Exponential Stability for Piezoelectric Beam With Memory,Magnetic Effect,and Time-Varying Delay

Abstract

The piezoelectric beam model is investigated with memory, magnetic effect, and time-varying delay, which describes the conservation law of piezoelectric viscoelastic beam in a magnetic field. Based on the global well-posedness demonstrated via Kato’s variable norm technique, the exponential stability of the considered piezoelectric beam system can be achieved by using some delicate energy estimates on transport terms for delay and memory combined with multiplier techniques.

Keywords

piezoelectric beam model memory time-varying delay exponential stability

1. Introduction

In 1880, Jacques Curie and Pierre Curie discovered that materials such as crystals and barium titanate would have opposite charges at their two ends when subjected to pressure. They named this special property the piezoelectric effect. Later, it was found that such materials would also undergo polarization deformation in an electric field, that is, they could convert electrical energy into mechanical energy, with the converse piezoelectric effect (Brunner et al., 2005; Chee et al., 1998; Pohl, 1987). Materials with this special physical property are called piezoelectric materials. Piezoelectric materials have shown great potential applications in numerous fields due to their unique electromechanical coupling characteristics (Dagdeviren et al., 2014; Erturk & Inman, 2008).

As a common and critical structure, a piezoelectric beam is of great significance in the study of its related equations. For example, in the field of engineering applications, piezoelectric beams have wide applications in the field of sensors and actuators (Brunner et al., 2005; Yeh et al., 2008). In addition, in terms of energy harvesting, piezoelectric beams are able to convert mechanical energy (such as vibration energy) from the environment into electrical energy (Erturk & Inman, 2008). Whether in broadening the scope of practical applications, improving energy efficiency, or in fields such as medical equipment, its operating mechanism is not clear, which attracts great attention.

In the early stage, the research mainly focused on utilizing the basic piezoelectric effect of beams to convert mechanical energy, so as to achieve application functions such as sensors and energy harvesters (Erturk & Inman, 2008). Therefore, most of these models were constructed around the basic piezoelectric principle, with an emphasis on describing the conventional mechanical and electrical conversions of piezoelectric beams (Yang, 2005). With the development of science and technology, piezoelectric beams may be in an environment with a magnetic field in some practical application scenarios, such as near certain magnetic field equipment. However, previous models did not fully take into account factors such as magnetic effects. For piezoelectric beams working in a magnetic field environment, it is difficult to accurately describe the changes in their physical behaviors and performances, see Pohl (1987), Vinogradov et al. (2004), and Yeh et al. (2008).

Morris and Özer (2013, 2014) derived a piezoelectric beam model with magnetic effects through a variational method equipped with boundary conditions for stretching and bending motions as

{\begin{aligned} ρ v_{t t} - α v_{x x} + γ β p_{x x} = 0, x \in (0, L), t > 0, \\ μ p_{t t} - β p_{x x} + γ β v_{x x} = 0, x \in (0, L), t > 0, \\ v (0, t) = α v_{x} (L, t) - γ β p_{x} (L, t) = 0, t \geq 0, \\ p (0, t) = p_{x} (L, t) - γ v_{x} (L, t) + V (t) = 0, t \geq 0, \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), x \in [0, L], \\ p (x, 0) = p_{0} (x), p_{t} (x, 0) = p_{1} (x), x \in [0, L], \end{aligned}

(1.1)

where

ρ

α

γ

β

, and

V

denote the mass density per unit volume, elastic stiffness, piezoelectric coefficient, magnetic permeability, the beam coefficient of impermeability, and the prescribed voltage on the beam electrodes, respectively. Here the functions

v

and

p

represent the transverse displacement of the beam and the total load of the electric displacement along the transverse direction at each point

x

, respectively. They observed that magnetic effects have a significant impact on the observability and stability of the system, more literature can be seen in Özer (2015) and Pazy (1983).

Ramos et al. (2018) considered the system of piezoelectric beams with the damping term $δ v_{t}$ given by

{\begin{aligned} ρ v_{t t} - α v_{x x} + γ β p_{x x} + δ v_{t} = 0, in (0, L) \times (0, T), \\ μ p_{t t} - β p_{x x} + γ β v_{x x} = 0, in (0, L) \times (0, T), \end{aligned}

(1.2)

which presented the total energy of the system is exponentially stable from theoretical analysis and numerical simulation, for further results, see Peng et al. (2019a, 2019b) and Ramos et al. (2019, 2021). In addition, Li et al. (2023) proved the Piezoelectric beam system with magnetic effects, time-varying delays, and weights has exponential stability by the multiplier technique, which has been verified by numerical simulations via the Chebyshev spectral method and the backward Euler method in Trefethen (2000). For more results about the Piezoelectric beam system with delay or memory for some elastic materials, we can refer to Chepyzhov and Pata (2006), Liu and Zheng (1996), Pata (2006), and Yang et al. (2020). Actually, frictional damping, delay, and memory can make the piezoelectric beam model stable or unstable, which is crucial in control and feedback for the considered system.

Here the main interest is the stability of a viscoelastic piezoelectric beam with memory, magnetic effect, and time-varying delay, which can be read as

{\begin{aligned} ρ v_{t t} - α v_{x x} + γ β p_{x x} + \int_{0}^{\infty} g (s) \partial_{x} [a (x) v_{x} (t - s)] d s \\ + μ_{1} (t) v_{t} (x) + μ_{2} (t) v_{t} (x, t - τ (t)) = 0, x \in (0, L), t > 0, \\ μ p_{t t} - β p_{x x} + γ β v_{x x} = 0, x \in (0, L), t > 0, \\ v (0, t) = p (0, t) = 0, t \geq 0, \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), x \in (0, L), \\ v_{t} (x, t - τ (0)) = f_{0} (x, t - τ (0)), x \in (0, L), 0 \leq t \leq τ (0), \\ p (x, 0) = p_{0} (x), p_{t} (x, 0) = p_{1} (x), x \in (0, L), \end{aligned}

(1.3)

where

ρ

α

γ

β

, and

μ

denote positive parameters.

v_{0}

v_{1}

p_{0}

p_{1}

, and

f_{0}

represent the general and delay initial conditions, respectively. The convolution term

\int_{0}^{\infty} g (s) \partial_{x} [a (x) v_{x} (t - s)] d s

reflects the memory effect on materials due to viscoelasticity, and the function

τ (t) : R^{+} \to R^{+}

denotes the time-varying delay.

The main results and features of our work are summarized as follows: (i)

Inspired by the piezoelectric beam model with magnetic effects and time-varying delay terms in Li et al. (2023) and Ramos et al. (2021), this study focuses on the impact of the beam’s past transverse displacement state on its current state. Through theoretical analysis and appropriate assumptions, the global well-posedness of system (1.3) is obtained, and its exponential stability is achieved.

(ii)

The first difficulty lies in dealing with the memory and delay terms. To overcome them, the Dafermos transformation and some properties of related transport equations have been used. Especially, the memory not only needs to satisfy condition (2.1), but also the memory kernel should be

\int_{0}^{\infty} g (s) d s < \frac{α - β γ^{2}}{‖ a (x) ‖_{\infty}}

to ensure the dissipation of the considered system. The second difficulty is that equivalent system (2.16) is nonautonomous, which leads to the Lumer–Phillips theorem cannot be used directly. Kato’s perturbation technique can be applied to overcome the above difficulty and the balance among coefficients in (1.3) is crucial to the stability, which has been constrained as

α_{1} = α - a (x) l - β γ^{2} \geq 0

to ensure the well-posedness and exponential stability via a multiplier approach.

