Exponential stability of Thermoelastic system with boundary time-varying delay

1.   Introduction

It is well known that the delay affects dynamical systems in many fields ^[1,2,3,4] and creates some instabilities ^[5,8]. Regarding the influence of the delay in hyperbolic equations, under some appropriate assumptions on the delay term, the authors of ^[8] proved that the delay term generates the stability mechanism in the following wave equation with a delayed velocity term and the mixed Dirichlet-Neumann boundary condition

where $\Omega \subset \mathbb{R}^n$ is a bounded set with a boundary $\Gamma$, which is divided into two parts $\Gamma_D$ and $\Gamma_N$. The exponential stability of (1.1) was obtained by introducing the suitable energies and using some observability inequalities when $\mu _2 < \mu_1$. For the case that one of the assumptions has not been satisfied, some instability examples are also given in ^[8].

One has that if the boundary condition on $\Gamma_N$ in (1.1) is replaced by

where $\tau(t)$ is the time-varying delay. The balance between the delay term and the damping term is quite different from the constant delay for the time-varying delay in hyperbolic equations ^[9,10,11]. Under some suitable assumptions, the exponential stability result of (1.1) with the boundary time-varying delay (1.2) was obtained in ^[12] for $\mu_2 < \sqrt{1-d}\mu_1$ and $\tau'(t)\leq d$. The exponential stability of (1.1) was also extended to a nonlinear version in ^[12] and distributed version in ^[2].

For the heat equation with a time-varying delay, the authors of ^[13] investigated the method of the construction of Lyapunov functionals in order to get the stability of the following abstract equation with an internal time-varying delay

where $A(t, \cdot)$, $F(\cdot)$ are appropriate partial differential operators and $h(t)$ is the time-varying delay. The exponential stability result was proved. If we consider the heat equation with constant delay or distributed delay, the two-dimensional parabolic functional-differential equation with both fixed and distributed delays is as can be found in ^[14]

The asymptotic stability of the zero solution for (1.3) was proved and the asymptotic stability result for the versions with a missing constant or distributed delays are also available in ^[14]. Moreover, the heat equation with a non-constant delay and nonlinear weights can be seen in ^[15].

If we put the time-varying delay on the boundary for the heat equation, the interested system can be described by

where $a > 0$, $\mu_0$ and $\mu_1$ are fixed nonnegative constants. The energy of (1.4) decays exponentially when $\tau'(t)\leq d$ and $\mu_1^2 < (1-d)\mu_0^2$ (see ^[16]).

We are interested in the effect of a time-varying delay in the boundary stabilization of a thermoelastic system in one dimension. The uniform stability of a thermoelastic system equipped with the boundary time-varying delay on the heat equation has been proved in ^[17]. Let us put the boundary time-varying delay on the wave equation, that is to say, let us equip the temperature field with a Dirichlet boundary and the velocity field with a dynamic boundary with a time-varying delay.

Our interested system can be specifically expressed by

where $u_0$, $u_1$, $\theta_0$ and $\sigma_0$ represent the general initial conditions, $\mu_1$ and $\mu_2$ are positive constants. $a$ is the density of the material medium, $b$ is the heat capacity per unit volume at constant strain, $k$ is the thermal conductivity, $\gamma$ is the product of the volume coefficient of thermal expansion and the bulk modulus, $d$ is the sum of Lamé modulus. In general, we assume that $a$, $b$, $k$, $|\gamma|$ and $d$ are all positive constants, which is reasonable (see ^[18] for the coupled partial differential equations of standard thermoacoustic wave propagation.) The function $\rho(t):\mathbb{R}^+ \rightarrow \mathbb{R}^+$ denotes the nonlinear time-varying delay, $u$ and $\theta$ are the displacement and the temperature difference relative to the reference value respectively. We study the asymptotic behavior of the solution of (1.5) and seek sufficient conditions on $\mu_2$ which guarantee the exponential stability of (1.5).

