we find

Inserting (3.5), (3.4), and

into (3.3)$_2$, (3.3)$_4$, (3.3)$_6$, (3.3)$_7$ and (3.3)$_8$, we get

where

and

We take the following variational formulation to solve (3.6):

where,

is a Hilbert space endowed with

As a part of this step, we provide definitions for both the bilinear form ${Q}: X\times X\rightarrow \mathbb{R}$ and the linear form $L: X\rightarrow \mathbb{R}$ as follows:

and

We can easily prove the continuity of ${Q}$ and $L$. Moreover, from (3.8) together with integration by parts, we arrive at

From this, we conclude the coercivity of ${Q}$. It follows from the Lax-Milgram lemma that (3.6) admits a unique solution satisfying

and

If we substitute $\varpi, \zeta,$ and $\psi$ into (3.3)$_1$, (3.3)$_3$ and (3.3)$_5$, we find

and

In addition, taking $(\bar{\zeta}, \bar{\psi}, \bar{\theta}, \bar{r})\equiv(0, 0, 0, 0) \in \left(\mathbb{J}^1_*(0, 1) \right)^2\times L^2(0, 1)\times H^1_0(0, 1),$ (3.7) becomes

for all $\bar{\varpi} \in \tilde{\mathbb{J}}^1_*(0, 1),$ which indicates that

The standard elliptic regularity implies that

We note that (3.9) remains valid for $\bar{\varphi}\in C^1([0, 1])\subset \tilde{\mathbb{J}}^1_*(0, 1),$ that is $\bar{\varphi}(1) = 0$. Then, we obtain

Integrating by parts, it follows that

Hence

Likewise, we show that

The standard elliptic regularity guarantees the existence of a unique $U \in D(\mathcal{A})$ which fulfills (3.2). Thereby, $\mathcal{A}$ is surjective.

As a consequence, we infer that $\mathcal{A}$ is a maximal dissipative operator. Then, the well-posedness result follows using Lumer-Philips theorem ^[15]. □

4.   Technical lemmas

The main purpose of this section is to establish the essential practical lemmas required to prove our stability results. To attain this goal, we apply a specific approach known as the multiplier technique, which enables us to prove the stability results of problem (2.8). Nevertheless, this method necessitates creating an appropriate Lyapunov functional equivalent to the energy and we will clarify on this in the next section. To simplify matters, we will employ $\chi_* > 0$ to represent a generic constant.

Lemma 4.1. Let $(\varpi, \phi, \psi, \theta, r, \eta^t, \mathcal{S})$ be the solution of (2.8) and (2.9), then, the energy functional satisfies

Proof. As a start, we multiply (2.8)$_1$, (2.8)$_2$, (2.8)$_3$, (2.8)$_4$ and (2.8)$_5$ by $\varpi_t, \; (3\psi_t-\phi_t), \; \psi_t, \; \theta$ and $r$ respectively, then, we integrate over $(0, 1)$ and use integration by parts together with boundary conditions (2.9) and (2.6) to find

It follows from the sixth equation in (2.8) and the integration by parts that

Applying Young's inequality, we find

Next, we multiply (2.8)$_7$ by $\mathcal{S} \vert \mu_2(\varsigma) \vert$ and integrate the result over $(0, 1)\times(0, 1)\times(\varsigma_1, \varsigma_2).$ We get

which, together with (4.2)–(4.4) and (T) gives us

We have then reached the desired result. □

Lemma 4.2. Consider the functional

then, it satisfies

Proof. First, we notice that

Next, we proceed by differentiating ${I}_1(t)$ and using both (2.8)$_2$ and relation (4.8), then, integrating by parts, we get

The last term in (4.9) can be rewritten as

Now, replacing (4.10) into (4.9), leads to

Finally, applying Young's inequality and making use of Lemma 2.1, we obtain (4.7). □

Lemma 4.3. Consider the functional

then, it satisfies

Proof. Simple calculations, using (2.8)$_1$, (2.8)$_4$ and integration by parts, we get

Now, thanks to Young, Poincaré's and Cauchy–Schwarz inequalities, we get, for any $\epsilon_2 > 0,$