The structure of this paper is organized as follows. In Section 2, the main results and the assumptions for (1.3) have been stated. The proof of well-posedness is given in Section 3. The exponential stability (1.3) has been proved in Section 4.

2. Main Results

2.1. Assumptions

The forced hypotheses in this paper are given as follows.

(A1)
The memory kernel $g : R^{+} \to [0, \infty)$ is a $C^{1} (R^{+}) ⋂ L^{1} (R^{+})$ function satisfying
$g (s) \geq 0 and g^{'} (s) \leq - ξ g (s), \forall s > 0,$
(2.1)
for some $ξ > 0$ . Moreover, there exists a positive constant $l$ such that
$0 < l = \int_{0}^{\infty} g (s) d s < \frac{α - β γ^{2}}{‖ a (x) ‖_{\infty}},$
(2.2)
where $α \geq β γ^{2}$ .
(A2)
The delay function $τ = τ (t)$ satisfies
$τ \in W^{2, \infty} ([0, T]), \forall T > 0.$
(2.3)
Moreover, there exist positive constants $τ_{0}$ , $τ_{1},$ and $d$ such that
$0 < τ_{0} \leq τ (t) \leq τ_{1}, \forall t \geq 0,$
(2.4)
and
$τ^{'} (t) \leq d < 1, \forall t \geq 0.$
(2.5)
(A3)
Let $a (x) \in C^{1} (\bar{Ω})$ , satisfying
$meas {x \in Γ | a (x) > 0} > 0.$
(2.6)
In addition, $a (x) + μ_{1} (t) > 0$ , and neither $a (x)$ nor $μ_{1}$ needs to satisfy the geometric control condition similar to the one in Cavalcanti et al. (2016).
(A4)
The function $μ_{1} : R^{+} \to (0, + \infty)$ is nonincreasing, which belongs to $C^{1} (R^{+})$ and satisfies
$| \frac{μ_{1}^{'} (t)}{μ_{1} (t)} | \leq M_{1},$
(2.7)
for some constant $M_{1} > 0$ and all $t \geq 0$ .
(A5)
The function $μ_{2} : R^{+} \to R$ is also a $C^{1} (R^{+})$ function. Furthermore, there exist constants $M_{2} > 0$ and $δ$ with $0 < δ < \sqrt{1 - d}$ , such that
$| μ_{2} (t) | \leq δ μ_{1} (t)$
(2.8)
and
$| μ_{2}^{'} (t) | \leq M_{2} μ_{1} (t) .$
(2.9)

2.2. Equivalent System

For a system with long-memory and time-varying delay, some transformations similar to those in Alabau et al. (2008), Dafermos (1970), Nicaise and Pignotti (2006), and Pata and Zucchi (2001) will be used to deal with these terms.

Displacement history . The following new variable will be introduced to deal with the relative displacement history as:

η^{t} (x, s) = v (x, t) - v (x, t - s), (x, s) \in Ω \times R_{+}, t \geq 0.

(2.10)

It is easy to verify that

\begin{aligned} {\begin{aligned} η_{t}^{t} (s) = - η_{s}^{t} (s) + v_{t} (t), t, s > 0, \\ η^{t} (0) = 0, t > 0, \\ η^{0} (s) = v_{0} (0) - v_{0} (- s), s > 0. \end{aligned} \end{aligned}

Delay term . Define another new variable transformation as:

z (x, y, t) = v_{t} (x, t - y τ (t)), (x, y, t) \in (0, L) \times (0, 1) \times (0, \infty),

(2.11)

which satisfies

τ (t) z_{t} (x, y, t) + (1 - τ^{'} (t) y) z_{y} (x, y, t) = 0.

(2.12)

By using the above transformations, system (1.3) can be rewritten as equivalent form:

{\begin{aligned} ρ v_{t t} - \partial_{x} [(α - a (x) l) v_{x}] + γ β p_{x x} - \int_{0}^{\infty} g (s) \partial_{x} [a (x) η_{x}^{t} (s)] d s \\ + μ_{1} (t) v_{t} (x) + μ_{2} (t) z (1) = 0, x \in (0, L), t > 0, \\ μ p_{t t} - β p_{x x} + γ β v_{x x} = 0, x \in (0, L), t > 0, \\ η_{t}^{t} (x, s) = - η_{s}^{t} (x, s) + v_{t} (x, t), x \in (0, L), t, s > 0, \\ τ (t) z_{t} (x, y, t) + (1 - τ^{'} (t) y) z_{y} (x, y, t) = 0, x \in (0, L), 00, \\ v (0, t) = p (0, t) = η^{t} (0, s) = 0, t \geq 0, \\ v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), η^{t} (x, 0) = 0, η^{0} (x, s) = η_{0} (x, s), x \in (0, L), \\ v_{t} (x, t - τ (0)) = f_{0} (x, t - τ (0)), x \in (0, L), 0 \leq t \leq τ (0), \\ p (x, 0) = p_{0} (x), p_{t} (x, 0) = p_{1} (x), x \in (0, L), \\ z (x, 0, t) = v_{t} (x, t), x \in (0, L), t > 0, \\ z (x, y, 0) = f_{0} (x, - y τ (0)), x \in (0, L), \end{aligned}

(2.13)

2.3. Some Notations and Settings

Though this paper denote $‖ \cdot ‖$ as $‖ \cdot ‖_{L^{2}}$ for simplification. The working spaces will be defined first.

The history space

M = {η : \int_{0}^{\infty} g (s) \int_{Ω} a (x) | η_{x} (s) |^{2} d x d s < \infty}

is a Hilbert space with an inner product

⟨ η, ξ ⟩_{M} = \int_{0}^{\infty} g (s) \int_{Ω} a (x) η_{x} (x, s) ξ_{x} (x, s) d x d s,

which induces the norm

‖ η ‖_{M}^{2} = \int_{0}^{\infty} g (s) \int_{Ω} a (x) | η_{x} (s) |^{2} d x d s .

Assume that

ξ (t) = \bar{ξ} μ_{1} (t),

(2.14)

and

\bar{ξ}

is a positive constant, such that

\frac{δ}{\sqrt{1 - d}} < \bar{ξ} < 2 - \frac{δ}{\sqrt{1 - d}} .

(2.15)

Define the delayed space by

Z = L^{2} ((0, L) \times (0, 1)),

which is a Hilbert space with the inner product

(z_{1}, z_{2})_{Z} = ξ (t) \int_{0}^{L} \int_{0}^{1} z_{1} (x, y, t) z_{2} (x, y, t) d y d x, z_{1}, z_{2} \in Z,

which induces the norm

‖ z ‖_{Z}^{2} = ξ (t) \int_{0}^{L} \int_{0}^{1} | z (x, y, t) |^{2} d y d x, z_{1}, z_{2} \in Z .

