Pointwise Stabilization in the Laminated Beam Model

Abstract

We consider two identical beams on top of each other with an adhesive in the middle. A slip occurs naturally in the structure. In this work, we take this slip into account and show that we can stabilize the system exponentially using pointwise controls applied on the axial force and the bending moment. The model consists of three coupled equations. The first two equations are related to the Timoshenko system and the third equation describes the dynamics of the slip. Our result improves previous results in the sense that it addresses the smaller dissipative effect, the point mechanism. To do this, we introduce a new method that allows us to show the exponential stability of systems of partial differential equations with point dissipation.

Keywords

laminated beam asymptotic behavior pointwise dissipation

1. Introduction

Due to the demand for structural design, the vibration of laminated beams has been one of the research topics in intelligent materials and structures. Hansen (1994; see also Hansen & Spies, 1997) proposed a laminated beam model for two Timoshenko’s beams given by

\begin{aligned} ρ A φ_{t t} & + κ \tilde{G} A (ψ - φ_{x})_{x} + γ_{1} δ (x - ξ) φ_{t} = 0, in (0, L) \times (0, T), \\ ρ I (3 s - ψ)_{t t} & - E I (3 s - ψ)_{x x} - κ \tilde{G} A (ψ - φ_{x}) + γ_{2} δ (x - ξ) (3 s - ψ)_{t} = 0, in (0, L) \times (0, T), \\ ρ I s_{t t} & - E I s_{x x} + κ \tilde{G} A (ψ - φ_{x}) + \frac{4}{3} σ s + \frac{4}{3} γ s_{t} = 0, in (0, L) \times (0, T), \end{aligned}

(1.1)

where the functions

φ, ψ,

and

s

, depending on

(x, t) \in (0, L) \times [0, T)

, model the transversal displacements, the shear angle, respectively, and

s

is proportional to the amount of slip along the interface. Moreover, we denote by

κ

a correction factor,

E

the Young’s modulus, and

G

the shear modulus. The parameters

ρ, A, I, σ

, and

γ

are the body’s density, area of the cross-section, the moment of inertia, adhesive stiffness, and adhesive damping parameter, respectively. Here,

δ (x - ξ)

is the Dirac mass

+ 1

at the point

x = ξ \in (0, L)

. Lastly,

γ_{i}

i = 1, 2

, denote positive damping coefficients.

Together with the interface, a thin adhesive layer joins the surfaces in such a way that a small slip is possible, providing friction. The friction, given by the small slip, is proportional to the slip rate which produces a partial dissipation in the model. Wang et al. (2005) showed that the partial dissipation obtained by the friction created by interfacial sliding, modeled linearly by the function $γ s_{t}$ , is not enough to stabilize the system exponentially. To make the model exponentially stable, the authors introduce dissipative mechanisms as feedback controls effective on the transverse displacements and shear angle.

In recent years, several works considered a natural question: What kind of stabilization mechanisms should be added to the model to achieve exponential stability? In Tatar (2015), the author proved the exponential stability of the laminated beam model by using dissipative boundary controls. In Lo and Tatar (2015), the authors studied a laminated beam model where, in addition to the dissipation produced by sliding, they introduced a memory effect effective on the bending moment. In this way, the resulting model is also partially dissipative, and therefore assuming that the propagation velocities are equal ( $ρ = G I_{ρ}$ ), the authors showed the exponential stability of the corresponding model.

Several authors considered partial dissipative laminated models (see, e.g., Apalara, 2021; Fayssal, 2024; Fayssal et al., 2024; Liu & Zhao, 2019; Lo & Tatar, 2016; Raposo, 2016; Raposo et al., 2024), where the exponential stability was proved provided the wave speeds of the system are equal. To different wave speeds, the authors found polynomial stability. When the wave speeds are different Liu et al. (2020) showed a lack of exponential stability.

Stability results for a memory-type laminated-thermoelastic system with Maxwell–Cattaneo heat conduction were considered in Mukiawa et al. (2020). A Timoshenko–Cattaneo system with viscoelastic Kelvin–Voigt damping and time delay was considered in Raposo et al. (2015). Stabilization of laminated beams with delay and time-varying delay has been considered in Mpungu et al. (2021) and Raposo et al. (2021). Finally, in Raposo et al. (2017), the authors considered the hybrid laminated Timoshenko beam model with dissipative dynamic boundary condition, and they proved that the system is, in general, polynomially stable.

Concerning pointwise damping, we have the work of Tucsnak (1998), where it is proved that the solution of the wave equation with irrational pointwise damping decays to zero polynomially. The above result was improved in Ammari et al. (2000), where a complete characterization of the positions of the pointwise actuator is given and for which the corresponding system is exponentially stable. Concerning pointwise dissipation to the Timoshenko and Bresse model, we refer to Muñoz Rivera and Naso (2022, 2023a), where the authors proved exponential stability to the corresponding semigroup. Moreover, as an application to a semilinear model with pointwise damping, Muñoz Rivera and Naso (2023b) found the exponential stability of Signorini’s problem, where a different method to obtain the existence and the exponential stability was introduced.

As far as we know, there is no work on laminated beam models with pointwise damping. So to fill this gap we will study this problem here.

Let us rewrite model (1.1) by introducing the effective rotation angle notation $ζ = 3 s - ψ$ , therefore system (1.1) can be written as

\begin{aligned} ρ_{1} φ_{t t} & - G (φ_{x} - 3 s + ζ)_{x} + γ_{1} δ (x - ξ) φ_{t} = 0, \\ ρ_{2} ζ_{t t} & - D ζ_{x x} + G (φ_{x} - 3 s + ζ) + γ_{2} δ (x - ξ) ζ_{t} = 0, \\ ρ_{3} s_{t t} & - 3 D s_{x x} - 3 G (φ_{x} - 3 s + ζ) + 4 σ s + γ_{3} s_{t} = 0, \end{aligned}

(1.2)

Note that the first two equations in (1.2) are related to the well-known Timoshenko system, and the third one describes the dynamic of the slip. Here we consider the following boundary and initial conditions, respectively

\begin{aligned} φ (0, t) & = φ (L, t) = ζ (0, t) = ζ (L, t) = s (0, t) = s (L, t) = 0, \forall t > 0, \end{aligned}

(1.3)

\begin{aligned} φ (x, 0) & = φ_{0} (x), ζ (x, 0) = ζ_{0} (x), s (x, 0) = s_{0} (x), \forall x \in (0, L), \\ φ_{t} (x, 0) & = φ_{1} (x), ζ_{t} (x, 0) = ζ_{1} (x), s_{t} (x, 0) = s_{1} (x), \forall x \in (0, L) . \end{aligned}

(1.4)

We consider all the parameters as positive constants with

ρ_{1} = ρ A, G = κ \tilde{G} A

ρ_{2} = ρ I

ρ_{3} = 3 ρ I = 3 ρ_{2}

D = E I

, and

γ_{3} = 4 γ

The main result of this paper is to show the exponential decay of system (1.1) provided

ξ \neq \frac{k}{j} L, \forall k, j \in N ∖ {0}, with k, and j co-prime .

(1.5)

To the best of our knowledge, this result is new in the context of laminated systems. The technique we use to show exponential stability is also new. This is because it is not possible to apply the methods developed in Ammari et al. (2000) and Tucsnak (1998) to show stability for

3 \times 3

systems with point damping. Note that we have not introduced additional dissipative mechanisms in the third equation. The remaining part of this document is organized as follows. In Section 2, we show the well-posedness of the model. Finally, in Section 3, we show the exponential stability of the model. Our result is based on Theorem 3.1 and the diagonalization theorem due to Neves et al. (1986).

