Asymptotic synchronization of a coupled system of wave equations on a rectangular domain with reflecting sides

Abstract

We show the sufficiency of Kalman’s rank condition for the uniqueness of solution to a coupled system of wave equations in a rectangular domain. The approach does not need any gap condition on the spectrum of the differential operator and the usual multiplier geometrical condition. Then, the study on the asymptotic synchronization by groups can be improved for the corresponding system.

Keywords

Fourier analysis uniqueness theorem Kalman’s rank condition asymptotic synchronization coupled system of wave equations

1. Introduction and main results

Let $Ω \subset R^{n}$ be a bounded domain with a smooth boundary $Γ = Γ_{1} \cup Γ_{0}$ such that ${\overline{Γ}}_{1} \cap {\overline{Γ}}_{0} = \emptyset$ and $mes (Γ_{1}) > 0$ . For fixing idea, we assume that $mes (Γ_{0}) > 0$ . Let A and D be symmetric and semi-positive definite matrices of order N. Consider the following coupled system of wave equations for the state variable $U = {(u^{(1)}, \dots, u^{(N)})}^{T}$ : $\begin{matrix} (1.1) & \{\begin{matrix} U^{″} - Δ U + A U = 0 & in R^{+} \times Ω, \\ U = 0 & on R^{+} \times Γ_{0}, \\ \partial_{ν} U + D U^{'} = 0 & on R^{+} \times Γ_{1}, \\ t = 0 : U = U_{0}, U^{'} = U_{1} & in Ω, \end{matrix} \end{matrix}$ where $\partial_{ν}$ denotes the outward normal derivative on the boundary $Γ_{1}$ .

In the previous work [14,15], by the frequency domain approach, we show that the asymptotic stability of system (1.1) is equivalent to the uniqueness of solution to the following overdetermined elliptic system $\begin{matrix} (1.2) & \{\begin{matrix} - Δ Φ + A Φ = β^{2} Φ & in Ω, \\ Φ = 0 & on Γ_{0}, \\ \partial_{ν} Φ = 0 & on Γ_{1} \end{matrix} \end{matrix}$ with the boundary observation $\begin{matrix} (1.3) & D Φ = 0 on Γ_{1} . \end{matrix}$ The following result is a simplified version of Theorem 2.6 in [15].

Theorem 1.1.
Let the pair $(A, D)$ satisfy Kalman’s rank condition $\begin{matrix} (1.4) & rank (D, A D, \dots, A^{N - 1}) = N . \end{matrix}$ Assume that there exists a constant $c > 0$ such that $‖ A - c I ‖$ is small enough. Assume furthermore that there exists a positive constant C, independent of β and f, such that the following uniform estimate $\begin{matrix} (1.5) & ‖ ϕ ‖_{L^{2} (Ω)} ⩽ C ‖ f ‖_{L^{2} (Ω)} \end{matrix}$ holds for any given solution ϕ to the overdetermined system: $\begin{matrix} (1.6) & \{\begin{matrix} β^{2} ϕ + Δ ϕ = f & in Ω, \\ ϕ = 0 & on Γ_{0}, \\ \partial_{ν} ϕ = 0 & on Γ_{1} \end{matrix} \end{matrix}$ associated with the boundary observation: $\begin{matrix} (1.7) & ϕ = 0 on Γ_{1} . \end{matrix}$ Then, the overdetermined system ( 1.2 )–( 1.3 ) has only the trivial solution.

Since the observability inequality (1.5) is often based on the multiplier method, the geometrical multiplier condition should be naturally required in applications.

The objective of this paper is to investigate the asymptotic stability of system (1.1) on a domain without the multiplier geometrical condition. More precisely, let Ω be a rectangular domain: $\begin{matrix} (1.8) & Ω = (0, 1) \times (0, a) \end{matrix}$ with the configuration of the boundary $\begin{matrix} (1.9) & Γ_{0} = {(0, y) \cup (1, y), 0 < y < a}, Γ_{1} = {(x, 0) \cup (x, a), 0 < x < 1} . \end{matrix}$ Since the horizontal rays of geometrical optics will be reflected between the two vertical boundaries $Γ_{0}$ , they can not hit the non-diffractive points on the boundary $Γ_{1}$ . So, $Γ_{1}$ can not be used to control the domain Ω in the sense of Bardos–Lebeau–Rauch in [1].

Fig. 1.
Rectangle with reflecting sides.

In this paper, we propose the time domain approach. By LaSalle’s invariance principle (see [2,4,6]), we can show that the asymptotic stability of system (1.1) is equivalent to the uniqueness of solution to the following problem $\begin{matrix} (1.10) & \{\begin{matrix} Φ^{″} - Δ Φ + A Φ = 0 & in R^{+} \times Ω, \\ Φ = 0 & on R^{+} \times Γ_{0}, \\ \partial_{ν} Φ = 0 & on R^{+} \times Γ_{1}, \\ t = 0 : Φ = Φ_{0}, Φ^{'} = Φ_{1} & in Ω \end{matrix} \end{matrix}$ with the boundary observation at the infinite horizon $\begin{matrix} (1.11) & D Φ = 0 on R^{+} \times Γ_{1} . \end{matrix}$ The following result does not need the multiplier geometrical condition and the gap condition on the spectrum of the system.
Theorem 1.2.
Let Ω be the rectangular domain given by ( 1.8 )–( 1.9 ). Let the pair $(A, D)$ satisfy Kalman’s rank condition ( 1.4 ). Assume that there exists a constant $c > 0$ such that $‖ A - c I ‖$ is small enough. Then the overdetermined system ( 1.10 )–( 1.11 ) has only the trivial solution.

On the other hand, in what follows, we will remove the smallness assumption of $‖ A - c I ‖$ and consider A from a small perturbation of $c I$ to an arbitrarily given matrix. This will be an interesting improvement from the point of view of perturbation.
Theorem 1.3.
Let Ω be the rectangular domain given by ( 1.8 )–( 1.9 ) as shown in Fig. 1 . Let the pair $(A, D)$ satisfy Kalman’s rank condition ( 1.4 ). Assume that the elements of A are rational numbers and a is an algebraic number. Then the overdetermined system ( 1.10 )–( 1.11 ) has only the trivial solution.