Let us take $U = (v, u, η^{t}, p, q, z)^{T}$ , where $u = v_{t} (x, t)$ , $q = p_{t} (x, t)$ . System (2.13) can be rewritten as a Cauchy problem

{\begin{aligned} \frac{d}{d t} U = L (t) U, \\ U (0) = (v_{0}, v_{1}, η_{0}, p_{0}, p_{1}, f_{0})^{T} . \end{aligned}

(2.16)

The phase space of the global solution to system (2.13) is defined as

H = H_{*}^{1} (Ω) \times L^{2} (Ω) \times M \times H_{*}^{1} (Ω) \times L^{2} (Ω) \times Z,

(2.17)

measured by norm

‖ (v, u, η^{t}, p, q, z) ‖_{H}^{2} = ρ ‖ u ‖^{2} + α_{1} ‖ v_{x} ‖^{2} + ‖ η^{t} ‖_{M}^{2} + μ ‖ q ‖^{2} + β ‖ γ v_{x} - p_{x} ‖^{2} + ‖ z ‖_{Z}^{2},

(2.18)

where

H_{*}^{1} (Ω) = {f \in H^{1} (0, L) | f (0) = 0}

and

α_{1} = α - a (x) l - β γ^{2} \geq 0

2.4. Main Results: Global Well-Posedness and Stability

Theorem 2.1 Global well-posedness

Assume that hypotheses (A1)–(A5) hold, system (1.3) with the initial data $U_{0} = (v_{0}, v_{1}, η_{0}, p_{0}, p_{1}, f_{0})^{T} \in H$ possesses a unique mild solution

(v, v_{t}, η^{t}, p, p_{t}, z)^{T} \in C ([0, \infty), H) .

If the initial data

U_{0} = (v_{0}, v_{1}, η_{0}, p_{0}, p_{1}, f_{0})^{T} \in D (L (0))

, then the solution is regular, which satisfies

(v, v_{t}, η^{t}, p, p_{t}, z)^{T} \in C ([0, \infty), D (L (0))) \cap C^{1} ([0, \infty), H) .

Proof.
See, for example, the detailed proof in Section 3.
Theorem 2.2
Let $U = (v, v_{t}, η^{t}, p, p_{t}, z)^{T}$ be the global solution of (2.13). Then the energy function defined by (3.2) is exponentially stable as $t \to \infty$ , that is, there exist positive constants $C$ and $λ$ such that
$E (t) \leq C E (0) e^{- λ t}, \forall t \geq 0.$
(2.19)
Proof.
See, for example, the details in Section 4.
3. Proof of Theorem 2.1

Kato’s variable norm technique is used to show the well-posedness of system (1.3), which is divided into the following subsections.

3.1. Some Lemmas

System (2.13) is equivalent to the Cauchy problem (2.16) with

L (t) U = (\begin{matrix} u \\ ρ^{- 1} {\partial_{x} [(α - a (x) l) v_{x}] - γ β p_{x x} + \int_{0}^{\infty} g (s) \partial_{x} [a (x) η_{x}^{t} (s)] d s - μ_{1} (t) u - μ_{2} (t) z (1)} \\ - η_{s}^{t} + u \\ q \\ μ^{- 1} [β p_{x x} - γ β v_{x x}] \\ - \frac{1 - τ^{'} (t) y}{τ (t)} z_{y} \end{matrix}) .

Indeed, for variable

η

, it is shown in Grasselli and Pata (2002) that

η_{t}^{t} (x, s) = u - η_{s}^{t}, η^{t} (0) = 0

can be considered as

η_{t}^{t} = u + L η^{t}

, that is,

L η^{t} = - η_{s}^{t}, η^{t} \in D (L),

is the generator of a translation semigroup with domain

D (L) = {η \in M | η_{s}^{t} \in M, η^{t} (0) = 0}

and is the infinitesimal generator of a right-translation semigroup given by

[T (t) η^{t}] (s) = {\begin{aligned} η^{t} (s - t), s > t, \\ 0, 0 < s \leq t . \end{aligned}

(3.1)

Furthermore, (3.1) can be used to extract a formula for

η^{t}

, that is,

η^{t} (s) = {\begin{aligned} η^{0} (s - t) + v (t) - v_{0} (0), s > t, \\ v (t) - v (t - s), 0 < s \leq t . \end{aligned}

Then the phase space $D (L (t))$ can be defined as

D (L (t)) = {\begin{aligned} (v, u, η^{t}, p, q, z) \in H, v, p \in H^{2} (Ω), u, q \in H_{*}^{1} (Ω) \\ \partial_{x} [(α - a (x) l) u_{x}] + \int_{0}^{\infty} g (s) \partial_{x} [a (x) η_{x}^{t} (s)] d s \in L^{2} (Ω) \\ η^{t} \in D (L), z_{y} \in Z, and z (0) = u \end{aligned}} .

(3.2)

Note that

D (L (t))

is independent of

t

, that is,

D (L (t)) = D (L (0)), \forall t > 0.

(3.3)

Next, consider the triplet ${(L, H), D (L (0))}$ forms a compact disc (CD) system.

Proposition 3.1 see Kato, 1985

A triplet ${L, H, D (L (0))}$ , consisting of a family $L = {L (t); t \in [0, T]}$ and a pair of real separable Banach space $D (L (0)) \subset H)$ , will be called a CD system if the following conditions are met.

(i)
$L = {L (t); t \in [0, T]}$ is a stable family of (negative) generators of $C_{0}$ semigroups on $H$ , with stability constants $M, β$ .
(ii)
The domain $D (L (t)) = D (L (0))$ of $L (t)$ is independent of $t$ . We regard $D (L (0))$ as a Banach space, embedded continuously and densely in $H$ .
(iii)
$L \in L i p_{} ([0, T]; L (D (L (0)); H))$ or equivalently, $\partial_{t} L \in L i p_{}^{\infty} ([0, T]; L (D (L (0)); H))$ , where $L i p_{}^{\infty} ([0, T]; L (D (L (0)); H))$ is the space of equivalent classes of essentially bounded, strongly measurable functions from $[0, T]$ into the set $L (D (L (0)); H)$ .

Let a triplet ${L, H, D (L (0))}$ be a CD system, then problem (2.16) has a unique solution
$U \in C ([0, T]; D (L (0))) \cap C^{1} ([0, T]; H) .$

Lemma 3.2
$D (L (0))$ is dense in $H$ .
Proof.
Let $\hat{U} = (\hat{v}, \hat{u}, \hat{η}, \hat{p}, \hat{q}, \hat{z})^{T} \in H$ be orthogonal to all elements of $D (L (0))$ .
$\begin{aligned} ⟨ \hat{U}, U ⟩_{H} & = \int_{Ω} ρ u \hat{u} d x + \int_{Ω} α_{1} v_{x} {\hat{v}}_{x} d x + \int_{0}^{\infty} g (s) \int_{Ω} a (x) η_{x}^{t} {\hat{η}}_{x}^{t} d x d s + \int_{Ω} μ q \hat{q} d x \\ + \int_{Ω} β (γ v_{x} - p_{x}) (γ {\hat{v}}_{x} - {\hat{p}}_{x}) d x + ξ (t) \int_{Ω} \int_{0}^{1} z \hat{z} d y d x, \end{aligned}$
(3.4)
for $U = (v, u, η, p, q, z)^{T} \in D (L (0))$ .