2. The Well-Posedness

To use the semigroup theory, we need to transform system (1.2) to a transmission problem. Here we consider the set $I =] 0, L [∖ {ξ}$ and denoting by $S = G (φ_{x} - 3 s + ζ)$ we get

\begin{aligned} ρ_{1} φ_{t t} - S_{x} & = 0 in I \times (0, T), \\ ρ_{2} ζ_{t t} - D ζ_{x x} + S & = 0 in I \times (0, T), \\ ρ_{3} s_{t t} - 3 D s_{x x} - 3 S + 4 σ s + γ_{3} s_{t} & = 0 in I \times (0, T), \end{aligned}

(2.1)

verifying initial conditions (1.4), boundary conditions (1.3). To facilitate notations, let us introduce the gap operator. Denoting the jump of

f

ξ

\begin{aligned} [[f]]_{ξ} := f (ξ^{-}) - f (ξ^{+}) . \end{aligned}

Therefore the corresponding transmission conditions on

ξ \in (0, L)

associated with (2.1) are given by

\begin{aligned} φ (ξ^{-}, t) & = φ (ξ^{+}, t), ζ (ξ^{-}, t) = ζ (ξ^{+}, t), s (ξ^{-}, t) = s (ξ^{+}, t), \end{aligned}

(2.2)

\begin{aligned} [[G φ_{x} (\cdot, t)]]_{ξ} & = - γ_{1} φ_{t} (ξ, t), [[D ζ_{x} (\cdot, t)]]_{ξ} = - γ_{2} ζ_{t} (ξ, t), [[s_{x} (\cdot, t)]]_{ξ} = 0. \end{aligned}

(2.3)

From now and on, we denote by

L^{2} = L^{2} (0, L), H^{1} = H^{1} (0, L)

, and so on. Instead, we denote by

H^{1} (I) = H^{1} (] 0, L [∖ ξ)

. Let us define the phase space

H

\begin{aligned} H := H_{0}^{1} \times L^{2} \times H_{0}^{1} \times L^{2} \times H_{0}^{1} \times L^{2} . \end{aligned}

(2.4)

For any

U := (φ, Φ, ζ, Θ, s, S)^{⊤}

, we define the norm in

H

‖ U ‖_{H}^{2} = \int_{0}^{L} [ρ_{1} | Φ |^{2} + ρ_{2} | Θ |^{2} + ρ_{3} | S |^{2} + G | φ_{x} - 3 s + ζ |^{2} + D | ζ_{x} |^{2} + 3 D | s_{x} |^{2} + 4 σ | s |^{2}] d x .

(2.5)

It is easy to see that

H

is a Hilbert space. Therefore system (1.2)–(2.3) can be written in the form of an abstract first-order evolutionary Cauchy problem

\begin{aligned} U_{t} = A U, U (0) = U_{0}, \end{aligned}

(2.6)

where

Φ = φ_{t}

Θ = ζ_{t}

S = s_{t}

, and

U_{0} = (φ_{0}, φ_{1}, ζ_{0}, ζ_{1}, s_{0}, s_{1})^{⊤}

A : D (A) \subseteq H \to H

is given by

\begin{aligned} A (\begin{matrix} φ \\ Φ \\ ζ \\ Θ \\ s \\ S \end{matrix}) & = (\begin{matrix} Φ \\ \frac{G}{ρ_{1}} (φ_{x} - 3 s + ζ)_{x} \\ Θ \\ \frac{D}{ρ_{2}} ζ_{x x} - \frac{G}{ρ_{2}} (φ_{x} - 3 s + ζ) \\ S \\ \frac{3 D}{ρ_{3}} s_{x x} - \frac{3 G}{ρ_{3}} (φ_{x} - 3 s + ζ) - \frac{4 σ}{3 ρ_{2}} s - \frac{γ_{3}}{3 ρ_{2}} S \end{matrix}) . \end{aligned}

(2.7)

Under the above notation, the transmission conditions (2.2)–(2.3) are rewritten as

\begin{aligned} [[φ]]_{ξ} & = [[ζ]]_{ξ} = [[s]]_{ξ} = 0, \end{aligned}

(2.8)

\begin{aligned} [[G φ_{x}]]_{ξ} & = - γ_{1} Φ (ξ), [[D ζ_{x}]]_{ξ} = - γ_{2} Θ (ξ), and [[s_{x}]]_{ξ} = 0, \end{aligned}

(2.9)

with domain

D (A) := {U \in H^{2}; verifying (2.8), (2.9)},

(2.10)

where

\begin{aligned} H^{2} := [H^{2} (I) \cap H_{0}^{1}] \times H_{0}^{1} \times [H^{2} (I) \cap H_{0}^{1}] \times H_{0}^{1} \times [H^{2} \cap H_{0}^{1}] \times H_{0}^{1} . \end{aligned}

Clearly

D (A)

is dense in

H

. A straightforward calculation leads to

Re (A U, U)_{H} = - γ_{1} | Φ (ξ) |^{2} - γ_{2} | Θ (ξ) |^{2} - \int_{0}^{L} γ_{3} | S |^{2} d x \leq 0.

(2.11)

According to the previous notation, we have

Theorem 2.1

The operator $A$ is the infinitesimal generator of a $C_{0}$ -semigroup of contractions $T (t) := e^{t A} : H \to H$ .

Proof.

From (2.11), $A$ is a dissipative operator. It is enough to show that $0 \in ϱ (A)$ the resolvent set of $A$ . That is, for any $F = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6})^{⊤} \in H$ , we will show there exists $U = (φ, Φ, ζ, Θ, s, S)^{⊤} \in D (A)$ such that

- A U = F .

(2.12)

From (2.7) we obtain

Φ = - f_{1} \in H_{0}^{1} (0, L), Θ = - f_{3} \in H_{0}^{1} (0, L), S = - f_{5} \in H_{0}^{1} (0, L) .

Using the resolvent equation (2.12), we get

\begin{aligned} - G (φ_{x} - 3 s + ζ)_{x} & = ρ_{1} f_{2} \in L^{2} (0, L), \\ - D ζ_{x x} + G (φ_{x} - 3 s + ζ) & = ρ_{2} f_{4} \in L^{2} (0, L), \\ - 3 D s_{x x} - 3 G (φ_{x} - 3 s + ζ) + 4 σ s & = 3 ρ_{2} f_{6} + γ_{3} f_{5} \in L^{2} (0, L), \end{aligned}

(2.13)

satisfying the transmission conditions (2.8) and (2.9) and the following boundary conditions:

φ (0) = φ (L) = ζ (0) = ζ (L) = s (0) = s (L) = 0.

Let us define the bilinear form

B

over

V = V_{0} \times H_{0}^{1} \times H_{0}^{1}

\begin{aligned} B ((φ, ζ, s), (\tilde{φ}, \tilde{ζ}, \tilde{s})) = \end{aligned}

\begin{aligned} = \int_{0}^{L} [G (3 s - ζ - φ_{x}) (3 \tilde{s} - \tilde{ζ} - {\tilde{φ}}_{x}) + D ζ_{x} {\tilde{ζ}}_{x} + 3 D s_{x} {\tilde{s}}_{x} + 4 σ s \tilde{s}] d x, \end{aligned}

and the linear form

F : V \to C

F (\tilde{φ}, \tilde{ζ}, \tilde{s}) = \int_{0}^{L} [ρ_{1} f_{2} \tilde{φ} + ρ_{2} f_{4} \tilde{ψ} + (ρ_{3} f_{6} + γ_{3} f_{5}) \tilde{s}] d x .