From the physical point, we can see that the system (1.5) is fixed at one end ($x = 0$) and controlled by a boundary controller at the other end ($x = L$). We consider the stability of the closed-loop system with a combination of a no delay term and a variable delay in the feedback loop (see ^[19,20]).

Inspired by the idea given in ^{[12,21,22,23,24]}, we introduce a new unknown variable, as follows, to represent the time-varying delay term:

which results in

It follows from (1.6) that System (1.5) can be written in the following equivalent form

In order to study the global well-posedness and stability of (1.7), we transform (1.7) into the abstract form given by (2.3), where the abstract non-autonomous operator $\mathcal{L}(t)$ can be seen in (2.4). The main results and features of our work are summarized as follows:

$(i)$ For the non-autonomous operator $\mathcal{L}$, we choose Kato's variable norm technique to replace the commonly used Lumer-Phillips theorem to guarantee the closure of $\mathcal{L}$.

$(ii)$ Suppose that $\rho(t)$ and $\rho'(t)$ are positive and limited; we can get the balance between the damping term and the time-varying delay term. We choose $\xi$ with $\frac{\mu_2}{\sqrt{1-\tilde{\rho_1}}} < \xi < 2\mu_1-\frac{\mu_2}{\sqrt{1-\tilde{\rho_1}}}$ as a weight for the energy space of the function $z$, where $\mu_2 < \sqrt{1-\tilde{\rho}_1}\mu_1$ and $\rho'(t)\leq \tilde{\rho}_1$. The balance is similar to that presented in ^[9,10,11,25].

$(iii)$ The exponential stability of (1.7) can be achieved by constructing a suitable Lyapunov functional, which needs some delicate estimates for the total energy and the time-varying delay term.

The rest of this paper is organized as follows. In Section 2, some preparations are presented, including some assumptions and the abstract form of (1.7). The global well-posedness of (1.7) is given in Section 3 and the exponential stability is done in the last section.

2.   Preliminaries

We assume that the nonlinear time-varying delay $\rho\in W^{2, {\infty}}([0, T])$ for all $T > 0$; there exist four constants $\rho_0$, $\rho_1$, $\tilde{\rho_0}$ and $\tilde{\rho_1}$, for all $t > 0$,

Moreover, assume that

For simplicity, the norms of the Lebesgue spaces $L^p(\Omega)$ are noted as $\|\cdot\|_p$, $1\leq p\leq {\infty}$. We abbreviate $z(L, \delta, t)$ as $z(L)$, $z(L, 0, t)$ as $z(L, 0)$ and $z(L, 1, t)$ as $z(L, 1)$ below.

If we denote

Problem (1.7) can be turned into the following Cauchy problem

where $(\cdot)^{\top}$ means the transpose of the vector $(\cdot)$ and

We consider the following Hilbert space

Assume that a constant $\xi > 0$ satisfies

Note that, such a constant $\xi$ exists from (2.2). Then for all $t\in\mathbb{R}^+$, $z(L, \cdot, t)$, $z_1(L, \cdot, t)$ and $z_2(L, \cdot, t) \in L^2(0, 1)$, the inner product and the corresponding norm can be defined as

and

respectively.

The phase space of (2.3) can be defined as follows

endowed with the inner product

for $U = (u, v, \theta, z(L))^{\top}$ and $\tilde{U} = (\tilde{u}, \tilde{v}, \tilde{\theta}, \tilde{z}(L))^{\top}\in \mathcal{H}$. The strong phase space $D(\mathcal{L}(t))$ can be defined as

where

Note that the domain of the operator $\mathcal{L}(t)$ is independent of the time $t$, that is,

It is worth noting that $\mathcal{L}$ is a non-autonomous operator because the fourth term in $\mathcal{L}$ is dependent on $t$. Moreover, we always assume that (2.1), (2.2) and (2.5) hold without any additional clarification.