Set $v = u = η = p = q = 0$ , and $z \in C_{0}^{\infty} (Ω \times (0, 1))$ . Then for $U = (0, 0, 0, 0, 0, z) \in D (L (0))$ , it follows from (3.4) that
$\int_{Ω} ξ (x) \int_{0}^{1} z \hat{z} d y d x = 0.$
Since $C_{0}^{\infty} (Ω \times (0, 1))$ is dense in $Z$ , one concludes that $\hat{z} = 0$ .

Similarly, based on the density of $C_{0}^{\infty}$ in $H_{}^{1}$ , $L^{2}$ , $M$ , it is proven that $D (L (0))$ is dense in $H$ .
Lemma 3.3
$L (t)$ generates a strongly continuous semigroup on $H$ .
Proof.
The equivalent norm for $U = (v, u, η, p, q, z)^{T} \in H$ can be considered as
$\begin{aligned} ‖ (v, u, η^{t}, p, q, z) ‖_{t}^{2} & = ρ ‖ u ‖^{2} + α_{1} ‖ v_{x} ‖^{2} + ‖ η^{t} ‖_{M}^{2} + μ ‖ q ‖^{2} \\ + β ‖ γ v_{x} - p_{x} ‖^{2} + ξ (t) τ (t) \int_{0}^{L} \int_{0}^{1} z^{2} d y d x . \end{aligned}$
(3.5)

$\begin{aligned} ⟨ L (t) U, U ⟩_{t} \\ = \int_{0}^{L} [- (α - a (x) l) + α_{1} + β γ^{2}] u_{x} v_{x} d x - μ_{1} (t) \int_{0}^{L} u^{2} d x - μ_{2} (t) \int_{0}^{L} z (1) u d x \\ - \frac{ξ (t)}{2} \int_{0}^{L} \int_{0}^{1} (1 - τ^{'} (t) y) \frac{\partial}{\partial y} z^{2} d y d x - \int_{0}^{\infty} g (s) \int_{0}^{L} a (x) η_{x} η_{s x} d x d s \\ = - μ_{1} (t) \int_{0}^{L} u^{2} d x - μ_{2} (t) \int_{0}^{L} z (1) u d x \\ - \frac{ξ (t)}{2} \int_{0}^{L} \int_{0}^{1} (1 - τ^{'} (t) y) \frac{\partial}{\partial y} z^{2} d y d x + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) \int_{0}^{L} a (x) | η_{x} |^{2} d x d s \\ = - μ_{1} (t) \int_{0}^{L} u^{2} d x - μ_{2} (t) \int_{0}^{L} z (1) u d x + \frac{ξ (t)}{2} \int_{0}^{L} u^{2} d x \\ - \frac{ξ (t) (1 - τ^{'} (t))}{2} \int_{0}^{L} z^{2} (1) d x - \frac{ξ (t) τ^{'} (t)}{2} \int_{0}^{L} \int_{0}^{1} z^{2} d y d x \\ + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) \int_{0}^{L} a (x) | η_{x} |^{2} d x d s . \end{aligned}$

The following inequality comes from Young’s inequality
$\begin{aligned} ⟨ L (t) U, U ⟩_{t} & \leq - (μ_{1} (t) - \frac{ξ (t)}{2} - \frac{| μ_{2} (t) |}{2 \sqrt{1 - d}}) \int_{0}^{L} u^{2} d x \\ - (\frac{ξ (t)}{2} - \frac{ξ (t) τ^{'} (t)}{2} - \frac{| μ_{2} (t) | \sqrt{1 - d}}{2}) \int_{0}^{L} z (1)^{2} d x \\ + \frac{ξ (t) | τ^{'} (t) |}{2 τ (t)} τ (t) \int_{0}^{L} \int_{0}^{1} z^{2} d y d x + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) \int_{0}^{L} a (x) | η_{x} |^{2} d x d s . \end{aligned}$
From (A5) and (2.14), one obtains
$\begin{aligned} ⟨ L (t) U, U ⟩_{t} & \leq - μ_{1} (t) (1 - \frac{\bar{ξ}}{2} - \frac{δ}{2 \sqrt{1 - d}}) \int_{0}^{L} u^{2} d x \\ - μ_{1} (t) (\frac{\bar{ξ} (1 - τ^{'} (t))}{2} - \frac{δ \sqrt{1 - d}}{2}) \int_{0}^{L} z (1)^{2} d x \\ + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) \int_{0}^{L} a (x) | η_{x} |^{2} d x d s + κ (t) ⟨ U, U ⟩_{t}, \end{aligned}$
(3.6)
where
$κ (t) = \frac{\sqrt{1 + τ^{'} {(t)}^{2}}}{2 τ (t)} .$
It follows from (A2) and (2.15) that
$⟨ L (t) U, U ⟩_{t} - κ (t) ⟨ U, U ⟩_{t} \leq 0,$
which means that the operator $\bar{L} (t) = L (t) - κ (t) I$ is dissipative.

Next, it is to be proved that $I - L$ is surjective. Indeed, let $F = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6})^{T} \in H$ , there exists a solution $U = (v, u, η^{t}, p, q, z)^{T} \in D (L (t))$ , satisfying
$(I - L) U = F,$
(3.7)
which is equivalent to
$\begin{matrix} v - u = f_{1}, \end{matrix}$
(3.8)

$\begin{matrix} ρ u - \partial_{x} [(α - a (x) l) v_{x}] + γ β p_{x x} - \int_{0}^{\infty} g (s) \partial_{x} [a (x) η_{x}^{t} (s)] d s + μ_{1} (t) u + μ_{2} (t) z (1) = ρ f_{2}, \end{matrix}$
(3.9)

$\begin{matrix} η^{t} - u + η_{s}^{t} = f_{3}, \end{matrix}$
(3.10)

$\begin{matrix} p - q = f_{4}, \end{matrix}$
(3.11)

$\begin{matrix} μ q - β p_{x x} + γ β v_{x x} = μ f_{5}, \end{matrix}$
(3.12)

$\begin{matrix} z + \frac{(1 - τ^{'} (t) y)}{τ (t)} z_{y} = f_{6} . \end{matrix}$
(3.13)
To solve (3.7), by multiplying (3.10) by $e^{s}$ and integrating over $(0, s)$ it yields
$η^{t} (s) = u (1 - e^{- s}) + \int_{0}^{s} e^{τ - s} f_{3} (τ) d τ .$
(3.14)
Furthermore, following the same approach as in Hu et al. (2024), the solution of (3.13) with initial data $z (x, 0, t) = u (x, t)$ can be written as:
$z (x, y) = u (x) e^{σ (y, t)} + τ (t) e^{σ (y, t)} \int_{0}^{y} \frac{f_{6} (x, s)}{1 - τ^{'} (t) s} e^{- σ (s, t)} d s,$
(3.15)
where
$σ (y, t) = \frac{τ (t)}{τ^{'} (t)} \ln (1 - τ^{'} (t) y) .$
In particular, when $y = 1$ , it follows from (3.8) that
$z (1) = v (x) e^{σ (1, t)} - f_{1} e^{σ (1, t)} + τ (t) e^{σ (1, t)} \int_{0}^{1} \frac{f_{6} (x, s)}{1 - τ^{'} (t) s} e^{- σ (1, t)} d s .$
Let
$z_{0} (x) = - f_{1} e^{σ (1, t)} + τ (t) e^{σ (1, t)} \int_{0}^{1} \frac{f_{6} (x, s)}{1 - τ^{'} (t) s} e^{- σ (1, t)} d s$
and
$z (1) = v (x) e^{σ (1, t)} + z_{0} (x) .$
(3.16)
Substituting (3.14), (3.16) in (3.9) and (3.11) in (3.12), one obtains
$\begin{aligned} [ρ + μ_{1} (t) + μ_{2} (t) e^{σ (1)}] v - \partial_{x} [(α - a (x) l) v_{x}] - k_{1} \partial_{x} [a (x) v_{x}] + γ β p_{x x} \\ = ρ f_{1} + ρ f_{2} - k_{1} \partial_{x} [a (x) f_{1 x}] + \int_{0}^{\infty} \int_{0}^{s} g (s) e^{τ - s} \partial_{x} [a (x) f_{3 x} (τ)] d τ d s \\ + μ_{1} (t) f_{1} - μ_{2} (t) z_{0} (x) \end{aligned}$
(3.17)
and
$μ p - β p_{x x} + γ β v_{x x} = μ f_{4} + μ f_{5},$
(3.18)
where $k_{1} = \int_{0}^{\infty} g (s) (1 - e^{- s}) d s$ .