It is easy to verify that

B

is symmetric, continuous, and coercive, and

F

is a continuous functional over

V

. So applying Lax–Milgram’s lemma, we obtain the existence and uniqueness of

(φ, ζ, s) \in V

such that

B ((φ, ζ, s), (\tilde{φ}, \tilde{ζ}, \tilde{s})) = F (\tilde{φ}, \tilde{ζ}, \tilde{s}), \forall (\tilde{φ}, \tilde{ζ}, \tilde{s}) \in V .

(2.14)

Taking

\tilde{ζ} = \tilde{s} = 0

in (2.14) we get

- \int_{0}^{L} G (3 s - ζ - φ_{x}) {\tilde{φ}}_{x} d x = \int_{0}^{L} ρ_{1} f_{2} \tilde{φ} d x, \forall \tilde{φ} \in V .

(2.15)

Taking

\tilde{φ} \in C_{0}^{\infty} (0, L)

in (2.15), after an integration by parts, we arrive at (2.13)

_{1}

in the distributional sense. Since

ζ

s \in H_{0}^{1}

, we get from (2.13)

_{1}

that

φ_{x x} \in L^{2} (0, L)

, so (2.13)

_{1}

is valid in the strong sense. Hence the solution verifies all the boundary conditions.

Finally, repeating the above procedure for $(0, \tilde{ζ}, 0), (0, 0, \tilde{s}) \in V$ in (2.14) we conclude that (2.13) $_{2}$ and (2.13) $_{3}$ are valid in the strong sense.

So we have that $0 \in ρ (A)$ , therefore the operator $A$ generates a $C_{0}$ -semigroup of contractions $T (t) = e^{t A}, t > 0,$ on the space $H$ for the evolution problem (2.6).

As a consequence of the above theorem, we have

Theorem 2.2

For any $U_{0} := (φ_{0}, φ_{1}, ζ_{0}, ζ_{1}, s_{0}, s_{1}) \in H$ there exists a unique mild solution to problems (2.1) and (2.2). Moreover if the initial data $U_{0} \in D (A)$ then the mild solution is a strong solution satisfying

U \in C^{1} (0, T; H) \cap C^{0} (0, T; D (A)) .

3. Decay of the Energy

Let us denote by $L (X)$ the Banach algebra of all bounded linear operators on a complex Banach space $X$ endowed with the operator norm, which again is denoted by $‖ \cdot ‖$ . For an operator $T \in L (X)$ , $σ (T)$ stands for its spectrum, while $ϱ (T) := C ∖ σ (T)$ is the resolvent set of $T$ .

The main tool we use in this paper is the following result.

Theorem 3.1

Let $S (t) = e^{A t}$ be a $C_{0}$ -semigroup of contractions over a Banach space and let $ω_{e s s} (S (t))$ be the essential growth bound of $S (t)$ . Then, $S (t)$ is exponentially stable if and only if

i R \subset ϱ (A) and ω_{ess} (S (t)) < 0,

(3.1)

Proof.

Here we use the method proposed by Engel and Nagel (2000: Corollary 2.11, p. 258) establishing that the type $ω$ of the semigroup $e^{A t}$ verifies

\begin{aligned} ω = max {ω_{ess}, ω_{σ} (A)}, \end{aligned}

(3.2)

where

ω_{σ} (A)

is the upper bound of the spectrum of

A

. Moreover, for any

c > ω_{ess}

, the set

I_{c} := σ (A) \cap {λ \in C : Re λ \geq c}

is finite.

From (3.1) and (3.2), to show exponential stability, it is enough to prove that $ω_{σ} (A) < 0$ . If $ω_{σ} (A) \leq ω_{ess}$ then we have nothing to prove. Let us suppose that $ω_{σ} (A) > ω_{ess}$ , so the set $I_{ω_{ess} + δ}$ is finite for $δ > 0$ small such that $ω_{ess} + δ < 0$ and $ω_{ess} + δ < ω_{σ} (A)$ , moreover $ω_{σ} (A) \in I_{ω_{ess} + δ}$ . From (3.1) and Hille-Yosida’s theorem, we have $\bar{C_{+}} \subset ϱ (A)$ hence $ω_{σ} (A) < 0$ , therefore the sufficient condition follows.

Reciprocally, let us suppose that the semigroup $S (t)$ is exponentially stable, in particular, it goes to zero. Then, by Batty and Duyckaerts (2008: Theorem 1.1), we have that $i R \subset ϱ (A)$ . Moreover, since the type $ω$ verifies (3.2), we have that

ω_{ess} \leq max {ω_{ess}, ω_{σ} (A)} = ω < 0.

Then, our conclusion follows.

In what follows, we use characterization due to Prüss (1984), reported here below.

Theorem 3.2

Let $S (t) = e^{A t}$ be a $C_{0}$ -semigroup of contractions on Hilbert space. Then $S (t)$ is exponentially stable if and only if

$i R \subset ϱ (A)$ ,

${lim sup}_{| λ | \to \infty} ‖ (i λ I - A)^{- 1} ‖_{L (H)} < + \infty .$

Let us consider the resolvent equation

\begin{aligned} i λ U - A U = F, \end{aligned}

Using an inner product with

U

over

H

, applying (2.11) and then taking the real part, we get

γ_{1} | Φ (ξ) |^{2} + γ_{2} | Θ (ξ) |^{2} + \int_{0}^{L} γ_{3} | S |^{2} d x = Re ⟨ U, F ⟩_{H} .

(3.3)

The resolvent equation

i λ U - A U = F

, in terms of its components is given by

\begin{aligned} i λ φ - Φ & = f_{1}, \end{aligned}

(3.4)

\begin{aligned} i λ ρ_{1} Φ + G (3 s - ζ - φ_{x})_{x} & = f_{2}, \end{aligned}

(3.5)

\begin{aligned} i λ ζ - Θ & = f_{3}, \end{aligned}

(3.6)

\begin{aligned} i λ ρ_{2} Θ - D ζ_{x x} - G (3 s - ζ - φ_{x}) & = f_{4}, \end{aligned}

(3.7)

\begin{aligned} i λ s - S & = f_{5}, \end{aligned}

(3.8)

\begin{aligned} i λ ρ_{3} S - 3 D s_{x x} + 3 G (3 s - ζ - φ_{x}) + 4 σ s + γ_{3} S & = f_{6}, \end{aligned}

(3.9)

verifying the transmission conditions (2.8) and (2.9) and the following boundary conditions:

φ (0) = φ (L) = ζ (0) = ζ (L) = s (0) = s (L) = 0.

Our starting point is to prove the strong stability of

T (t)

Lemma 3.3

Under the above conditions we have $i R \subset ϱ (A)$ .

Proof.