As an immediate consequence of Theorems 1.2 and 1.3, we have the following result.
Theorem 1.4.
Let Ω be the rectangular domain given by ( 1.8 )–( 1.9 ). Let the pair $(A, D)$ satisfy Kalman’s rank condition ( 1.4 ). Assume that there exists a constant $c > 0$ such that $‖ A - c I ‖$ is small enough, or that the elements of A are rational numbers and the constant a is an algebraic number. Then system ( 1.1 ) is asymptotically stable.

The paper is organized as follows. In Section 2, we first establish the unique continuation for a non-harmonic series at the infinite horizon. Compared with the Ingham’s inequality on a finite interval, the uniqueness at the infinite horizon (see Theorem 2.1) does not require any gap condition on the spectrum of the differential operator $- Δ + A$ . The Section 3 will be devoted to the proof of Theorems 1.2 and 1.3. In Section 4, we improve the study on the asymptotic synchronization by p-groups of system (1.1), which was left in [15].
2. Uniqueness theorem for non-harmonic series

Let Ω be the rectangular domain given by (1.8)–(1.9). Let $λ_{n} > 0$ be an eigenvalue of $- Δ$ with the multiplicity $s_{n}$ and $e_{n}^{(s)}$ with $1 ⩽ s ⩽ s_{n}$ be the associated eigen-functions: $\begin{matrix} (2.1) & \{\begin{matrix} - Δ e_{n}^{(s)} = λ_{n} e_{n}^{(s)} & in Ω, \\ e_{n}^{(s)} = 0 & on Γ_{0}, \\ \partial_{ν} e_{n}^{(s)} = 0 & on Γ_{1} . \end{matrix} \end{matrix}$

Let $δ_{l} ⩾ 0$ be an eigenvalue of A with the multiplicity $μ_{l}$ , associated with the eigenvectors $ω_{l}^{(μ)}$ : $\begin{matrix} A ω_{l}^{(μ)} = δ_{l} ω_{l}^{(μ)}, 1 ⩽ l ⩽ d, 1 ⩽ μ ⩽ μ_{l} . \end{matrix}$

Define the differential operator corresponding to problem (1.10) by $\begin{matrix} (2.2) & A_{0} (\begin{matrix} Φ \\ Ψ \end{matrix}) = (\begin{matrix} Ψ \\ Δ Φ - A Φ \end{matrix}) \end{matrix}$ with the domain of definition $\begin{matrix} (2.3) & D (A_{0}) = \{\begin{matrix} (Φ, Ψ) \in {(H_{Γ_{0}}^{1} (Ω) \times H_{Γ_{0}}^{1} (Ω))}^{N} such that \\ Δ Φ - A Φ \in {(L^{2} (Ω))}^{N} and \partial_{ν} Φ = 0 on Γ_{1} \end{matrix}\}, \end{matrix}$ where $H_{Γ_{0}}^{1} (Ω)$ denotes the subspace of $H^{1} (Ω)$ , composed of functions with vanishing trace on $Γ_{0}$ .

To be clear, recall that $N^{*}$ denotes the set of all the positive integers and $Z^{*}$ the set of all the non-zero integers.

For any given $n \in N^{*}$ and any given l with $1 ⩽ l ⩽ d$ , we take $\begin{matrix} (2.4) & {‖ e_{n}^{(s)} ‖}_{L^{2} (Ω)} = 1, {‖ ω_{l}^{(μ)} ‖}_{R^{N}} = 1 \end{matrix}$ and define $\begin{matrix} (2.5) & e_{- n}^{(s)} = e_{n}^{(s)}, s_{- n} = s_{n}, β_{\pm n, l} = \pm \sqrt{λ_{n} + δ_{l}} . \end{matrix}$ Then $β_{n, l}$ is an eigenvalue of $A_{0}$ , associated with the eigenvectors $\begin{matrix} E_{n, l}^{(s, μ)} = (\begin{matrix} \frac{1}{i β_{n, l}} e_{n}^{(s)} ω_{l}^{(μ)} \\ e_{n}^{(s)} ω_{l}^{(μ)} \end{matrix}), 1 ⩽ s ⩽ s_{n}, 1 ⩽ μ ⩽ μ_{l} . \end{matrix}$ Since $A_{0}$ is a skew-adjoint operator with the compact resolvent, the family of the eigenvectors ${E_{n, l}^{(s, μ)}}$ for all $n \in Z^{*}$ , $1 ⩽ l ⩽ d$ , $1 ⩽ s ⩽ s_{n}$ and $1 ⩽ μ ⩽ μ_{l}$ constitutes a Hilbert basis in the space ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . Then, for any given initial data $(Φ_{0}, Φ_{1}) \in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , we can decompose it as $\begin{matrix} (\begin{matrix} Φ_{0} \\ Φ_{1} \end{matrix}) = \sum_{n \in Z^{*}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} a_{n, l}^{(s, μ)} E_{n, l}^{(s, μ)}) \end{matrix}$ with $\begin{matrix} ‖ Φ_{0} ‖_{{(H^{1} (Ω))}^{N}}^{2} + ‖ Φ_{1} ‖_{{(L^{2} (Ω))}^{N}}^{2} = \sum_{n \in Z^{*}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} {| a_{n, l}^{(s, μ)} |}^{2}) . \end{matrix}$ Accordingly, the solution to problem (1.10) is given by $\begin{matrix} Φ = \sum_{n \in Z^{*}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} e_{n}^{(s)} ω_{l}^{(μ)}) e^{i β_{n, l} t} in R^{+} \times Ω, \end{matrix}$ and the D-observation (1.11) becomes $\begin{matrix} (2.6) & \sum_{n \in Z^{*}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} e_{n}^{(s)} |_{Γ_{1}} D ω_{l}^{(μ)}) e^{i β_{n, l} t} = 0 on R^{+} \times Γ_{1} . \end{matrix}$ Noting (2.4) and the trace embedding, the above series (2.6) converges in ${(C^{0} (0, + \infty; L^{2} (Γ_{1})))}^{N}$ .