In order to solve (3.17), one considers bilinear form
$B : (H_{}^{1} (0, L) \times H_{}^{1} (0, L))^{2} \to R$
given by
$\begin{aligned} B ((v, p), (\tilde{v}, \tilde{p})) \\ = (ρ + μ_{1} (t) + μ_{2} (t) e^{σ (1)}) \int_{0}^{L} v \tilde{v} d x + \int_{0}^{L} (α - a (x) l) v_{x} {\tilde{v}}_{x} d x + k_{1} \int_{0}^{L} a (x) v_{x} {\tilde{v}}_{x} d x \\ - γ β \int_{0}^{L} p_{x} {\tilde{v}}_{x} d x + μ \int_{0}^{L} p \tilde{p} d x + β \int_{0}^{L} p_{x} {\tilde{p}}_{x} d x - γ β \int_{0}^{L} v_{x} {\tilde{p}}_{x} d x . \end{aligned}$

According to the definition of the norm of phase space, it is not difficult to show that $B$ is linear, continuous, and coercive. Moreover, it suffices to prove that the right-hand side of (3.17) is an element of $(L^{2} (0, L) \times L^{2} (0, L))$ .

For the remaining terms, $- μ_{2} (t) z_{0} (x)$ and $\int_{0}^{\infty} \int_{0}^{s} g (s) e^{τ - s} [a (x) f_{3 x} (τ)]_{x} d τ d s$ have given detailed proof in Cavalcanti et al. (2016) and Hu et al. (2024).

By Lax–Milgram theorem, equation (3.17) has a weak solution
$(\tilde{v}, \tilde{p}) \in (H_{}^{1} (Ω) \times H_{}^{1} (Ω)) .$
In addition, it follows from (3.9) and (3.12) that
$\begin{aligned} (\tilde{v}, \tilde{p}) \in (H^{2} (Ω) \times H^{2} (Ω)), \\ (\tilde{v}, \tilde{u}, \tilde{η^{t}}, \tilde{p}, \tilde{q}, \tilde{z})^{T} \in D (L (t)) . \end{aligned}$
Therefore, the operator $I - L (t)$ is surjective for all $t > 0$ . And $κ (t) > 0$ means that $\tilde{L} (t)$ is maximal for all $t > 0$ .
Lemma 3.4
The family $L (t) = {L (t); t \in [0, T]}$ is stable.
Proof.
It is difficult to verify the stability condition directly. Here the proposition in Kato (2011) is adopted, claiming that
$\frac{‖ U ‖_{t}}{‖ U ‖_{s}} \leq e^{c} | t - s |, \forall t, s \in [0, T],$
(3.19)
where $U = (v, u, η^{t}, p, q, z)^{T}$ , $c$ is a positive constant, and $‖ \cdot ‖_{t}$ is the norm (3.5). For $s, t \in [0, T]$ and $0 \leq s \leq t \leq T$ :
$\begin{aligned} ‖ U ‖_{t}^{2} - e^{c (t - s)} ‖ U ‖_{s}^{2} \\ = (1 - e^{c (t - s)}) {\int_{0}^{L} [ρ | u |^{2} + α_{1} | v_{x} |^{2} + μ | q |^{2} + β (γ v_{x} - p_{x})^{2}] d x + ‖ η^{t} ‖_{M}^{2}} \\ + (ξ (t) τ (t) - ξ (s) τ (s) e^{c (t - s)}) \int_{0}^{L} \int_{0}^{1} z^{2} d y d x . \end{aligned}$
(3.20)
Since $τ (t) \in W^{2, \infty} ([0, T]) ↪ C^{1} ([0, T])$ holds, $τ (t)$ can be written as
$τ (t) = τ (s) + τ^{'} (r) (t - s), r \in (s, t) .$
The function $ξ (t)$ is a nonincreasing function and $ξ > 0$ , which implies that
$ξ (t) τ (t) \leq ξ (s) τ (s) + ξ (s) τ^{'} (r) (t - s),$
(3.21)
which is equivalent to
$\frac{ξ (t) τ (t)}{ξ (s) τ (s)} \leq 1 + \frac{| τ^{'} (r) |}{τ (s)} (t - s) .$
(3.22)
Using (A2), one deduces that
$\frac{ξ (t) τ (t)}{ξ (s) τ (s)} \leq 1 + c | t - s | \leq e^{c | t - s |} .$
It is clear that $1 - e^{c | t - s |} \leq 0$ , which proves (3.19). Hence, this lemma is proved.
Lemma 3.5
$\partial_{t} \tilde{L} (t) \in L i p_{}^{\infty} ([0, T], B (D (L (0)), H))$ .
Proof.
From (A2), one can obtain
$κ^{'} (t) = \frac{τ^{'} (t) τ^{″} (t)}{2 τ (t) \sqrt{1 + τ^{'} {(t)}^{2}}} - \frac{τ^{'} (t) \sqrt{1 + τ^{'} {(t)}^{2}}}{2 τ {(t)}^{2}},$
which is bounded on $[0, T]$ by (2.4) and (2.5) for all $T > 0$ . Moreover,
$L (t) = (\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{\partial_{x} [(α - a (x) l) \partial_{x}]}{ρ} & - \frac{μ_{1} (t)}{ρ} & \frac{\int_{0}^{\infty} g (s) \partial_{x} [a (x) \partial_{x}] d s}{ρ} & - \frac{γ β \partial_{x x}}{ρ} & 0 & - \frac{μ_{2} (t)}{ρ} |_{y = 1} \\ 0 & 1 & - \frac{\partial}{\partial s} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ - \frac{γ β \partial_{x x}}{μ} & 0 & 0 & \frac{β \partial_{x x}}{μ} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{1 - τ^{'} (t) y}{τ (t)} \frac{\partial}{\partial y} \end{matrix}),$
which implies
$\begin{aligned} \frac{d}{d t} L (t) U = (\begin{matrix} 0 \\ - \frac{μ_{1}^{'} (t) v + μ_{2}^{'} (t) z (1)}{ρ} \\ 0 \\ 0 \\ 0 \\ \frac{τ^{″} (t) τ (t) y - τ^{'} (t) (τ^{'} (t) y - 1)}{τ {(t)}^{2}} z_{y} \end{matrix}) . \end{aligned}$

Since $κ^{'} (t)$ is bounded on $[0, T]$ by (A2), and considering (A4) and (A5), we have
${\tilde{L}}_{t} (t) \in L i p_{}^{\infty} ([0, T], B (D (L (0)), H)),$
which implies that the result in this lemma is true.