Since $D (A)$ has compact embedding over the phase space $H$ and since $0 \in ϱ (A)$ , then $A^{- 1}$ is a compact operator. Therefore the spectrum of $A$ consists only of eigenvalues. Then, we will show that there are no imaginary eigenvalues. By contradiction, let us suppose that there exists an eigenvector $U \neq 0$ verifying

\begin{aligned} A U = i λ U for some λ \in R . \end{aligned}

From (3.3) we get

\begin{aligned} γ_{1} | Φ (ξ) |^{2} + γ_{2} | Θ (ξ) |^{2} + \int_{0}^{L} γ_{3} | S |^{2} d x = 0. \end{aligned}

This implies that

φ (ξ) = ζ (ξ) = s = S = 0

. Hence using (3.9) and recalling that

f_{i} = 0

we arrive at

S = G (φ_{x} - 3 s + ζ) = 0.

(3.10)

Using (3.5), since

f_{2} = 0

, we get that

Φ = 0

. So, applying (3.4) and

f_{1} = 0

, we have that

φ = 0

. Hence, getting that

s = φ = 0

and substituting into (3.10), we find that

ζ = 0

. Then we have that

U = 0

, but this is a contradiction and our conclusion follows.

For the sake of simplicity, here and in what follows we shall employ the same symbol $C$ for different constants, even in the same formula.

Remark 3.4

Here assumption (1.5) is not necessary because system (2.1) is already strongly stable due to the frictional damping $γ s_{t}$ created by interfacial slip.

To show the exponential stability of $T (t)$ associated with system (2.1)–(2.3), thanks to Theorem 3.1, we only have to prove that the essential type $ω_{ess} (T)$ of $T (t)$ is negative.

To show that, let us introduce the operator $B$

B : H \to H, B U = (0, 0, 0, \frac{G}{ρ_{2}} (3 s - ζ), 0, - 3 \frac{G}{ρ_{3}} (3 s - ζ) - \frac{4 σ}{ρ_{3}} s) .

It is easy to verify that

B

is a compact operator over

H

. Hence the operator

A_{0} = A - B

is the infinitesimal generator of a

C_{0}

-semigroup denoted by

T_{1} (t) = e^{t A_{0}}

. Note that

T_{1} (t)

defines the solution of the system

\begin{aligned} ρ_{1} φ_{t t} - G (φ_{x} - 3 s + ζ)_{x} & = 0 in I \times (0, T), \\ ρ_{2} ζ_{t t} - D ζ_{x x} + G φ_{x} & = 0 in I \times (0, T), \\ ρ_{3} s_{t t} - 3 D s_{x x} - 3 G φ_{x} + γ_{3} s_{t} & = 0 in I \times (0, T), \end{aligned}

(3.11)

with boundary conditions (1.3) and verifying the initial and transmission conditions (1.4), (2.2), and (2.3), respectively. Under the above notations, we establish the following lemma.

Lemma 3.5

The difference $T (t) - T_{1} (t)$ is a compact operator. Hence the corresponding essential types are equal.

Proof.

Equation $U_{t} - A U = 0$ can be written as $U_{t} - A_{0} U = B U .$ Then the solution can be written as

U (t) = e^{A_{0} t} U_{0} + \int_{0}^{t} e^{A_{0} (t - s)} B U (s) d s .

(3.12)

Recalling the definition of

U (t)

and

T_{1} (t)

, equation (3.12) implies

T (t) U_{0} = T_{1} (t) U_{0} + \int_{0}^{t} e^{A_{0} (t - s)} B e^{A s} U_{0} d s .

Since

B

is a compact operator then the composition

e^{A_{0} (t - s)} B e^{A s}

is also a compact operator. Therefore,

T (t) - T_{1} (t)

is a compact operator over

H

Hence, to prove the exponential decay of $T (t)$ we only need to show that the essential type of $T_{1}$ is negative.

We now analyze the first-order system associated with (3.11). In what follows we assume that

\frac{G}{ρ_{1}} \neq \frac{D}{ρ_{2}} .

(3.13)

Using the Riemann invariants associated with system (3.11) we get

\begin{array}{ll} p_{1} = \sqrt{ρ_{1}} φ_{t} - \sqrt{G} φ_{x}, & q_{1} = \sqrt{ρ_{1}} φ_{t} + \sqrt{G} φ_{x}, \\ p_{2} = \sqrt{ρ_{2}} ζ_{t} - \sqrt{D} ζ_{x}, & q_{2} = \sqrt{ρ_{2}} ζ_{t} + \sqrt{D} ζ_{x}, \\ p_{3} = \sqrt{ρ_{3}} s_{t} - \sqrt{3 D} s_{x}, & q_{3} = \sqrt{ρ_{3}} s_{t} + \sqrt{3 D} s_{x}, \end{array}

we have that

φ_{t} = \frac{q_{1} + p_{1}}{2 \sqrt{ρ_{1}}}, φ_{x} = \frac{q_{1} - p_{1}}{2 \sqrt{G}}, ζ_{t} = \frac{q_{2} + p_{2}}{2 \sqrt{ρ_{2}}}, ζ_{x} = \frac{q_{2} - p_{2}}{2 \sqrt{D}}, s_{t} = \frac{q_{3} + p_{3}}{2 \sqrt{ρ_{3}}}, s_{x} = \frac{q_{3} - p_{3}}{2 \sqrt{3 D}} .

Therefore, the evolution problem can be written as

\begin{aligned} p_{1, t} + k_{1} p_{1, x} - c_{1} (q_{2} - p_{2}) - c_{3} (q_{3} - p_{3}) & = 0, \end{aligned}

(3.14)

\begin{aligned} q_{1, t} - k_{1} q_{1, x} - c_{1} (q_{2} - p_{2}) - c_{3} (q_{3} - p_{3}) & = 0, \end{aligned}

(3.15)

\begin{aligned} p_{2, t} + k_{2} p_{2, x} - c_{2} (q_{1} - p_{1}) & = 0, \end{aligned}

(3.16)

\begin{aligned} q_{2, t} - k_{2} q_{2, x} - c_{2} (q_{1} - p_{1}) & = 0, \end{aligned}

(3.17)

\begin{aligned} p_{3, t} + k_{3} p_{3, x} - c_{4} (q_{1} - p_{1}) + c_{5} (q_{3} + p_{3}) & = 0, \end{aligned}

(3.18)

\begin{aligned} q_{3, t} - k_{3} q_{3, x} - c_{4} (q_{1} - p_{1}) + c_{5} (q_{3} + p_{3}) & = 0, \end{aligned}

(3.19)

where

\begin{aligned} k_{1} & = \sqrt{\frac{G}{ρ_{1}}}, k_{2} & = \sqrt{\frac{D}{ρ_{2}}}, k_{3} & = \sqrt{\frac{3 D}{ρ_{3}}}, \\ c_{1} & = \frac{G}{2 \sqrt{D ρ_{1}}}, c_{2} & = - \frac{1}{2} \sqrt{\frac{G}{ρ_{2}}}, c_{3} & = - \frac{3 G}{2 \sqrt{ρ_{1} 3 D}}, c_{4} & = \frac{3 G}{2 \sqrt{ρ_{3} G}} c_{5} & = \frac{γ_{3}}{2 ρ_{3}}, \end{aligned}

(3.20)

verifying the following boundary conditions:

q_{1} (0) + p_{1} (0) = 0, q_{1} (L) + p_{1} (L) = 0, q_{j} (0) + p_{j} (0) = 0, q_{j} (L) + p_{j} (L) = 0, j = 2, 3,

(3.21)

and the following transmission conditions:

\begin{aligned} [[q_{i} (\cdot, t) + p_{i} (\cdot, t)]]_{ξ} = 0, i = 1, 2, \end{aligned}