Theorem 2.1.
Let the pair $(A, D)$ satisfy Kalman’s rank condition ( 1.4 ). Assume that for any given $\begin{matrix} (2.7) & β_{m, k} = β_{n, l} with (m, k) \neq (n, l), \end{matrix}$ we have $\begin{matrix} (2.8) & \int_{Γ_{1}} e_{m}^{(r)} e_{n}^{(s)} d Γ = 0, 1 ⩽ r ⩽ s_{m}, 1 ⩽ s ⩽ s_{n} . \end{matrix}$ Then, the overdetermined system ( 1.10 )–( 1.11 ) has only the trivial solution.
Proof.
For any given $ϵ > 0$ small enough, let $N_{ϵ} > 0$ such that $\begin{matrix} (2.9) & \sum_{| n | > N_{ϵ}} \sum_{1 ⩽ l ⩽ d} \sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} {| a_{n, l}^{(s, μ)} |}^{2} ⩽ ϵ . \end{matrix}$ Let $\begin{matrix} (\begin{matrix} Φ_{0}^{ϵ} \\ Φ_{1}^{ϵ} \end{matrix}) = \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} a_{n, l}^{(s, μ)} E_{n, l}^{(s, μ)}) . \end{matrix}$ The corresponding solution $Φ^{ϵ}$ to system (1.1) is given by $\begin{matrix} Φ^{ϵ} = \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} e_{n}^{(s)} ω_{l}^{(μ)}) e^{i β_{n, l} t} on R^{+} \times Ω, \end{matrix}$ and the corresponding observation (1.11) is written as $\begin{matrix} (2.10) & D Φ^{ϵ} = \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} (\sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} e_{n}^{(s)} |_{Γ_{1}} D ω_{l}^{(μ)}) e^{i β_{n, l} t} on R^{+} \times Γ_{1} . \end{matrix}$ By trace embedding, there exists a positive constant $c_{0}$ such that $\begin{array}{l} \int_{Γ_{1}} {| D (Φ - Φ^{ϵ}) |}^{2} d Γ ⩽ c_{0} ({‖ Φ_{0} - Φ_{0}^{ϵ} ‖}_{H^{1} (Ω)}^{2} + {‖ Φ_{1} - Φ_{1}^{ϵ} ‖}_{L (Ω)}^{2}) . \end{array}$ By (1.11) and (2.9), we have $\begin{matrix} (2.11) & \int_{Γ_{1}} {| D Φ^{ϵ} |}^{2} d Γ ⩽ c_{0} \sum_{| n | > N_{ϵ}} \sum_{1 ⩽ l ⩽ d} \sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} {| a_{n, l}^{(s, μ)} |}^{2} ⩽ c_{0} ϵ . \end{matrix}$

On the other hand, we write (2.10) as $\begin{matrix} (2.12) & D Φ^{ϵ} = \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} A_{n, l} e^{i β_{n, l} t}, \end{matrix}$ where $\begin{matrix} (2.13) & A_{n, l} = \sum_{s = 1}^{s_{n}} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} e_{n}^{(s)} |_{Γ_{1}} D ω_{l}^{(μ)} . \end{matrix}$ Then, it follows that $\begin{matrix} (2.14) & {| D Φ^{ϵ} |}^{2} = \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} | A_{n, l} |^{2} + \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l) \end{array}} A_{n, l} {\bar{A}}_{m, k} e^{i (β_{n, l} - β_{m, k}) t} . \end{matrix}$ But $\begin{array}{l} \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l) \end{array}} A_{n, l} {\bar{A}}_{m, k} e^{i (β_{n, l} - β_{m, k}) t} \\ (2.15) & = \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l), β_{m, k} = β_{n, l} \end{array}} A_{n, l} {\bar{A}}_{m, k} + \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l), β_{m, k} \neq β_{n, l} \end{array}} A_{n, l} {\bar{A}}_{m, k} e^{i (β_{n, l} - β_{m, k}) t} . \end{array}$ Noting (2.8) and (2.13), the integration on $Γ_{1}$ of the first term on the right-hand side of (2.15) vanishes. Then, it follows that $\begin{array}{l} (2.16) & \begin{matrix} \int_{0}^{T} \int_{Γ_{1}} {| D Φ^{ϵ} |}^{2} d Γ d t = & \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} T \int_{Γ_{1}} | A_{n, l} |^{2} d Γ \\ + \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l), β_{m, k} \neq β_{n, l} \end{array}} \frac{e^{i (β_{n, l} - β_{m, k}) T} - 1}{i (β_{n, l} - β_{m, k})} \int_{Γ_{1}} A_{n, l} {\bar{A}}_{m, k} d Γ . \end{matrix} \end{array}$ Let $\begin{matrix} (2.17) & C_{ϵ} = \sum_{\begin{array}{c} | n |, | m | ⩽ N_{ϵ}, 1 ⩽ l, k ⩽ d \\ (m, k) \neq (n, l), β_{m, k} \neq β_{n, l} \end{array}} \frac{2}{| β_{n, l} - β_{m, k} |} \int_{Γ_{1}} | A_{n, l} {\bar{A}}_{m, k} | d Γ . \end{matrix}$ It follows from (2.11) and (2.16) that $\begin{array}{l} \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} \int_{Γ_{1}} | A_{n, l} |^{2} d Γ \\ ⩽ \frac{C_{ϵ}}{T} + \frac{1}{T} \int_{0}^{T} \int_{Γ_{1}} {| D Φ^{ϵ} |}^{2} d Γ d t ⩽ \frac{C_{ϵ}}{T} + c_{0} ϵ . \end{array}$ Taking $T \to + \infty$ , we get $\begin{matrix} \sum_{| n | ⩽ N_{ϵ}} \sum_{1 ⩽ l ⩽ d} \int_{Γ_{1}} | A_{n, l} |^{2} d Γ ⩽ c_{0} ϵ \end{matrix}$ for any given $ϵ > 0$ small enough. Then, noting (2.13), we get $\begin{matrix} \sum_{s = 1}^{s_{n}} (\sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} D ω_{l}^{(μ)}) e_{n}^{(s)} |_{Γ_{1}} = 0, n \in Z^{}, 1 ⩽ l ⩽ d . \end{matrix}$