The proof is then completed. □

Lemma 4.4. Consider the functional

then, it satisfies

Proof. We differentiate ${I}_3$, using (2.8)$_1$ together with integration by parts, to get

Notice that

By Young, Poincaré's, and Cauchy-Schwarz inequalities, we easily prove (4.13). □

Lemma 4.5. Consider the functional

then, it satisfies

Proof. We proceed by differentiating the functional ${I}_4$ and using Eq (2.8)$_2$ together with integration by parts, which leads to

By virtue of Young's inequality and (2.4), we have

This completes the proof of (4.15). □

Lemma 4.6. Consider the functional

then, it satisfies the estimate

Proof. Simple computations using Eq (2.8)$_3$ and integration by parts, yield

Employing Young's inequality, we conclude (4.18). □

Lemma 4.7. Consider the functional

then, it satisfies

where $m_1$ is a positive constant.

Proof. Taking the derivative of ${I}_6$ and using (2.8)$_7$ and $\mathcal{S}(x, 0, t) = \psi_t,$ we have

From $e^{-\varsigma }\leq e^{-\varsigma p}\leq 1,$ where $0 < p < 1,$ we arrive at

Since $-e^{-\varsigma}$ is an increasing function, then

Hence, if we denote $m_1 = e^{-\varsigma_2}$ and use (T), we easily prove (4.21). □

5.   Stability results

Let us now prove our stability result by using the lemmas in Section 4.

Proof of Theorem 2.2. We proceed by introducing a Lyapunov functional

where constants $N, N_j > 0, \; j = 1, \cdots, 6,$ will be chosen later.

From (5.1), we write

Thanks to Young, Cauchy-Schwarz and Poincaré's inequalities, we get

i.e.,

Now, differentiating the Lyapunov functional ${L}(t),$ using (4.1), (4.7), (4.11), (4.13), (4.15), (4.18), and (4.21), and fixing

We find

Next, we choose our coefficients in (5.3), in a way that, they all except the last two become negative. We start by selecting $N_6$ big enough so that

then, we take $N_3$ fairly wide, such that

after that, we choose $N_2$ large enough, so that

now, we select $N_1$ sufficiently large such that

We can now select $N$ large enough so that we have (5.2) and

Hence, relation (5.3) becomes

Now, exploiting (2.13) and Poincaré's inequality, we obtain

from which

where $\vartheta_5 > 0.$ Thereby, if we combine (5.5) and (5.4), we have

Next, we multiply (5.6), by

we find

Now, we estimate the last term in (5.7) and use both (A$_2$) and (2.14), we find

We insert (5.8) in (5.7) and set $\epsilon_0 = \frac{\vartheta_6 {E}(0)}{2\vartheta_7'},$ we get

We consider now the functional

It is clear that

moreover, noticing that ${E}^\prime(t)\leq0, \; {G}^{\prime \prime}(t) > 0,$ we obtain

Next, we present the functional

where, $\alpha_2$ is a positive constant, therefore,

We integrate over $(0, t)$ to find

from which, we deduce that

where, $\alpha_1$ and $\alpha_3$ are positive constants. The proof is then completed. □

6.   Conclusions

The article is about the laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperature effects introduced in (1.1). By the semigroup approach, we established the existence and uniqueness of the solution which can be considered as the first main result. In addition, as a second novelty, a general decay result for the solution unusually with no constraints regarding the speeds of wave propagation is found. This last new result is considered, as far as we know, the first similar result in the literature for such a system, where we succeed to improve the earlier works known for the case of finite history, to the case of infinite history. The relaxation function becomes intended to satisfy a broader class of relaxation functions.

We mention here that the distributed delay in our system makes a good interaction between the past history and the other damping terms of system (1.1). This type of damping gives more information and qualitative properties on the solution and also its impact on stability is very important as it is shown in the requirement of Theorem 2.2. Of course, the other terms (both temperatures and micro-temperature effects) act as balances in the stability of the system.

Author contributions

Fares Yazid and Fatima Siham Djerad: Writing—original draft preparation; Abdelkader Moumen and Moheddine Imsatfia, Tayeb Mahrouz: Writing—review and editing; Keltoum Bouhali: Supervision. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through small group research project under grant number RGP1/21/45.

Conflict of interest

The authors declare that there is no conflict of interest.