(3.22)

\begin{aligned} [[q_{i} (\cdot, t) - p_{i} (\cdot, t)]]_{ξ} = - γ_{i} υ_{i} (p_{i} (ξ, t) + q_{i} (ξ, t)), i = 1, 2, \end{aligned}

(3.23)

with

υ_{1} = 1 / \sqrt{ρ_{1} G}

and

υ_{2} = 1 / \sqrt{ρ_{2} D}

. Let us denote by

K = (\begin{matrix} k_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & - k_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & - k_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & - k_{3} \end{matrix}), C = (\begin{matrix} 0 & 0 & c_{1} & - c_{1} & c_{3} & - c_{3} \\ 0 & 0 & c_{1} & - c_{1} & c_{3} & - c_{3} \\ c_{2} & - c_{2} & 0 & 0 & 0 & 0 \\ c_{2} & - c_{2} & 0 & 0 & 0 & 0 \\ c_{4} & - c_{4} & 0 & 0 & c_{5} & c_{5} \\ c_{4} & - c_{4} & 0 & 0 & c_{5} & c_{5} \end{matrix}), U = (\begin{matrix} p_{1} \\ q_{1} \\ p_{2} \\ q_{2} \\ p_{3} \\ q_{3} \end{matrix}),

hence system (3.14)–(3.17) can be written as

U_{t} + K U_{x} + C U = 0, U (0) = U_{0} .

(3.24)

Note that system (3.11) is equivalent to (3.24). Let us denote by

T_{2} (t)

the semigroup defined by (3.24) over

H_{6} := [L^{2} (0, L)]^{6}

. Recalling the definition of

C

we get

C_{0} := diag (C) = (0, 0, 0, 0, c_{5}, c_{5})

Let us introduce $T_{3} (t) : H_{6} \to H_{6}$ the semigroup defined by the diagonal system

{\tilde{U}}_{t} + K {\tilde{U}}_{x} + C_{0} U = 0, \tilde{U} (0) = U_{0},

(3.25)

verifying the boundary conditions (3.21) and the transmission conditions (3.22) and (3.23).

Remark 3.6

To use the same notation of Neves et al. (1986), we rewrite

diag (K) := (d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, d_{6}) = (k_{1}, - k_{1}, k_{2}, - k_{2}, k_{3}, - k_{3}) .

The positive values of the diagonal of

K

are given by

d_{1}, d_{3}, d_{5}

. By hypotheses (3.13) and recalling the definition of

k_{i}

: (3.20) we get that

k_{1} \neq k_{2}

that is

d_{1} \neq d_{3}

. Moreover the matrix

K

and

C = (c_{i j})_{6 \times 6}

verifies the property:

k_{2} = k_{3} \Rightarrow d_{3} = d_{5} \Rightarrow c_{3, 5} = c_{5, 3} = 0.

Hence the hypotheses of Neves et al. (1986: Theorem A) are valid.

At this point, we use the result due to Neves et al. (1986) which in our case implies the following result.

Theorem 3.7

Under the above notations, the difference $T_{3} (t) - T_{2} (t)$ is a compact operator over $H_{6}$ , provided condition (3.13) holds.

Proof.

The result follows from Neves et al. (1986: Theorem A) and Remark 3.6.

System (3.25) is completely decoupled and, for $j = 1, 2$ , it can be written as

\begin{aligned} p_{j, t} + k_{i} p_{j, x} = 0, q_{j, t} - k_{j} q_{j, x} = 0, \\ q_{j} (0) + p_{j} (0) = 0, q_{j} (L) + p_{j} (L) = 0, j = 1, 2, \end{aligned}

(3.26)

together with

\begin{aligned} q_{j} (ξ^{-}) + p_{j} (ξ^{-}) & = q_{j} (ξ^{+}) + p_{j} (ξ^{+}), \end{aligned}

(3.27)

\begin{aligned} q_{j} (ξ^{-}) - q_{j} (ξ^{+}) + p_{j} (ξ^{+}) - p_{j} (ξ^{-}) & = - γ_{j} ν_{j} (p_{j} (ξ^{+}) + q_{j} (ξ^{+})) . \end{aligned}

(3.28)

For

j = 3

, we have

\begin{aligned} p_{3, t} + k_{3} p_{3, x} + c_{5} p_{3} = 0, q_{3, t} - k_{3} q_{3, x} + c_{5} q_{3} = 0, \\ q_{3} (0) + p_{3} (0) = 0, q_{3} (L) + p_{3} (L) = 0, \end{aligned}

(3.29)

with (3.27) and (3.28).

The semigroup $T_{3} (t) : H_{6} \to H_{6}$ is generated by the operator

\begin{aligned} A & = (\begin{matrix} A_{1} & 0 & 0 \\ 0 & A_{2} & 0 \\ 0 & 0 & A_{3} \end{matrix}), \end{aligned}

(3.30)

where

0

is the

2 \times 2

null matrix and

A_{j}

, for

j = 1, 2

, is given by

A_{j} U_{2} = k_{j} K U_{2, x}, U_{2} = (\begin{matrix} p \\ q \end{matrix}), K = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}),

and, for

j = 3

, we have

A_{3} U_{2} = k_{3} K U_{2, x} + c_{5} U_{2} .

Denoting by

I = (0, ζ) \cup (ζ, L)

the corresponding domains are given by

D (A_{j}) = {(\begin{matrix} p \\ q \end{matrix}) \in H_{2} : p, q \in H^{1} (I), verifying (3.26), (3.27), (3.28),},

for

j = 1, 2

and

D (A_{3}) = {(\begin{matrix} p \\ q \end{matrix}) \in H_{2} : p, q \in H^{1} (I), verifying (3.29),},

with

H_{2} := [L^{2} (0, L)]^{2}

. The resolvent system

i λ U_{2} + A_{j} U_{2} = F

is given by

i λ U_{2} + k_{j} K U_{2, x} = F,

(3.31)

where

K = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), F = (\begin{matrix} f_{1} \\ f_{2} \end{matrix}), U_{2} = (\begin{matrix} p \\ q \end{matrix}) .

Hence for

j = 1, 2

, the system can be rewritten as

U_{2, x} + \frac{i λ}{k_{j}} K U_{2} = \frac{1}{k_{j}} K F, j = 1, 2.

(3.32)

Instead for

j = 3

, we have

U_{2, x} + \frac{1}{k_{3}} (i λ + c_{5}) {K U}_{2} = \frac{1}{k_{3}} K F,

(3.33)

In terms of the components the above system (3.32) and (3.33) can be written as

\begin{aligned} p_{x} + \frac{i λ}{k_{j}} p & = \frac{1}{k_{j}} f_{1}, q_{x} - \frac{i λ}{k_{j}} q = - \frac{1}{k_{j}} f_{2} \end{aligned}

(3.34)

\begin{aligned} p_{3, x} + \frac{1}{k_{3}} (i λ + c_{5}) p_{3} & = \frac{1}{k_{3}} f_{1}, q_{3, x} - \frac{1}{k_{3}} (i λ + c_{5}) q_{3} = - \frac{1}{k_{3}} f_{2}, \end{aligned}

(3.35)

verifying the boundary conditions (3.26) and the transmission conditions (3.27) and (3.28).

Lemma 3.8

The operator $A$ , infinitesimal generator of $T_{3}$ given in (3.30), is dissipative over the phase space $H_{6}$ .

Proof.