Since $e_{n}^{(1)}, \dots, e_{n}^{(s_{n})}$ are the eigenfunctions associated with the same eigenvalue $λ_{n}$ , by Carleman’s uniqueness theorem (see [5,7]), their traces on $Γ_{1}$ are linearly independent, then we get $\begin{matrix} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} D ω_{l}^{(μ)} = 0, n \in Z^{}, 1 ⩽ l ⩽ d, 1 ⩽ s ⩽ s_{n} . \end{matrix}$ Since $ω_{l}^{(1)}, \dots, ω_{l}^{(μ_{l})}$ are the eigenvectors associated with the same eigenvalue $δ_{l}$ , by Lemma 2.4 in [12], Kalman’s rank condition (1.4) implies that $\begin{matrix} \sum_{μ = 1}^{μ_{l}} \frac{a_{n, l}^{(s, μ)}}{i β_{n, l}} ω_{l}^{(μ)} = 0, n \in Z^{}, 1 ⩽ l ⩽ d, 1 ⩽ s ⩽ s_{n} . \end{matrix}$ Then, by the linear independence of $ω_{l}^{(1)}, \dots, ω_{l}^{(μ_{l})}$ , we have $\begin{matrix} a_{n, l}^{(s, μ)} = 0, n \in Z^{}, 1 ⩽ l ⩽ d, 1 ⩽ s ⩽ s_{n}, 1 ⩽ μ ⩽ μ_{l} . \end{matrix}$ Then $Φ_{0} \equiv Φ_{1} \equiv 0$ . The proof is thus achieved. □
Remark 2.2.
If there exist positive constants a and γ such that the sequence ${β_{n, l}}$ satisfies the following gap condition $\begin{matrix} (2.18) & | β_{n + 1, l} - β_{n, l} | ⩾ \frac{C}{n^{γ}} \end{matrix}$ for all $n \in Z^{*}$ and $1 ⩽ l ⩽ d$ , then, using the generalized Ingham’s inequality in [8] and proceeding as in [12], we can show the uniqueness of solution to system (1.1) with the D-observation on a finite interval $\begin{matrix} (2.19) & D Φ = 0 on Γ_{1} \times (0, T) . \end{matrix}$ However, the gap condition (2.18) restricts the applicability of Ingham’s type approach, essentially to one-dimensional problems. Fortunately, the D-observation (1.11) at the infinite horizon $(0, + \infty)$ does not require any gap condition on the sequence ${β_{n, l}}$ . Moreover, even the case that $β_{m, k} = β_{n, l}$ for some $(m, k) \neq (n, l)$ is also tolerated under the orthogonality condition (2.8). This shows the main difference between the two methods.

3. Proof of the main results

The well-posedness of system (1.1) is a standard problem which can be found in [15].

For the sake of completeness, we give a brief reminder here.We endow the space ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ with the inner product: $\begin{matrix} (3.1) & ((U, V), (\tilde{U}, \tilde{V})) = \int_{Ω} {⟨ \nabla U, \nabla \tilde{U} ⟩ + (A U, \tilde{U}) + (V, \tilde{V})} d x, \end{matrix}$ where $(\cdot, \cdot)$ denotes the inner product of $R^{N}$ , while $⟨ \cdot, \cdot ⟩$ denotes the inner product of $M^{N \times N} (R)$ .

Let $\begin{matrix} D (A) = \{\begin{matrix} (U, V) \in {(H_{Γ_{0}}^{1} (Ω) \times H_{Γ_{0}}^{1} (Ω))}^{N} such that \\ Δ U - A U \in {(L^{2} (Ω))}^{N} and \partial_{ν} U + D V = 0 on Γ_{1} \end{matrix}\} . \end{matrix}$ We define the linear unbounded operator $A : D (A) \to {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ by $\begin{matrix} A (U, V) = (V, Δ U - A U) . \end{matrix}$ Then system (1.1) can be formally transformed into the following abstract formulation: $\begin{matrix} {(U, V)}^{'} = A (U, V) . \end{matrix}$

Proposition 3.1.
The operator $A$ is m-accretive, therefore, generates a semigroup of contractions $e^{t A}$ on the space ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . Moreover, the resolvent of $A$ is compact.
Proof.
First, it is easy to check the accretiveness of $A$ : $\begin{matrix} (A (U, V), (U, V)) = - \int_{Γ_{1}} (D V, V) d Γ ⩽ 0 . \end{matrix}$ Now for any given $(E, F) \in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , we solve the equation $(I - A) (U, V) = (E, F)$ . A straightforward computation gives $\begin{matrix} (3.2) & V = U - E \end{matrix}$ and $\begin{matrix} (3.3) & \{\begin{matrix} U - Δ U + A U = E + F & in Ω, \\ U = 0 & on Γ_{0}, \\ \partial_{ν} U + D U = D E & on Γ_{1} . \end{matrix} \end{matrix}$ By Lax–Milgram’s Lemma, problem (3.3) admits a unique solution $U \in {(H_{Γ_{0}}^{1} (Ω))}^{N}$ . Therefore, $Im (I - A) = {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . By Hill–Yosida’s Theorem (see [17]), the operator $A$ generates a semi-group of contractions on the space ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . Moreover, there exists a positive constant C such that $\begin{matrix} (3.4) & ‖ U ‖_{{(H_{Γ_{0}}^{1} (Ω))}^{N}} ⩽ C (‖ D E ‖_{{(H^{- 1 / 2} (Γ_{1}))}^{N}} + ‖ E + F ‖_{{({(H_{Γ_{0}}^{1} (Ω))}^{'})}^{N}}), \end{matrix}$ where ${(H_{Γ_{0}}^{1} (Ω))}^{'}$ is the dual space of $H_{Γ_{0}}^{1} (Ω)$ with respect to the pivot space $L^{2} (Ω)$ . Because of the compactness of the embedding $L^{2} (Ω) \subset H_{Γ_{0}}^{1} (Ω)$ and that of $H_{Γ_{0}}^{1} (Ω) \subset H^{- 1 / 2} (Γ_{1})$ , the map $(E, F) \to (U, V)$ , defined by (3.2)–(3.3), is compact in ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . □

We first clarify the following statement given in Introduction.
Proposition 3.2.
System ( 1.1 ) is asymptotically stable if and only if system ( 1.10 )–( 1.11 ) with initial data $(Φ_{0}, Φ_{1}) \in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ has only the trivial solution.
Proof.
Since the resolvent of $A$ is compact, for any given initial data $(U_{0}, U_{1}) \in D (A)$ , the positive trajectory starting from $(U_{0}, U_{1})$ , defined by $\begin{matrix} (3.5) & O^{+} (U_{0}, U_{1}) = ⋃_{t ⩾ 0} {e^{t A} (U_{0}, U_{1})} \end{matrix}$ is bounded for the graph norm in $D (A)$ , therefore, relatively compact in ${(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . Then the ω-limit set $ω (U_{0}, U_{1}) \subset D (A)$

For any given initial data $(Ψ_{0}, Ψ_{1}) \in ω (U_{0}, U_{1}) \subset D (A)$ , denote by $(Ψ (t), Ψ^{'} (t)) = e^{t A} (Ψ_{0}, Ψ_{1})$ the corresponding trajectory, namely, we have $\begin{matrix} (3.6) & \{\begin{matrix} Ψ^{″} - Δ Ψ + A Ψ = 0 & in R^{+} \times Ω, \\ Ψ = 0 & on R^{+} \times Γ_{0}, \\ \partial_{ν} Ψ + D Ψ^{'} = 0 & on R^{+} \times Γ_{1}, \\ t = 0 : Ψ = Ψ_{0}, Ψ^{'} = Ψ_{1} & in Ω . \end{matrix} \end{matrix}$ By LaSalle’s invariance principle (cf. Theorem 9.2.3 in [2] or Theorem 1 in [4], see also [6,18] for applications), $ω (U_{0}, U_{1})$ is invariant for $e^{t A}$ and the restriction of $e^{t A}$ on $ω (U_{0}, U_{1})$ is an isometry for all $t ⩾ 0$ , therefore, the boundary damping should vanish: $\begin{matrix} (3.7) & \int_{Γ_{1}} {(D Ψ^{'}, Ψ^{'})}_{R^{N}} d Γ = 0, \forall t > 0 . \end{matrix}$ Since D is symmetric and semi-positive, we get thus the boundary dissipation $\begin{matrix} (3.8) & D Ψ^{'} = 0 on R^{+} \times Γ_{1} . \end{matrix}$ By the contraction of $e^{t A}$ , system (1.1) is asymptotically stable if and only if $ω (U_{0}, U_{1})$ is reduced to $(0, 0)$ for all the initial data $(U_{0}, U_{1}) \in D (A)$ , therefore, if and only if the overdetermined system (3.6)–(3.8) with the initial data $(Ψ_{0}, Ψ_{1}) \in D (A)$ has only the trivial solution. However (see [16]), setting $Φ = Ψ^{'}$ , we may check the equivalence between the uniqueness of solution to system (3.6)–(3.8) with the initial data $(Ψ_{0}, Ψ_{1}) \in D (A)$ and the uniqueness of solution to system (1.10)–(1.11) with the initial data $(Φ_{0}, Φ_{1}) \in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ . □

We are now going to apply Theorem 2.1. First we consider the corresponding eigen-problem of system (2.1) as follows $\begin{matrix} \{\begin{matrix} - Δ ϕ = λ ϕ & in Ω, \\ ϕ = 0 & on Γ_{0}, \\ \partial_{ν} ϕ = 0 & on Γ_{1}, \end{matrix} \end{matrix}$ where Ω is the rectangular domain with reflecting sides given by (1.8)–(1.9). By a straightforward computation as in [3], we get the following eigenvalues and the associated eigenvectors: $\begin{matrix} (3.9) & λ_{m, n} = (m^{2} + \frac{n^{2}}{a^{2}}) π^{2}, ϕ_{m, n} = sin (m π x) cos (\frac{n π y}{a}) \end{matrix}$ for all $(m, n) \in N^{} \times N$ . In particular, on the boundary $Γ_{1}$ we have $\begin{matrix} (3.10) & ϕ_{m, n} = \{\begin{matrix} sin (m π x) & on (x, y) = (0, 1) \times {0}, \\ {(- 1)}^{n} sin (m π x) & on (x, y) = (0, 1) \times {a} . \end{matrix} \end{matrix}$ According to (2.5), we put $\begin{matrix} (3.11) & β_{m, n, k}^{2} = λ_{m, n} + δ_{k} = (m^{2} + \frac{n^{2}}{a^{2}}) π^{2} + δ_{k}, \end{matrix}$ where $δ_{k}$ ( $1 ⩽ k ⩽ d$ ) stands for the distinct eigenvalues of A.

We next check that the following condition $\begin{matrix} λ_{m, n} + δ_{k} = λ_{p, q} + δ_{l} with (m, n, k) \neq (p, q, l) \end{matrix}$ implies that $\begin{matrix} \int_{Γ_{1}} ϕ_{m, n} ϕ_{p, q} d Γ = 0 . \end{matrix}$ Noting (3.10)–(3.11), it is sufficient to check that the equality $\begin{matrix} (3.12) & (m^{2} - p^{2} + \frac{n^{2} - q^{2}}{a^{2}}) π^{2} = δ_{l} - δ_{k} with (m, n, k) \neq (p, q, l) \end{matrix}$ implies that $m \neq p$ . Proof of Theorem 1.2.
When $k = l$ , it is easy to see that equality (3.12) implies that $m \neq p$ . So we consider only the case that $k \neq l$ . Define the set $\begin{matrix} C_{a} = {m^{2} - p^{2} + \frac{n^{2} - q^{2}}{a^{2}}, (m, n), (p, q) \in N^{} \times N} . \end{matrix}$

If $a^{2}$ is a rational number, then $C_{a}$ is a discrete set and the elements in $C_{a}$ have gaps. When $‖ A - c I ‖$ is small enough, equality (3.12) can not take place.