For the complete proof of well-posedness, the energy of (1.3) needs to be defined as
$\begin{aligned} E (t) & = \frac{1}{2} \int_{0}^{L} [ρ v_{t}^{2} + α_{1} v_{x}^{2} + μ p_{t}^{2} + β (γ v_{x} - p_{x})^{2}] d x \\ + \frac{ξ (t) τ (t)}{2} \int_{0}^{L} \int_{0}^{1} z^{2} d y d x + \frac{1}{2} ‖ η^{t} ‖_{M}^{2} . \end{aligned}$
(3.23)
Lemma 3.6
The energy $E (t)$ is nonincreasing along any solution $(v, u, η^{t}, p, q, z)^{T}$ . In addition, there exists a positive constant $\tilde{c}$ such that
$‖ (u, v, η^{t}, p, q, z) ‖_{H}^{2} \leq \tilde{c} E (t), for all t \geq 0.$

Proof.
By multiplying (2.13) $_{1}$ by $v_{t}$ , (2.13) $_{2}$ by $p_{t}$ and integrating each of them by parts over $[0, L]$ , it follows from $v_{t} = η_{t}^{t} + η_{s}^{t}$ , $η^{t} (0) = 0$ that
$\begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{0}^{L} ρ v_{t}^{2} d x + \frac{1}{2} \int_{0}^{L} (α - a (x) l) \frac{d}{d t} v_{x}^{2} d x + \int_{0}^{L} \int_{0}^{\infty} g (s) a (x) η_{x}^{t} (s) v_{t x} d s d x \\ - γ β \int_{0}^{L} p_{x} v_{x t} d x + \int_{0}^{L} μ_{2} (t) z (1) v_{t} d x = 0 \end{aligned}$
(3.24)
and
$\begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{0}^{L} μ p_{t}^{2} d x - β \int_{0}^{L} (γ v_{x} - p_{x}) p_{x t} d x = 0. \end{aligned}$
(3.25)
Multiplying (2.13) $_{3}$ by $ξ (t) z$ and integrating over $[0, L] \times [0, 1]$ , one obtains
$\frac{τ (t) ξ (t)}{2} \int_{0}^{L} \int_{0}^{1} \frac{d}{d t} z^{2} d y d x + \frac{ξ (t)}{2} \int_{0}^{L} \int_{0}^{1} (1 - τ^{'} (t) y) \frac{\partial}{\partial y} z^{2} d y d x = 0,$
(3.26)
it follows from (3.24) to (3.26) that
$\begin{aligned} \frac{d}{d t} E (t) & = - μ_{1} (t) \int_{0}^{L} v_{t}^{2} d x + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) ‖ η^{t} ‖_{V_{a}}^{2} d s - μ_{2} (t) \int_{0}^{L} z (1) v_{t} d x + \frac{ξ (t)}{2} \int_{0}^{L} v_{t}^{2} d x \\ - \frac{ξ (t)}{2} \int_{0}^{L} z^{2} (1) d x + \frac{ξ (t) τ^{'} (t)}{2} \int_{0}^{L} z^{2} (1) d x + \frac{ξ^{'} (t) τ (t)}{2} \int_{0}^{L} \int_{0}^{1} z^{2} d y d x \\ \leq - μ_{1} (t) (1 - \frac{\bar{ξ}}{2} - \frac{δ}{2 \sqrt{1 - d}}) \int_{0}^{L} v_{t}^{2} d x + \frac{1}{2} \int_{0}^{\infty} g^{'} (s) ‖ η^{t} ‖_{V_{a}}^{2} d s \\ - μ_{1} (t) (\frac{\bar{ξ} (1 - τ^{'} (t))}{2} - \frac{δ \sqrt{1 - d}}{2}) \int_{0}^{1} z (1)^{2} d x . \end{aligned}$
(3.27)
Thus, $E (t)$ is nonincreasing, and there exists a positive constant $\tilde{c}$ such that
$‖ U ‖_{H}^{2} \leq \tilde{c} E (t), for all t \geq 0.$
The proof of lemma is complete.
3.2. Proof of Theorem 2.1

The assumptions (i)–(iii) of Proposition 3.1 are verified by Lemmas 3.2–3.5. Thus, the Cauchy problem

{\begin{aligned} \frac{d}{d t} \tilde{U} = \tilde{L} (t) \tilde{U}, \\ U (0) = U_{0}, \end{aligned}

(3.28)

has a unique solution

\tilde{U} \in C ([0, + \infty), D (L (0))) \cap C^{1} ([0, + \infty), H),

for

U_{0} \in D (L (0))

. The solution of (2.16) is given by

U (t) = e^{\int_{0}^{t} κ (s) d s} \tilde{U} (t) .

In fact,

\begin{aligned} \frac{d}{d t} U (t) & = κ (t) e^{\int_{0}^{t} κ (s) d s} \tilde{U} (t) + e^{\int_{0}^{t} κ (s) d s} {\tilde{U}}_{t} (t) \\ = e^{\int_{0}^{t} κ (s) d s} (κ (t) + \tilde{L} (t)) \tilde{U} (t) \\ = L (t) e^{\int_{0}^{t} κ (s) d s} \tilde{U} (t) \\ = L (t) U (t) . \end{aligned}

Then the unique solution of (2.16) can be found as

U (t) = S_{t} (0) U_{0}, t \in (0, t_{max}),

where

S_{t} (0)

is an evolution system generated by

L (t)

. Lemma 3.6 guarantees that the local solution cannot blow up in a finite time, so we get the global solution, which concludes the proof.

4. Proof of Theorem 2.2

This section is dedicated to studying the exponential stability of the solution using the multiplier technique. The details of the proof of Theorem 2.2 is divided into several lemmas.

4.1. Technical Estimates

Denote the following functions $I_{i} (t) : H \to R (i = 1, 2, 3, 4)$ by

\begin{aligned} I_{1} (t) & = ρ \int_{0}^{L} v v_{t} d x + γ μ \int_{0}^{L} v p_{t} d x, \\ I_{2} (t) & = ρ \int_{0}^{L} v_{t} (γ v - p) d x + γ μ \int_{0}^{L} p_{t} (γ v - p) d x, \\ I_{3} (t) & = ρ \int_{0}^{L} v v_{t} d x + μ \int_{0}^{L} p_{t} p d x, \\ I_{4} (t) & = \bar{ξ} τ (t) \int_{0}^{L} \int_{0}^{1} e^{- 2 τ (t) y} z^{2} d y d x . \end{aligned}