Because of (3.30) it is enough to show that $A_{i}$ is a dissipative operator over $H_{2}$ for $i = 1, 2, 3$ . Here we prove only for $i = 1$ , the proof to $i = 2, 3$ is similar. For the sake of simplicity, the index $1$ is not written in $p$ and $q$ . Note that

\begin{aligned} {((\begin{matrix} p \\ q \end{matrix}), A_{1} (\begin{matrix} p \\ q \end{matrix}))}_{H_{2}} & = {((\begin{matrix} p \\ q \end{matrix}), (\begin{matrix} k_{1} p_{x} \\ - k_{1} q_{x} \end{matrix}))}_{H_{2}} \\ = \int_{0}^{ξ} (k_{1} p_{x} p - k_{1} q_{x} q) d x + \int_{ξ}^{L} (k_{1} p_{x} p - k_{1} q_{x} q) d x \\ = \frac{k_{1}}{2} (| p (ξ^{-}) |^{2} - | p (0) |^{2} - | q (ξ^{-}) |^{2} + | q (0) |^{2}) \\ + \frac{k_{1}}{2} (| p (L) |^{2} - | p (ξ^{+}) |^{2} - | q (L) |^{2} + | q (ξ^{+}) |^{2}) . \end{aligned}

Using the boundary conditions (3.26) and the transmission conditions (3.27) and (3.28), we get

\begin{aligned} {((\begin{matrix} p \\ q \end{matrix}), A_{1} (\begin{matrix} p \\ q \end{matrix}))}_{H_{2}} & = \frac{k_{1}}{2} (| p (ξ^{-}) |^{2} - | q (ξ^{-}) |^{2}) + \frac{k_{1}}{2} (- | p (ξ^{+}) |^{2} + | q (ξ^{+}) |^{2}) \\ = \frac{k_{1}}{2} [(p (ξ^{-}) - q (ξ^{-})) + (- p (ξ^{+}) + q (ξ^{+}))] (p (ξ) + q (ξ)) . \end{aligned}

Here we used the continuity of the sum

p + q

ξ

. Finally from (3.28) we get

\begin{aligned} {((\begin{matrix} p \\ q \end{matrix}), A_{1} (\begin{matrix} p \\ q \end{matrix}))}_{H_{2}} & = \frac{1}{2} γ_{1} k_{1} ν_{1} {| p (ξ) + q (ξ) |}^{2}, \end{aligned}

(3.36)

Similarly, we can prove that

\begin{aligned} {((\begin{matrix} p \\ q \end{matrix}), A_{3} (\begin{matrix} p \\ q \end{matrix}))}_{H_{2}} & = c_{5} \int_{0}^{L} [| p (s) |^{2} + | q ((s) |^{2}] d s, \end{aligned}

(3.37)

and our conclusion follows.

Lemma 3.9

The infinitesimal generator $A$ of $T_{3}$ given in (3.30), verifies

i R \subset ϱ (A)

provided

ξ \neq (n / m) L

\forall n, m \in N ∖ {0}

and

n

and

m

co-prime.

Proof.

Since system (3.25) is fully decoupled it is enough to show that $i R \subset ϱ (A_{i})$ for $i = 1, 2, 3$ . Because of the compacity of the resolvent family associated with $A_{i}$ it is enough to find that there are no imaginary eigenvalues. On the contrary, suppose that there exists $0 \neq W \in D (A_{1})$ such that $A_{1} W = i λ W$ , $λ \in R$ . Since $A_{1}$ (the cases $i = 1, 2$ are identical) is dissipative we get

{| p (ξ) + q (ξ) |}^{2} = 0,

(3.38)

which implies that

p (ξ^{+}) - q (ξ^{+}) = p (ξ^{-}) - q (ξ^{-}) .

For

W = (p, q)

, we have that

A_{1} W = i λ W

implies

p_{x} + i \frac{λ}{k_{1}} p = 0, q_{x} - i \frac{λ}{k_{1}} q = 0.

Using the boundary condition at

x = 0

p (0) + q (0) = 0

, we solve the above system as

p (x) = p (0) e^{- i (λ / k_{1}) x}, q (x) = - p (0) e^{i (λ / k_{1}) x} .

From the boundary condition at

x = L

p (L) + q (L) = 0

we get

p (L) + q (L) = p (0) (e^{- (i (λ / k_{1}) L} - e^{(i λ / k_{1}) L}) = 0, and this implies λ = m k_{1} (π / L) .

(3.39)

At the transmission point

x = ξ

we get

p (ξ) = p (0) e^{- i (λ / k_{1}) ξ}, q (ξ) = - p (0) e^{i (λ / k_{1}) ξ},

and consequently using (3.38) we arrive at

p (ξ) + q (ξ) = p (0) (e^{- i (λ / k_{1}) ξ} - e^{i (λ / k_{1}) ξ}) = 0.

This implies that

e^{i (2 λ / k_{1}) ξ} = 1

hence

(λ / k_{1}) ξ = n π

. Substitution of

λ

given in (3.39) yields

ξ = \frac{n}{m} L,

but this is contrary to our hypotheses. So

p (0) = 0

, hence

W = 0

, and this is a contradiction. Finally, suppose that there exists

0 \neq W \in D (A_{3})

such that

A_{3} W = i λ W

λ \in R

. Since

A_{3}

is dissipative we get

\int_{0}^{L} [| p (s) |^{2} + | q (s) |^{2}] d s = 0.

(3.40)

But this implies

W = 0

, which is a contradiction.

Let us introduce the function $F_{ξ} (λ)$ :

F_{ξ} (λ) = \sin^{2} (\frac{λ}{k_{1}} L) + \frac{γ_{1}^{2}}{k_{1}^{2}} \sin^{2} (\frac{λ}{k_{1}} ξ) \sin^{2} (\frac{λ}{k_{1}} (L - ξ)),

and let us denote by

A = {ξ \in (0, L) : inf F_{ξ} (R) > 0}, i = 1, 2.

Lemma 3.10

For any $ξ \neq (n / m) L$ , $\forall n, m \in N$ , with $n$ and $m$ co-prime we have

I := inf F_{ξ} (R) > 0.

Proof.

First let us consider any bounded interval $[0, N]$ and let us denote by

I_{N} := inf F_{ξ} ([0, N]) .

We show that

I_{N} > 0

. By contradiction, let us suppose that

I = 0

. So, there exists a sequence of elements

λ_{n} \in [0, N]

such that

\sin^{2} (\frac{λ_{n}}{k_{1}} L) + \frac{γ_{1}^{2}}{k_{1}^{2}} \sin^{2} (\frac{λ_{n}}{k_{1}} ξ) \sin^{2} (\frac{λ_{n}}{k_{1}} (L - ξ)) \to I = 0.

Since

λ_{n} \leq N

there exists a convergent subsequence (we still denote in the same way) such that

λ_{n} \to λ^{*}

and that

\sin^{2} (\frac{λ^{*}}{k_{1}} L) + \frac{γ_{1}^{2}}{k_{1}^{2}} \sin^{2} (\frac{λ^{*}}{k_{1}} ξ) \sin^{2} (\frac{λ^{*}}{k_{1}} (L - ξ)) = 0.

Hence we have that

\frac{λ^{*}}{k_{1}} L = j π, \frac{λ^{*}}{k_{1}} ξ = ν π, j, ν \in N,

(3.41)

\frac{λ^{*}}{k_{1}} L = j π, \frac{λ^{*}}{k_{1}} (L - ξ) = ν π, j, μ \in N .