If $a^{2}$ is an irrational number, by Dirichlet’s theorem on Diophantine approximation (Corollary 1B in [19]), the set $C_{a}$ is dense in $R$ , so there is no positive gap for the elements in $C_{a}$ . However, since $k \neq l$ , as $‖ A - c I ‖$ is small enough, we have $\begin{matrix} 0 < | δ_{k} - δ_{l} | < \frac{π^{2}}{a^{2}}, \end{matrix}$ then, equality (3.12) implies well $m \neq p$ . The proof is thus achieved. □
Proof of Theorem 1.3.
When the elements of A are rational numbers, the characteristic polynomial of A has only rational coefficients, then the roots $δ_{k}$ ( $1 ⩽ k ⩽ d$ ) should be algebraic numbers. On the other hand, since a as well as $\frac{1}{a^{2}}$ are all algebraic numbers, then $\begin{matrix} λ_{m, n} - λ_{p, q} = (m^{2} - p^{2} + \frac{n^{2} - q^{2}}{a^{2}}) π^{2} \end{matrix}$ is a transcendent number or 0. Then the equality $\begin{matrix} λ_{m, n} - λ_{p, q} = δ_{l} - δ_{k} \end{matrix}$ implies that $k = l$ , and it is easy to see that $m \neq p$ . The proof is complete. □

4. Asymptotic synchronization by groups

In [15], we established a general theory on the asymptotic synchronization for linear dissipative systems, in particular, for a coupled system of wave equations, when Ω is a smooth bounded domain with the geometrical multiplier condition.

In this section, we will continue the study when Ω is a rectangular domain with reflecting sides. We first recall briefly the usual notations.

Let $p ⩾ 1$ be an integer such that $\begin{matrix} 0 = n_{0} < n_{1} < \dots < n_{p} = N \end{matrix}$ with $n_{r} - n_{r - 1} ⩾ 2$ for $r = 1, \dots, p$ . We re-arrange the components of the state variable U into p groups $\begin{matrix} (u^{(1)}, \dots, u^{(n_{1})}), (u^{(n_{1} + 1)}, \dots, u^{(n_{2})}), \dots, (u^{(n_{p - 1} + 1)}, \dots, u^{(n_{p})}) . \end{matrix}$

Let $S_{r}$ be a full row-rank matrix of order $(n_{r} - n_{r - 1} - 1) \times (n_{r} - n_{r - 1})$ : $\begin{matrix} S_{r} = (\begin{matrix} 1 & - 1 \\ 1 & - 1 \\ ⋱ & ⋱ \\ 1 & - 1 \end{matrix}), 1 ⩽ r ⩽ p . \end{matrix}$ Define the $(N - p) \times N$ matrix $C_{p}$ of synchronization by p-groups as $\begin{matrix} C_{p} = (\begin{matrix} S_{1} \\ S_{2} \\ ⋱ \\ S_{p} \end{matrix}) . \end{matrix}$

Let $\begin{matrix} Ker (C_{p}) = Span {e_{1}, \dots, e_{p}}, \end{matrix}$ where $\begin{matrix} e_{r} = {(0, \dots, 0, \overset{(n_{r - 1} + 1)}{1}, \dots, \overset{(n_{r})}{1}, 0, \dots, 0)}^{T}, 1 ⩽ r ⩽ p . \end{matrix}$

Definition 4.1.

System (1.1) is asymptotically synchronizable by p-groups, if for any given initial data $(U_{0}, U_{1}) \in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N}$ , the corresponding solution U satisfies $\begin{array}{l} (4.1) & (u^{(k)} - u^{(l)}, {u^{(k)}}^{'} - {u^{(l)}}^{'}) \to (0, 0) in H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω) \end{array}$ as $t \to + \infty$ for all $n_{r - 1} + 1 ⩽ k, l ⩽ n_{r}$ and $1 ⩽ r ⩽ p$ , or equivalently $\begin{matrix} (4.2) & C_{p} (U, U^{'}) \to (0, 0) in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N - p} as t \to + \infty . \end{matrix}$

Let us recall some known results. First, by Theorem 4.7 in [15], if system (1.1) is asymptotically synchronizable by p-groups, then, $\begin{matrix} rank (D, A D, \dots, A^{N - 1}) ⩾ N - p . \end{matrix}$

Inversely, by Theorem 4.8 in [15], if system (1.1) is asymptotically synchronizable by p-groups under the minimum rank condition $\begin{matrix} (4.3) & rank (D, A D, \dots, A^{N - 1}) = N - p, \end{matrix}$ then the matrix A satisfies the condition of $C_{p}$ -compatibility: $\begin{matrix} (4.4) & A Ker (C_{p}) \subseteq Ker (C_{p}), \end{matrix}$ or equivalently, by Proposition 4.2 in [15], there exists a symmetric and semi-positive definite matrix ${\bar{A}}_{p}$ of order $(N - p)$ , such that $\begin{matrix} (4.5) & {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} A = {\bar{A}}_{p} {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} . \end{matrix}$

Respectively, the control matrix D satisfies the condition of strong $C_{p}$ -compatibility, if we have $\begin{matrix} (4.6) & Ker (C_{p}) \subseteq Ker (D), \end{matrix}$ or equivalently, by Proposition 4.4 in [15], there exists a symmetric and semi-positive definite matrix R of order $(N - p)$ , such that $\begin{matrix} (4.7) & D = C_{p}^{T} R C_{p} . \end{matrix}$ Thus, setting $\begin{matrix} (4.8) & {\overline{D}}_{p} = {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} D C_{p}^{T} {(C_{p} C_{p}^{T})}^{- 1 / 2}, \end{matrix}$ we have $\begin{matrix} (4.9) & {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} D = {\overline{D}}_{p} {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} . \end{matrix}$

Under the conditions of $C_{p}$ -compatibility, applying ${(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p}$ to system (1.1) and setting $W = {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} U$ , we get the following reduced system $\begin{matrix} (4.10) & \{\begin{matrix} W^{″} - Δ W + {\bar{A}}_{p} W = 0 & in R^{+} \times Ω, \\ W = 0 & on R^{+} \times Γ_{0}, \\ \partial_{ν} W + {\overline{D}}_{p} W^{'} = 0 & on R^{+} \times Γ_{1} . \end{matrix} \end{matrix}$ Obviously, the asymptotic synchronization by p-groups of system (1.1) is equivalent to the asymptotic stability of the reduced system (4.10). Moreover, we have

Theorem 4.2.