Lemma 4.1
If $U (t) = (v (t), v_{t} (t), η^{t} (s), p (t), p_{t} (t), z (t))^{T}$ is a solution of (2.13), then function $I_{1}$ satisfies the estimative
$\begin{aligned} \frac{d}{d t} I_{1} (t) & \leq - \frac{α_{1}}{2} \int_{0}^{L} v_{x}^{2} d x + ε_{1} \int_{0}^{L} p_{t}^{2} d x + c_{1} \int_{0}^{L} z^{2} (1) d x \\ + c_{1} (1 + \frac{1}{ε_{1}}) \int_{0}^{L} v_{t}^{2} d x + \frac{c_{1}}{ε_{1}} ‖ η^{t} ‖_{M}^{2}, \end{aligned}$
(4.1)
for some constants $ε_{1} > 0$ and $c_{1} > 0$ .
Proof.
Taking the derivative of $I_{1} (t)$ and using (2.13) $_{1, 2}$ , one derives
$\begin{aligned} \frac{d}{d t} I_{1} (t) & = - α_{1} \int_{0}^{L} v_{x}^{2} d x + ρ \int_{0}^{L} v_{t}^{2} d x + γ μ \int_{0}^{L} p_{t} v_{t} d x - μ_{1} (t) \int_{0}^{L} v_{t} v d x \\ - μ_{2} (t) \int_{0}^{L} z (1) v d x - \int_{0}^{L} \int_{0}^{\infty} g (s) a (x) η_{x}^{t} v_{x} d s d x . \end{aligned}$
According to (A4) and (A5), by Poincaré’s inequality, the estimate of $I_{1} (t)$ is obtained as
$\begin{aligned} \frac{d}{d t} I_{1} (t) & \leq - α_{1} \int_{0}^{L} v_{x}^{2} d x + ρ \int_{0}^{L} v_{t}^{2} d x + γ μ \int_{0}^{L} p_{t} v_{t} d x + μ_{1} (0) \int_{0}^{L} | v_{t} v | d x \\ + δ μ_{1} (0) \int_{0}^{L} | z (1) v | d x - \int_{0}^{L} \int_{0}^{\infty} g (s) a (x) η_{x}^{t} v_{x} d s d x \\ \leq - \frac{α_{1}}{2} \int_{0}^{L} v_{x}^{2} d x + ε_{1} \int_{0}^{L} p_{t}^{2} d x + c_{1} \int_{0}^{L} z^{2} (1) d x \\ + c_{1} (1 + \frac{1}{ε_{1}}) \int_{0}^{L} v_{t}^{2} d x + \frac{c_{1}}{ε_{1}} ‖ η^{t} ‖_{M}^{2} . \end{aligned}$
The lemma has been proved.
Lemma 4.2
Along the solutions, for arbitrary $ε_{2} > 0$ , $I_{2} (t)$ satisfies the following inequality:
$\begin{aligned} \frac{d}{d t} I_{2} (t) & \leq - \frac{γ μ}{2} \int_{0}^{L} p_{t}^{2} d x + 4 α_{1} ε_{2} \int_{0}^{L} (γ v_{x} - p_{x})^{2} d x + \frac{c_{2}}{ε_{2}} \int_{0}^{L} v_{x}^{2} d x + \frac{c_{2}}{ε_{2}} \int_{0}^{L} z^{2} (1) d x \\ + c_{2} (1 + \frac{1}{ε_{2}}) \int_{0}^{L} v_{t}^{2} d x + \frac{c_{2}}{ε_{2}} ‖ η^{t} ‖_{M}^{2} . \end{aligned}$
(4.2)
Proof.
Analogously, the derivative of $I_{2} (t)$ is given from (2.13) $_{1, 2}$ ,
$\begin{aligned} \frac{d}{d t} I_{2} (t) & = - α_{1} \int_{0}^{L} v_{x} (γ v_{x} - p_{x}) d x + ρ γ \int_{0}^{L} v_{t}^{2} d x - ρ \int_{0}^{L} v_{t} p_{t} d x + γ^{2} μ \int_{0}^{L} p_{t} v_{t} d x \\ - γ μ \int_{0}^{L} p_{t}^{2} d x - \int_{0}^{L} g (s) \int_{0}^{\infty} a (x) η_{x}^{t} (s) d s (γ v_{x} - p_{x}) d x \\ - μ_{1} (t) \int_{0}^{L} v_{t} (γ v - p) d x - μ_{2} (t) \int_{0}^{L} z (1) (γ v - p) d x . \end{aligned}$
From (A4) and (A5), one has
$\begin{aligned} \frac{d}{d t} I_{2} (t) & \leq - α_{1} \int_{0}^{L} v_{x} (γ v_{x} - p_{x}) d x + ρ γ \int_{0}^{L} v_{t}^{2} d x - ρ \int_{0}^{L} v_{t} p_{t} d x + γ^{2} μ \int_{0}^{L} p_{t} v_{t} d x \\ - γ μ \int_{0}^{L} p_{t}^{2} d x - \int_{0}^{L} g (s) \int_{0}^{\infty} a (x) η_{x}^{t} (s) d s (γ v_{x} - p_{x}) d x \\ + μ_{1} (0) \int_{0}^{L} | v_{t} (γ v - p) | d x + δ μ_{1} (0) \int_{0}^{L} | z (1) (γ v - p) | d x . \end{aligned}$
Estimate (4.2) is derived from Young’s and Poincaré’s inequalities.
Lemma 4.3
If $U$ is a solution of (2.13), then there exists $ε_{3} > 0$ , such that
$\begin{aligned} \frac{d}{d t} I_{3} (t) & \leq - \frac{α_{1}}{2} \int_{0}^{L} v_{x}^{2} d x - β \int_{0}^{L} (γ v_{x} - p_{x})^{2} d x + μ \int_{0}^{L} p_{t}^{2} d x \\ + c_{3} \int_{0}^{L} v_{t}^{2} d x + c_{3} \int_{0}^{L} z^{2} (1) d x + \frac{c_{3}}{ε_{3}} ‖ η^{t} ‖_{M}^{2} . \end{aligned}$
(4.3)
Proof.
A similar technique as above leads to
$\begin{aligned} \frac{d}{d t} I_{3} (t) & = - α_{1} \int_{0}^{L} v_{x}^{2} d x - β γ^{2} \int_{0}^{L} v_{x}^{2} d x + 2 β γ \int_{0}^{L} p_{x} v_{x} d x \\ - β \int_{0}^{L} p_{x}^{2} d x + ρ \int_{0}^{L} v_{t}^{2} d x + μ \int_{0}^{L} p_{t}^{2} d x \\ - μ_{1} (t) \int_{t}^{L} v_{t} v d x - μ_{2} (t) \int_{0}^{L} z (1) v d x - \int_{0}^{\infty} g (s) \int_{0}^{L} a (x) η_{x}^{t} v_{x} d x d s \\ \leq - α_{1} \int_{0}^{L} v_{x}^{2} d x - β \int_{0}^{L} (γ v_{x} - p_{x})^{2} d x + ρ \int_{0}^{L} v_{t}^{2} d x + μ \int_{0}^{L} p_{t}^{2} d x \\ + μ_{1} (0) \int_{0}^{L} | v_{t} v | d x + δ μ_{1} (0) \int_{0}^{L} | β (1) v | d x + | \int_{0}^{\infty} g (s) \int_{0}^{L} a (x) η_{x}^{t} v_{x} d x d s | \\ \leq - \frac{α_{1}}{2} \int_{0}^{L} v_{x}^{2} d x - β \int_{0}^{L} (γ v_{x} - p_{x})^{2} d x + μ \int_{0}^{L} p_{t}^{2} d x \\ + c_{3} \int_{0}^{L} v_{t}^{2} d x + c_{3} \int_{0}^{L} z^{2} (1) d x + \frac{c_{3}}{ε_{3}} ‖ η^{t} ‖_{M}^{2} . \end{aligned}$
The proof has been finished.
Lemma 4.4
Let $U (t) = (v (t), v_{t} (t), η^{t} (s), p (t), p_{t} (t), z (t))^{T}$ be solution of (2.13). Then function $I_{4} (t)$ satisfies
$\frac{d}{d t} I_{4} (t) \leq - 2 I_{4} (t) + \bar{ξ} \int_{0}^{L} v_{t}^{2} d x .$
(4.4)
Proof.
By simple calculation, the proof of function $I_{4} (t)$ is similar as in Kirane et al. (2011), here the detail is omitted.
4.2. Proof of Theorem 2.2