(3.42)

Let us suppose that (3.41) holds (the other is similar), dividing the identities we get

\frac{ξ}{L} = \frac{ν}{j},

but this is contradictory to our hypothesis

ξ \neq (n / m) L

with

n, m \in N ∖ {0}

, therefore

I_{N} > 0

. Finally, we prove that

I := inf F_{ξ} (R) > 0.

Note that

F_{ξ}

is an almost periodic function, then taking

ϵ = I_{N} / 2

we get that there exists a translation

T > 0

for which we have

| F_{ξ} (λ) - F_{ξ} (λ + T) | \leq ϵ \forall λ \in R .

Therefore,

F_{ξ} (λ + T) \geq F_{ξ} (λ) - ϵ \geq \frac{I_{N}}{2} .

This implies that for any subinterval of the form

[m, m + T]

we have that

F_{ξ} (λ) \geq I_{N} / 2

, with

m \in N

. The proof is now complete.

In that follows we will show that $A_{i}$ , $i = 1, 2, 3$ , is the infinitesimal generator of contractions semigroup that decays exponentially to zero.

Lemma 3.11

The semigroup $e^{A_{i} t}$ is exponentially stable over $H_{2}$ for $i = 1, 2$ .

Proof.

We only prove it for $i = 1$ , the other is similar. For convenience, we denote $p_{1} = p$ and $q_{1} = q$ . We use Theorem 3.2 to show the exponential stability. Because of Lemma 3.9, it is enough to show that the resolvent operator is uniformly bounded over the imaginary axes. Note that the solution of the resolvent system (3.32) is given by

\begin{aligned} p (x) & = p (0) e^{- i (λ / k_{1}) x} + \frac{1}{k_{1}} \int_{0}^{x} e^{- i (λ / k_{1}) (x - s)} f_{1} (s) d s, x \in [0, ξ], \end{aligned}

(3.43)

\begin{aligned} q (x) & = - p (0) e^{i (λ / k_{1}) x} - \frac{1}{k_{1}} \int_{0}^{x} e^{i (λ / k_{1}) (x - s)} f_{2} (s) d s, x \in [0, ξ] . \end{aligned}

(3.44)

Similarly, over

[ξ, L]

we have that

\begin{aligned} p (x) & = p (ξ^{+}) e^{- i (λ / k_{1}) (x - ξ)} + \frac{1}{k_{1}} \int_{ξ}^{x} e^{- i (λ / k_{1}) (x - s)} f_{1} (s) d s, x \in [ξ, L], \end{aligned}

(3.45)

\begin{aligned} q (x) & = q (ξ^{+}) e^{i \frac{λ}{k_{1}} (x - ξ)} - \frac{1}{k_{1}} \int_{ξ}^{x} e^{i \frac{λ}{k_{1}} (x - s)} f_{2} (s) d s, x \in [ξ, L] . \end{aligned}

(3.46)

The above solution verifies equation (3.31) and also the boundary condition at

x = 0

. Using (3.43) and (3.44), we get

p (ξ^{-}) = p (0) e^{- (i λ / k_{1}) ξ} + J_{1}, q (ξ^{-}) = - p (0) e^{i (λ / k_{1}) ξ} + J_{2}, x \in [0, ξ]

(3.47)

where

J_{1} = \frac{1}{k_{1}} \int_{0}^{ξ} e^{- (i λ / k_{1}) (x - s)} f_{1} (s) d s, J_{2} = - \frac{1}{k_{1}} \int_{0}^{ξ} e^{(i λ / k_{1}) (x - s)} f_{2} (s) d s .

(3.48)

Now we adjust

q (ξ^{+})

and

p (ξ^{+})

such that the transmission conditions (3.28) hold for

i = 1

\begin{aligned} p (ξ^{+}) + q (ξ^{+}) & = p (ξ^{-}) + q (ξ^{-}), \\ p (ξ^{+}) - q (ξ^{+}) & = p (ξ^{-}) - q (ξ^{-}) + \frac{γ_{1}}{k_{1}} (p (ξ^{+}) + q (ξ^{+})) . \end{aligned}

Solving the above system, we get

p (ξ^{+}) = p (ξ^{-}) + \frac{γ_{1}}{2 k_{1}} (p (ξ^{-}) + q (ξ^{-})), q (ξ^{+}) = q (ξ^{-}) - \frac{γ_{1}}{2 k_{1}} (p (ξ^{-}) + q (ξ^{-})) .

Applying (3.47) we obtain

\begin{aligned} p (ξ^{+}) & = p (0) e^{- i (λ / k_{1}) ξ} - \frac{γ_{1}}{2 k_{1}} p (0) (e^{i (λ / k_{1}) ξ} - e^{- i (λ / k_{1}) ξ}) + J_{1} + \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2}), \\ q (ξ^{+}) & = - p (0) e^{i (λ / k_{1}) ξ} + \frac{γ_{1}}{2 k_{1}} p (0) (e^{i (λ / k_{1}) ξ} - e^{- i (λ / k_{1}) ξ}) + J_{2} - \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2}) . \end{aligned}

Hence, with this choice, the transmission conditions (3.27) and (3.28) hold. Finally, we take

p (0)

such that the boundary conditions at

x = L

hold.

\begin{aligned} p (ξ^{+}) e^{- i (λ / k_{1}) (x - ξ)} & = p (0) e^{- i (λ / k_{1}) x} - \frac{γ_{1}}{2 k_{1}} p (0) (e^{i (λ / k_{1}) ξ} - e^{- i (λ / k_{1}) ξ}) e^{- (i λ / k_{1}) (x - ξ)} + (J_{1} + \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2})) e^{- (i λ / k_{1}) (x - ξ)}, \\ q (ξ^{+}) e^{i (λ / k_{1}) (x - ξ)} & = - p (0) e^{(i λ / k_{1}) x} + \frac{γ_{1}}{2 k_{1}} p (0) (e^{i (λ / k_{1}) ξ} - e^{- i (λ / k_{1}) ξ}) e^{(i λ / k_{1}) (x - ξ)} + (J_{2} - \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2})) e^{(i λ / k_{1}) (x - ξ)} . \end{aligned}

Using (3.45) and (3.46) we get that

q (L) + p (L) = 0

implies

0 = - p (0) (e^{i (λ / k_{1}) L} - e^{- i (λ / k_{1}) L}) + \frac{γ_{1}}{2 k_{1}} p (0) (e^{i (λ / k_{1}) ξ} - e^{- i (λ / k_{1}) ξ}) (e^{(i λ / k_{1}) (L - ξ)} - e^{(- i λ / k_{1}) (L - ξ)}) + G,

where

\begin{aligned} G & = (J_{1} + \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2})) e^{- (i λ / k_{1}) (L - ξ)} + (J_{2} - \frac{γ_{1}}{2 k_{1}} (J_{1} + J_{2})) e^{(i λ / k_{1}) (L - ξ)} \\ + \frac{1}{k_{1}} \int_{ξ}^{L} e^{- i (λ / k_{1}) (L - s)} f_{1} (s) d s - \frac{1}{k_{1}} \int_{ξ}^{L} e^{i (λ / k_{1}) (L - s)} f_{2} (s) d s . \end{aligned}

p (0)

has to be chosen such that

\begin{aligned} 0 & = - 2 i p (0) \sin (\frac{λ}{k_{1}} L) + 2 i^{2} \frac{γ_{1}}{k_{1}} p (0) \sin (\frac{λ}{k_{1}} ξ) \sin (\frac{λ}{k_{1}} (L - ξ)) + G \\ = - 2 i p (0) [\sin (\frac{λ}{k_{1}} L) + i \frac{γ_{1}}{k_{1}} \sin (\frac{λ}{k_{1}} ξ) \sin (\frac{λ}{k_{1}} (L - ξ))] + G . \end{aligned}

The existence of

p (0)

and hence the existence solution

U_{2} \in D (A_{1})

will depend on

\sin (\frac{λ}{k_{1}} L) + i \frac{γ_{1}}{k_{1}} \sin (\frac{λ}{k_{1}} ξ) \sin (\frac{λ}{k_{1}} (L - ξ)) \neq 0.