3.   Global well-posedness

In this section, the well-posedness of Cauchy problem (2.3) is established by using the variable norm technique of Kato for the non-autonomous operators and estimating the total energy.

Theorem 3.1. Assume the initial data $U_0 = (u_0, u_1, \theta_0, \sigma_0)^{\top}\in \mathcal{D}(\mathcal{L}(0))$ and fix $\xi$ such that (2.5) holds. Then, Problem (2.3) possesses a unique solution

In order to show the global well-posedness of Problem (2.3), we first introduce the following proposition (^[26]) which is a strong version of Theorem 1.9 by Kato in ^[27].

Proposition 3.1. Let $\mathcal{L}(t)$ be an operator with the dense domain $\mathcal{D}(\mathcal{L}(t))$ in $\mathcal{H}$ and the domain be independent of $t$. Assume the following:

1) For $T > 0$, $\mathcal{L} = \{\mathcal{L}(t)|t\in[0, T]\}$ is a stable family of generators in $\mathcal{H}$, and the stability constants are independent of $t$.

2) $\mathcal{L}_t$ belongs to $L^{\infty}_{\ast}([0, T], B(\mathcal{D}(\mathcal{L}(0), \mathcal{H}))$. ($B(X, Y)$ is the set of all bounded linear operators on $X$ to $Y$; $L^{\infty}_{\ast}(I, Z)$ denotes the equivalence classes of strongly measurable, essentially bounded functions on $I$ to $Z$.)

Then for all initial data in $\mathcal{D}(\mathcal{L}(0)$, there exists a unique solution

for Problem (2.3).

In this proposition, the triplet $\{\mathcal{L}, \mathcal{H}, \mathcal{D}(\mathcal{L}(0))\}$, with $\mathcal{L} = \{\mathcal{L}(t)\; |\; t\in[0, T]\}$, for some fixed $T > 0$, forms a constant domain system (see ^[27]). This proposition guarantees the existence and uniqueness of (2.3); the same application of Proposition 3.1 can be found in ^[12,22]. In order to apply the above proposition, we next prove the density of $D(\mathcal{L}(0))$ in $\mathcal{H}$ in the following lemma.

Lemma 3.2. $D(\mathcal{L}(0))$ is dense in $\mathcal{H}$.

Proof. Establish that $\hat{U} = (\hat{u}, \hat{v}, \hat{\theta}, \hat{z}(L))^{\top}\in\mathcal{H}$ is orthogonal to all elements of $D(\mathcal{L}(0))$, that is, for all $U = (u, v, \theta, z(L))^{\top}\in D(\mathcal{L}(0))$, it holds that

Let $u = v = \theta = 0$ and $z(L)\in C^{\infty}_0(0, 1)$; the following result holds by (3.1) for $U = (0, 0, 0, z(L))^{\top}\in D(\mathcal{L}(0))$:

Due to the density of $C^{\infty}_0(0, 1)$ in $L^2(0, 1)$ and the fact that $\rho(t) > 0$, it follows that $\hat{z}(L) = 0$ for all $(\delta, t)\in(0, 1)\times\mathbb{R}^+$. In the same way, let $u = v = z = 0$; (3.1) implies that

where $\theta \in C^{\infty}_0(0, L)$. Since $C^{\infty}_0(0, L)$ is dense in $L^2(0, L)$, $\hat{\theta} = 0$ can be obtained. Similarly $\hat{v} = 0$ can be obtained. Finally, set $v = \theta = z(L) = 0$; there is

We find that $(u, 0, 0, 0)^{\top}\in D(\mathcal{L}(0))$ if and only if $u\in \mathcal{D}(\Delta) = \{E(\Delta, L^2(0, L))\cap V, u_x(L) = 0\}$, where $\mathcal{D}(\Delta)$ is the domain of the Laplace operator with the mixed boundary conditions. Since $\mathcal{D}(\Delta)$ is dense in $V$, $\hat{u} = 0$ can be obtained. Thus, $D(\mathcal{L}(0))$ is dense in $\mathcal{H}$.