Let Ω be the rectangular domain given by ( 1.8 )–( 1.9 ). Let A satisfy the condition of $C_{p}$ -compatibility ( 4.4 ), respectively, D satisfy the condition of strong $C_{p}$ -compatibility ( 4.6 ). Assume that A satisfies the rank condition ( 4.3 ) and that there exists a constant $c > 0$ such that $‖ A - c I ‖$ is small enough. Then system ( 1.1 ) is asymptotically synchronizable by p-groups.

Proof.

Since the reduced matrices ${\bar{A}}_{p}$ and ${\overline{D}}_{p}$ are still symmetric and semi-positive definite, the asymptotic stability of the reduced system (4.10) can be treated by Theorem 1.4.

First, using (4.5) and (4.8), we have $\begin{matrix} {\bar{A}}_{p} {\overline{D}}_{p} = {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} A D C_{p}^{T} {(C_{p} C_{p}^{T})}^{- 1 / 2} . \end{matrix}$ Successively, noting (4.5), we have $\begin{matrix} {\bar{A}}_{p}^{2} {\overline{D}}_{p} = {\bar{A}}_{p} ({\bar{A}}_{p} {\overline{D}}_{p}) = {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} A^{2} D C_{p}^{T} {(C_{p} C_{p}^{T})}^{- 1 / 2} \end{matrix}$ and so on. Then by Cayley–Hamilton’s theorem, we get $\begin{array}{l} rank ({\overline{D}}_{p}, \bar{A} {\overline{D}}_{p}, \dots, {\bar{A}}_{p}^{N - p - 1} {\overline{D}}_{p}) \\ = rank {(C_{p} C_{p}^{T})}^{- 1 / 2} C_{p} (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) \\ (4.11) & = rank C_{p} (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) . \end{array}$

Now, noting (4.4), it is easy to see that $A^{T} {Ker (C_{p})}^{⊥} \subseteq {Ker (C_{p})}^{⊥}$ . Since $A^{T} = A$ and ${Ker (C_{p})}^{⊥} = Im (C_{p}^{T})$ , we get $A Im (C_{p}^{T}) \subseteq Im (C_{p}^{T})$ . By condition (4.7), we have $Im (D) \subset Im (C_{p}^{T})$ . Then, we successively get $Im (A D) \subseteq A Im (C_{p}^{T}) \subseteq Im (C_{p}^{T})$ , $Im (A^{2} D) \subseteq A Im (C_{p}^{T}) \subseteq Im (C_{p}^{T})$ and so on. It follows that $\begin{array}{l} Ker (C_{p}) \cap Im (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) \\ (4.12) & \subseteq Ker (C_{p}) \cap Im (C_{p}^{T}) = Ker (C_{p}) \cap {Ker (C_{p})}^{⊥} = {0} . \end{array}$ By Proposition 2.7 in [13], we get $\begin{array}{l} rank C_{p} (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) \\ (4.13) & = rank (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) . \end{array}$

Next, we write the transposition of the matrix on the right-hand side of (4.13) as $\begin{matrix} (\begin{matrix} C_{p} D \\ C_{p} D A \\ ⋮ \\ C_{p} D A^{N - 1} \end{matrix}) = (\begin{matrix} C_{p} \\ C_{p} \\ ⋱ \\ C_{p} \end{matrix}) (\begin{matrix} D \\ D A \\ ⋮ \\ D A^{N - 1} \end{matrix}) . \end{matrix}$ By (4.7), it is easy to see $\begin{matrix} Ker (\begin{matrix} C_{p} \\ C_{p} \\ ⋱ \\ C_{p} \end{matrix}) \cap Im (\begin{matrix} D \\ D A \\ ⋮ \\ D A^{N - 1} \end{matrix}) = {0} . \end{matrix}$ Similarly as for (4.13), we get $\begin{matrix} rank (\begin{matrix} C_{p} D \\ C_{p} D A \\ ⋮ \\ C_{p} D A^{N - 1} \end{matrix}) = rank (\begin{matrix} D \\ D A \\ ⋮ \\ D A^{N - 1} \end{matrix}), \end{matrix}$ namely, $\begin{array}{l} rank (D C_{p}^{T}, A D C_{p}^{T}, \dots, A^{N - 1} D C_{p}^{T}) \\ (4.14) & = rank (D, A D, \dots, A^{N - 1} D) . \end{array}$

Finally, combining (4.11)–(4.14) and using the rank condition (4.3), we obtain $\begin{array}{l} rank ({\overline{D}}_{p}, {\bar{A}}_{p} {\overline{D}}_{p}, \dots, {\bar{A}}_{p}^{N - p - 1} {\overline{D}}_{p}) = N - p . \end{array}$ By Theorem 1.4, the reduced system (4.10) is asymptotically stable, namely, the original system (1.1) is asymptotically synchronizable by p-groups. □

The following result does not require the condition of $C_{p}$ -compatibility for A and the condition of strong $C_{p}$ -compatibility for D, so, improves Theorem 4.11 given in [15].

Theorem 4.3.