The above expression identically vanishes if and only if

\sin (\frac{λ}{k_{1}} L) = 0, \sin (\frac{λ}{k_{1}} (L - ξ)) = 0, or \sin (\frac{λ}{k_{1}} L) = 0, \sin (\frac{λ}{k_{1}} ξ) = 0,

But the above identity implies

\frac{λ}{k_{1}} L = j π, \frac{λ}{k_{1}} (L - ξ) = m π, j, m \in Z,

and consequently

λ = \frac{j}{L} k_{1} π, ξ = \frac{j - m}{j} L, j, m \in Z, with j \neq 0 .

But this is not possible because it contradicts our hypothesis about

ξ

. Therefore we can write

2 i p (0) = \frac{G}{\sin ((λ / k_{1}) L) + i \frac{γ_{1}}{k_{1}} \sin ((λ / k_{1}) ξ) \sin ((λ / k_{1}) (L - ξ))},

and we find that

| p (0) | \leq c \frac{‖ F ‖_{H_{2}}}{\sqrt{\sin^{2} ((λ / k_{1}) L) + (γ_{1}^{2} / k_{1}^{2}) \sin^{2} ((λ / k_{1}) ξ) \sin^{2} ((λ / k_{1}) (L - ξ))}} = c \frac{‖ F ‖_{H_{2}}}{\sqrt{F_{ξ} (λ)}} .

Using Lemma 3.10 we get that there exists a positive constant

c

such that

| p (0) | \leq c ‖ F ‖_{H_{2}},

from where it follows:

‖ U_{2} ‖_{H_{2}} = {‖ (\begin{matrix} p \\ q \end{matrix}) ‖}_{H_{2}} = ‖ (i λ I - A_{1})^{- 1} F ‖_{H_{2}} \leq c ‖ F ‖_{H_{2}} .

Applying Theorem 3.2 we get the exponential stability of the semigroup

e^{t A_{1}}

Lemma 3.12

The semigroup $e^{A_{3} t}$ is exponentially stable over $H_{2}$ for $i = 1, 2$ .

Proof.

As in Lemma 3.11, we use Theorem 3.2 to show the exponential stability. Because of Lemma 3.9, it is enough to show that the resolvent operator is uniformly bounded over the imaginary axes. The solution of system (3.33) is given by

\begin{aligned} p (x) & = p (0) e^{- (1 / k_{1}) (i λ + c_{5}) x} + \frac{1}{k_{1}} \int_{0}^{x} e^{- (1 / k_{1}) (i λ + c_{5}) (x - s)} f_{1} (s) d s, x \in [0, L], \end{aligned}

(3.49)

\begin{aligned} q (x) & = - p (0) e^{(1 / k_{1}) (i λ + c_{5}) x} - \frac{1}{k_{1}} \int_{0}^{x} e^{(1 / k_{1}) (i λ + c_{5}) (x - s)} f_{2} (s) d s, x \in [0, L] . \end{aligned}

(3.50)

To show that

U_{2} \in D (A_{3})

, we only need to verify that the boundary condition

p (L) + q (L) = 0

. Indeed summing up (3.49) with (3.50), we have to find

p (0)

such that

0 = p (L) + q (L) = p (0) (e^{- (1 / k_{1}) (i λ + c_{5}) L} - e^{(1 / k_{1}) (i λ + c_{5}) L}) + F .

It is not difficult to see that

‖ (e^{- (1 / k_{1}) (i λ + c_{5}) L} - e^{(1 / k_{1}) (i λ + c_{5}) L}) ‖ \geq a > 0.

(3.51)

This because

e^{- (1 / k_{1}) (i λ + c_{5}) L} - e^{(1 / k_{1}) (i λ + c_{5}) L} = e^{(1 / k_{1}) (i λ + c_{5}) L} (e^{- (2 / k_{1}) (i λ + c_{5}) L} - 1)

and

‖ e^{- (2 / k_{1}) (i λ + c_{5}) L} - 1 ‖ \geq 1 - ‖ e^{- (2 / k_{1}) (i λ + c_{5}) L} ‖ = 1 - e^{- 2 c_{5} / k_{1}} := c_{0} > 0.

So we write

p (0) = \frac{F}{(e^{- (1 / k_{1}) (i λ + c_{5}) L} - e^{- (1 / k_{1}) (i λ + c_{5}) L})},

where

\begin{aligned} F = \frac{1}{k_{1}} \int_{0}^{x} e^{- (1 / k_{1}) (i λ + c_{5}) (x - s)} f_{1} (s) d s - \frac{1}{k_{1}} \int_{0}^{x} e^{(1 / k_{1}) (i λ + c_{5}) (x - s)} f_{2} (s) d s . \end{aligned}

Substitution of

p (0)

into (3.49) and (3.50) and using inequality (3.51) we get

‖ U_{2} ‖_{H_{2}} = {‖ (\begin{matrix} p \\ q \end{matrix}) ‖}_{H_{2}} = {‖ (i λ I - A_{3})^{- 1} F ‖}_{H_{2}} \leq c ‖ F ‖_{H_{2}} .

Using Theorem 3.2, the exponential stability follows.

Theorem 3.13

The semigroup $T_{3}$ is exponentially stable, provided that $ξ$ verifies hypotheses of Lemma 3.10.

Proof.

Immediate consequence of Lemma 3.11 and Lemma 3.12.

We are now in a position to state the main result of the paper.

Theorem 3.14

Under the hypothesis of Lemma 3.10, system (2.1)–(2.3) with (1.3)–(1.4) is exponentially stable, that is to say the semigroup $T (t) = e^{t A}$ associated with system (1.2)–(1.4) is exponentially stable.

Proof.

Theorem 3.5 implies that $T - T_{1}$ is a compact operator over $H_{6}$ , hence $ω_{ess} (T (t)) = ω_{ess} (T_{1} (t))$ . Since $T_{1}$ and $T_{2}$ are different representation of the same system we get $ω_{ess} (T_{1} (t)) = ω_{ess} (T_{2} (t)) .$ Moreover from Theorem 3.7 the operator $T_{2} (t) - T_{3} (t)$ is a compact operator over $H_{6}$ hence $ω_{ess} (T_{2} (t)) = ω_{ess} (T_{3} (t))$ . Finally, from Theorem 3.13 we get

ω_{ess} (T (t)) = ω_{ess} (T_{3} (t)) \leq ω (T_{3} (t)) = max {ω_{ess}, ω_{σ} (A)} < 0.

The above inequality together with Lemma 3.3 and Theorem 3.1 implies exponential stability.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: J.E. Muñoz Rivera was financially funded by CNPq project 307947/2022-0. Project Fondecyt 1230914. M.G. Naso has been partially supported by Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM).

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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