Based on the result described by (2.6) and Lemma 3.2, let us check the assumptions in Proposition 3.1 one by one. We first give a lemma which ensures that the operator $\widetilde{\mathcal{L}}$ generates a $C_0$-semigroup in $\mathcal{H}$.

Lemma 3.3. Let $\widetilde{\mathcal{L}}(t) = \mathcal{L}(t)-\kappa(t)I$ ($\kappa(t)$ will be defined below). Then the operator $\widetilde{\mathcal{L}}(t)$ generates a $C_0$-semigroup in $\mathcal{H}$ for a fixed $t$.

Proof. For $U = (u, v, \theta, z(L))^{\top}\in \mathcal{D}(\mathcal{L}(0))$, we calculate directly $(\mathcal{L}(t)U, U)_{\mathcal{H}}$ for a fixed $t$ as follows

By Green's formula, for the boundary conditions of $u$ and $\theta$, the following holds:

and

Moreover, we find that

and

From the above estimates, we have

Using Young's inequality, we get

Let $\kappa(t) = \frac{\sqrt{1+(\rho'(t))^2}}{2\rho(t)}$; there is

Therefore, it follows that

Due to the range of $\xi$ in (2.5), one can obtain

Now we define a new operator

then, $\widetilde{\mathcal{L}}(t)$ is dissipative.

Now, we shall show that $I-\mathcal{L}(t)$ is surjective for a fixed $t > 0$. For all $F = (f_1, f_2, f_3, f_4(L))^{\top}\in \mathcal{H}$, we seek a solution $U = (u, v, \theta, z(L))^{\top}\in D(\mathcal{L}(t))$ such that

Equivalently, we want to find a $U$ that satisfies the following equations

Equation (3.3) implies $v = u-f_1$. The solution of the fourth equation in (3.3) with the initial data $z(L, 0) = v(L)$ can be found as

where $\sigma(\delta, t) = \frac{\rho(t)}{\rho'(t)}\ln(1-\delta \rho'(t))$. Then we get

In particular, we obtain that

where $z_0(L) = -f_1(L)e^{\sigma(1, t)}+\rho(t)e^{\sigma(1, t)} \int^1_0\frac{f_4(L)}{1-s\rho'(t)}e^{-\sigma(\delta, t)}\; ds$. Thus, due to $v = u-f_1$, the Eqs $(3.3)_2$ and $(3.3)_3$ become

where $\ell_1 = a(f_1+f_2)$ and $\ell_2 = bf_3+\gamma f_{1x}\in L^2(0, L)$. The bilinear and linear forms of the variational equations corresponding to (3.5) can be defined as

where the bilinear form $\mathcal{B}: (V\times H^1_0(0, L))^2 \mapsto \mathbb{R}$ is given by

and the linear form $\mathcal{G}: V\times H^1_0(0, L) \mapsto \mathbb{R}$ is defined as

For $(u, \theta)\in V\times H^1_0(0, L)$, the norm of $V\times H^1_0(0, L)$ is

Similar forms to (3.6) can be seen in (19) and (41) in ^[16]. Similar to ^[16], it is easy to show that $\mathcal{B}$ is bounded and coercive in $V\times H^1_0(0, L)$. Therefore, by the Lax-Milgram theorem, there is a unique solution $(u, \theta)\in V\times H^1_0(0, L)$ for (3.5). Then, we can obtain

Given the boundedness of $\rho(t)$, $v(L)$ and $f_4(L)$, we find that $z(L, \delta)\in L^2(0, 1)$ by (3.4). From the fourth equation in (3.3), we have $z_{\delta}(L, \delta)\in L^2(0, 1)$ and $z(L, \delta)\in H^1(0, 1)$.