Let $U (t) = (v (t), v_{t} (t), η^{t} (s), p (t), p_{t} (t), z (t))^{T}$ be solution of (2.13). Set

ψ (t) = N E (t) + \sum_{i = 1}^{4} N_{i} I_{i}, (i = 1, 2, 3, 4),

(4.5)

where

N_{i}

is a positive real number. By (3.27) there exists a positive constant

K

such that

\frac{d}{d t} E (t) \leq - K (\int_{0}^{L} v_{t}^{2} d x + \int_{0}^{L} z^{2} (1) d x + ‖ η^{t} ‖_{M}^{2}) .

Then, we get

\begin{aligned} | ψ (t) - N E (t) | & \leq N_{1} (ρ \int_{0}^{L} | v v_{t} | d x + γ μ \int_{0}^{L} | v p_{t} | d x) \\ + N_{2} (ρ \int_{0}^{L} | v_{t} (γ v - p) | d x + γ μ \int_{0}^{L} | p_{t} (γ v - p) | d x) \\ + N_{3} (ρ \int_{0}^{L} | v v_{t} | d x + μ \int_{0}^{L} | p_{t} p | d x) \\ + N_{4} | \bar{ξ} τ (t) \int_{0}^{L} \int_{0}^{1} e^{- 2 τ (t) y} z^{2} d y d x | . \end{aligned}

(4.6)

A straightforward application of Young’s and Poincaré’s inequalities, and

τ (t) \leq τ_{1}

yield the estimate

\begin{aligned} | ψ (t) - N E (t) | & \leq {\bar{γ}}_{3} {\int_{0}^{L} [v_{t}^{2} + p_{t}^{2} + v_{x}^{2} + (γ v_{x} - p_{x})^{2} + \int_{0}^{1} z^{2} d y] d x + ‖ η^{t} ‖_{M}^{2}} \\ \leq γ_{3} E (t), \end{aligned}

(4.7)

for constant

γ_{3} > 0

. Then, choosing

N

large enough that

γ_{1} = N - γ_{3}

γ_{2} = N + γ_{3}

, one deduces

γ_{1} E (t) \leq ψ (t) \leq γ_{2} E (t), \forall t \geq 0.

(4.8)

Now, it follows from derivative $ψ (t)$ and (4.1)–(4.4) that the derivative estimate of (4.6) is given

\begin{aligned} \frac{d}{d t} ψ (t) & = N \frac{d}{d t} E + N_{1} \frac{d}{d t} I_{1} + N_{2} \frac{d}{d t} I_{2} + N_{3} \frac{d}{d t} I_{3} + N_{4} \frac{d}{d t} I_{4} \\ \leq - [N K - c_{1} (1 + \frac{1}{ε_{1}}) - c_{2} (1 + \frac{1}{ε_{2}}) - c_{3} - N_{4} \bar{ξ}] \int_{0}^{L} v_{t}^{2} d x \\ - [N K - c_{1} N_{1} - \frac{N_{2} c_{2}}{ε_{2}} - c_{3} N_{3}] \int_{0}^{L} z^{2} (1) d x \\ - [N K - \frac{N_{1} c_{1}}{ε_{1}} - \frac{N_{2} c_{2}}{ε_{2}} - \frac{N_{3} c_{3}}{ε_{3}}] ‖ η^{t} ‖_{M}^{2} \\ - [\frac{α_{1} N_{1}}{2} - \frac{N_{2} c_{2}}{ε_{2}} + \frac{α_{1} N_{3}}{2}] \int_{0}^{L} v_{x}^{2} d x \\ - [- N_{1} ε_{1} + \frac{N_{2} γ μ}{2} - N_{3} μ] \int_{0}^{L} p_{t}^{2} d x \\ - [- 4 α_{1} ε_{1} N_{2} + β N_{3}] \int_{0}^{L} (γ v_{x} - p_{x})^{2} d x - N_{4} I_{4} (t) . \end{aligned}

(4.9)

Set

\begin{aligned} N_{1} & = \frac{16 μ α_{1}}{γ β}, N_{2} = \frac{2}{γ}, N_{3} = \frac{1}{4}, N_{4} = \frac{1}{2} \\ ε_{1} & = \frac{μ}{2 N_{1}}, ε_{2} = \frac{16 c_{2} β}{α_{1}^{2} μ + α_{1} γ β}, ε_{3} = 1. \end{aligned}

(4.10)

Combining (4.9) and (4.10) gives the following estimate:

\frac{d}{d t} ψ \leq - m \int_{0}^{L} [v_{t}^{2} + z^{2} (1) + v_{x}^{2} + p_{t}^{2} + (γ u_{x} - p_{x})^{2}] d x - m ‖ η^{t} ‖_{M}^{2}, \forall t \geq 0.

In view of (3.23), one can observe that

\frac{d}{d t} ψ (t) \leq - m E (t),

for all

t \geq 0.

It follows from (4.8) that

\frac{d}{d t} ψ (t) \leq - λ ψ (t), \forall t \geq 0,

which leads to

ψ (t) \leq ψ (0) e^{- λ t}, \forall t \geq 0.

The desired result (2.19) follows by using estimates (4.8). Then, the proof of Theorem 2.2 is finished.

Remark 4.5

In comparison with Li’s model presented in Li et al. (2023), our system (1.3) takes into account historical memory and delay in the longitudinal displacement of the beam. This addition renders the model more realistic and endows it with greater physical significance for practical applications.

Remark 4.6

During the subsequent research, we transformed the target system into its equivalent abstract form (2.16) and identified that $L (t)$ represents a nonautonomous operator within this form. For nonautonomous systems of this kind, the commonly used Lumer–Phillips theorem cannot be directly applied. Consequently, in this paper, we employed Kato’s variable norm technique to demonstrate the well-posedness of system (1.3), meaning that we had to prove the system to be a CD system.

Remark 4.7

Regarding the main result, by utilizing multiplier techniques, we successfully demonstrated that the solution of problem (1.3) is exponentially stable. This not only enriches the theoretical understanding of such systems but also provides a more robust foundation for potential engineering and scientific applications related to beam dynamics with memory and delay effects.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Workshop of Outstanding Foreign Scientists in Henan Province (No. GZS2024007), Postgraduate Education Reform and Quality Improvement Project of Henan Province (No. YJS2023JC23), the international communication project from the Department of Science and Technology in Henan province (No. 242102521039) and the international communication project from the Chinese Educational Ministry (No. HZKY20220270). The work of the last author was partially supported by NSFC (No. 12171087).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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