Let Ω be the rectangular domain given by ( 1.8 )–( 1.9 ). Assume that there exists a constant $c > 0$ such that $‖ A - c I ‖$ is small enough. If system ( 1.1 ) is asymptotically synchroizable by p-groups, then there exist functions $u_{1}, \dots, u_{p}$ in $C^{0} (0, + \infty; H_{Γ_{0}}^{1} (Ω)) \cap C^{1} (0, + \infty; L^{2} (Ω))$ , such that, setting $\begin{matrix} (4.15) & u = \sum_{r = 1}^{p} u_{r} e_{r}, \end{matrix}$ we have $\begin{matrix} (4.16) & (U - u, U^{'} - u^{'}) \to (0, 0) in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N} as t \to + \infty . \end{matrix}$

Proof.

By Theorem 4.7 in [15], the asymptotic synchronization by p-groups of system (1.1) implies $\begin{matrix} (4.17) & rank (D, A D, \dots, A^{N - 1} D) = N - q \end{matrix}$ with $q ⩽ p$ . Then, by Lemma 2.4 in [12], we have $\begin{matrix} V = Span {{\tilde{e}}_{1}, \dots, {\tilde{e}}_{q}} \subseteq Ker (D), \end{matrix}$ where V is the largest invariant subspace of A, contained in $Ker (D)$ . Let ${\tilde{C}}_{q}$ be a full row-rank matrix of order $(N - q) \times N$ , defined by $\begin{matrix} Ker ({\tilde{C}}_{q}) = V . \end{matrix}$ A satisfies the corresponding condition of ${\tilde{C}}_{q}$ -compatibility (4.4), and D satisfies the corresponding condition of strong ${\tilde{C}}_{q}$ -compatibility (4.6). Then, by Theorem 4.2, we get $\begin{matrix} (4.18) & {\tilde{C}}_{q} (U, U^{'}) \to (0, 0) in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N - q} as t \to + \infty . \end{matrix}$

Since the matrix A is symmetric, we may assume that ${\tilde{e}}_{1}, \dots, {\tilde{e}}_{q}$ are the orthonormal eigenvectors of A. It follows that $\begin{matrix} U = \tilde{u} + {\tilde{C}}_{q}^{T} {({\tilde{C}}_{q} {\tilde{C}}_{q}^{T})}^{- 1} {\tilde{C}}_{q} U, \end{matrix}$ where $\begin{matrix} \tilde{u} = \sum_{r = 1}^{q} {\tilde{u}}_{r} {\tilde{e}}_{r} with {\tilde{u}}_{r} = (U, {\tilde{e}}_{r}), 1 ⩽ r ⩽ q . \end{matrix}$ Then, noting (4.18), we have $\begin{matrix} (4.19) & (U - \tilde{u}, U^{'} - {\tilde{u}}^{'}) \to (0, 0) in {(H_{Γ_{0}}^{1} (Ω) \times L^{2} (Ω))}^{N} as t \to + \infty . \end{matrix}$

Next, multiplying (1.1) by ${\tilde{e}}_{s}$ for $s = 1, \dots, q$ , we get $\begin{matrix} (4.20) & \{\begin{matrix} {\tilde{u}}_{s}^{″} - Δ {\tilde{u}}_{s} + λ_{s} {\tilde{u}}_{s} = 0 & in R^{+} \times Ω, \\ {\tilde{u}}_{s} = 0 & on R^{+} \times Γ_{0}, \\ \partial_{ν} {\tilde{u}}_{s} = 0 & on R^{+} \times Γ_{1} . \end{matrix} \end{matrix}$ For any given $r = 1, \dots, q$ , taking $\begin{matrix} t = 0 : {‖ ({\tilde{u}}_{r}, {\tilde{u}}_{r}^{'}) ‖}_{H^{1} (Ω) \times L^{2} (Ω)} = 1, {\tilde{u}}_{s} = {\tilde{u}}_{s}^{'} = 0, s \neq r, \end{matrix}$ it follows that $\begin{array}{l} ‖ C_{p} {\tilde{e}}_{r} ‖ & = ‖ C_{p} {\tilde{e}}_{r} ({\tilde{u}}_{r}, {\tilde{u}}_{r}^{'}) ‖ \\ ⩽ ‖ C_{p} (U - {\tilde{u}}_{r} {\tilde{e}}_{r}, U^{'} - {\tilde{u}}_{r}^{'} {\tilde{e}}_{r}) ‖ + ‖ C_{p} (U, U^{'}) ‖ \to 0 as t \to + \infty . \end{array}$ We thus get $C_{p} {\tilde{e}}_{r} = 0$ for $r = 1, \dots, q$ , namely, $Ker ({\tilde{C}}_{q}) \subseteq Ker (C_{p})$ . We can then write $\begin{matrix} {\tilde{e}}_{r} = \sum_{s = 1}^{p} α_{s r} e_{s}, r = 1, \dots, q . \end{matrix}$ Accordingly, we have $\begin{matrix} \tilde{u} = \sum_{r = 1}^{q} {\tilde{u}}_{r} {\tilde{e}}_{r} = \sum_{r = 1}^{q} \sum_{s = 1}^{p} {\tilde{u}}_{r} α_{s r} e_{s} = \sum_{s = 1}^{p} (\sum_{r = 1}^{q} α_{s r} {\tilde{u}}_{r}) e_{s} . \end{matrix}$ Then, inserting the expression $\begin{matrix} (4.21) & u_{s} = \sum_{r = 1}^{q} α_{s r} {\tilde{u}}_{r}, s = 1, \dots, p \end{matrix}$ into (4.19), we get the convergence (4.16), called the asymptotic synchronization by p-groups in the pinning sense. However, when $q < p$ , it follows from (4.21) that the functions $u_{1}, \dots, u_{p}$ are linearly dependent. □

Remark 4.4.

Instead of the smallness assumption of $‖ A - c I ‖$ , Theorems 4.2 and 4.3 remain true if A has rational elements and a is an algebraic number. The previous approach is quite flexible and can be easily applied to other types of wave equations [9,10,20].

Footnotes

Acknowledgement

This work is partially supported by National Natural Sciences Foundation of China under Grant 11831011.

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