Moreover, for all $(\phi, w)\in C^{\infty}_0(0, L)\times C^{\infty}_0(0, L) \subset V\times H^1_0(0, L)$, there is

Then, $\theta_{xx}$ and $u_{xx}\in L^2(0, L)$; we have $u\in E(\Delta, L^2(0, L))\cap V$ and $\theta \in E(\Delta, L^2(0, L))\cap H^1_0(0, L)$.

Next, we verify the boundary conditions of $u$. From (3.7), we have

Since $((\ell_1-au-\gamma \theta_x), (\ell_2-b\theta-\gamma u_x))\in L^2(0, L)\times L^2(0, L)$, $u_x$ and $\theta_x$ have weak derivations in $L^2(0, L)$ from (3.8). Then we have

and

Bringing the above two equations into (3.6), we find $u_x(L) = -\mu_1v(L)-\mu_2z(L, 1)$.

Therefore, we find a solution $U = (u, v, \theta, z(L))^{\top}\in \mathcal{D}(\mathcal{L}(t))$ for (3.3). Again, given $\kappa(t) > 0$, $\kappa(t)I-\mathcal{L}(t)$ is surjective because of the boundedness of $\kappa(t)$, so $\widetilde{\mathcal{L}}(t)$ is maximal for a fixed $t > 0$. The proof of Lemma 3.3 is completed.

The stability of the operator $\widetilde{\mathcal{L}}$ in $\mathcal{H}$ can be obtained in the following lemma; a similar process can be seen in ^[12,16,22].

Lemma 3.4. $\widetilde{\mathcal{L}} = \{\widetilde{\mathcal{L}}(t), \; t \in [0, T]\}$ is a stable family in $\mathcal{H}$ for all $T > 0$.

Proof. For $U = (u, v, \theta, z(L))^{\top}\in\mathcal{H}$, let $\|U\|_t = \int^L_0 (du^2_x+av^2+b\theta^2)\; dx +d\xi\rho(t)\int^1_0z^2(L)\; d\delta$ for $0\leq s \leq t\leq T$. We claim that

A direct computation deduces that

Since $\rho \in W^{2, {\infty}}([0, T]) \hookrightarrow C^1([0, T])$, there exists a constant $a\in(s, t)$, such that

Then it follows from (2.1) that

which results in $\rho(t)-\rho(s)e^{\frac{\widetilde{\rho}_1}{\rho_0}|t-s|}\leq0$. Combining $1-e^{\frac{\widetilde{\rho}_1}{\rho_0}|t-s|}\leq0$, we find that (3.9) holds.

Lemma 3.4 and (3.9) imply that the family $\mathcal{L} = \{\mathcal{L}(t), t \in [0, T]\}$ is a stable family of generators in $\mathcal{H}$ by Proposition 1.1 in ^[27]. Due to the boundedness of $\kappa$, we find that $\widetilde{\mathcal{L}}(t) = \mathcal{L}(t)-\kappa(t)I$ is stable in $\mathcal{H}$ from Proposition 1.2 in ^[27].

By applying Proposition 3.1, a property of $\widetilde{\mathcal{L}}$ can be verified in the following lemma, which is similar with Theorem 2.3 in ^[12].

Lemma 3.5. $\widetilde{\mathcal{L}}_t\in L^{\infty}_{\ast}([0, T], B(D(\mathcal{L}(0)), \mathcal{H})).$

Proof. It follows from $\kappa(t) = \frac{\sqrt{1+(\rho'(t))^2}}{2\rho(t)}$ that

The boundedness of $\kappa'(t)$ can be obtained from (2.1). By (2.4), we get

A direct computation and (2.1) yield that

is also bounded for $t\in \mathbb{R}^+$. For $U\in \mathcal{D}(\mathcal{L}(0))$, $\widetilde{\mathcal{L}}_t(t)U = (\mathcal{L}_t(t)-\kappa'(t)I)U$ is bounded from the definition of the norm of the operator. Thus, we have proved this lemma.

We define the energy of Problem (2.3) as follows

Some properties of the energy $E(t)$ can be illustrated in the following lemma.

Lemma 3.6. The energy $E(t)$ is non-increasing. Moreover, there exists a constant $c_0 > 0$ such that

Proof. Multiplying the first equation in (1.7) by $u_t$ and the second one by $\theta$, then adding the two results together, we obtain

By the formula of derivation calculus, we have

It follows from $z_t = -\frac{1-\delta\rho(t)} {\rho(t)}z_{\delta}$ that

Putting (3.14) into (3.13) and combing (3.10), (3.12) and (3.13), we get

Using Young's inequality, we find that

Due to (2.1) and (2.5), there exists a positive constant $c_0$ which satisfies

Then (3.11) holds.

Proof of Theorem 3.1. From Lemmas 3.2–3.5, (2.6) and Proposition 3.1, the problem

has a unique solution:

Then the solution of (2.3) can be given as

In fact, by a simple computation, we have

Lemma 3.6 guarantees that the local solutions can not blow-up in finite time; therefore we have $t_{max} = \infty$. Then the proof of Theorem 3.1 is completed.

4.   Exponential stability

In this section, the exponential stability of Problem (2.3) can be obtained by the energy method.

Theorem 4.1. For all solutions of Problem (2.3), there exist two positive constants $C_1$ and $C_2$ such that

The proof of Theorem 4.1 depends on the following three lemmas. First, a functional can be defined as

where we assume that there exists an $x_0\in\mathbb{R}$ such that $m(x)$ is a standard multiplier:

The applications of $m(x)$ can be found in ^[12]. Moreover, it holds that $m(0)\leq0$ and $m(L)\geq \alpha_0 > 0$. An estimate of $\varphi$ can be obtained in the following lemma.

Lemma 4.1. For all $\varepsilon > 0$, there exists a consant $C_{\varepsilon} > 0$ such that, for all $t > 0$,

Proof. We can write

Using the boundary conditions of $u$ and the first equation in (1.7), we have

and

For all $\varepsilon > 0$, by Young's inequality, we obtain

and

Then we arrive at

From (4.3) and (4.4), the desired result is given.

If we add a perturbation to the function $z$, the functional with respect to $z$ can be expressed by

the same choice for $J(t)$ can be seen in ^[12]. The following lemma gives a property of the functional $J$.

Lemma 4.2. For the above functional, for all $t > 0$, it holds that

Proof. By a direct computation, we obtain

By a similar process in (3.2), we deal with the second item on the right of (4.7). To be specific, by the relation $z_t(L) = -\frac{1-\delta\rho'(t)}{\rho(t)}z_{\delta}(L)$, it holds that

Taking (4.8) into (4.7) and combing the two integrations, we obtain

Therefore, (4.6) can be obtained from the above three equalities.

Let

where the constant $M$ large enough will be fixed later, $\phi$ and $J$ can be seen in (4.1) and (4.5) respectively. We can find the equivalence of $E(t)$ and $G(t)$ from the following lemma.

Lemma 4.3. For the constant $M$, there exist $c_1$ and $c_2 > 0$ such that

Proof. From Young's inequality and the boundedness of $\rho$ by (2.1), there exists a positive constant $c_0'$ such that

Taking $c_1 = M-c_0'$ and $c_2 = M+c_0'$, we get this lemma with $M > c_0'$.

Proof of Theorem 4.1. From (3.11), (4.2) and (4.6), we have

Now we choose $0 < \varepsilon < d$ and $M > 0$ such that

Due to Poincaré's inequality and (4.10), there exists $c_3 > 0$ such that

Consequently, it follows from (4.9) that

Let $C_1 = \frac{c_2}{c_1}$ and $C_2 = \frac{c_3}{c_1}$; then, Theorem 4.1 is completed.

Acknowledgments

This work was partially supported by the Cultivation Fund of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003) and Key Project of Henan Education Department (No. 22A110011).

Conflict of interest

The authors declare that there is no conflicts of interest.