On a partially synchronizable system for a coupled system of wave equations in one dimension

1.   Introduction

The research on the exact boundary synchronization of coupled system of wave equations mainly focuses on the case of the same wave speed ^{[1,3,4,5,6,7,8]}. One of the important issues is that the system can still maintain synchronization after removal of all of the boundary controls. However, it is clear that not all systems can have this property, and only the systems that satisfy the compatibility conditions have such properties.

In ^[2], Lei et al. put forward the concept of a synchronizable system, that is, if a system has a synchronized solution, then it is called a synchronizable system. They studied the coupled system of wave equations with different wave speeds and obtained all of the synchronized initial values which can make the system have synchronized solutions.

Previous studies on the synchronizable system focused on the framework of weak solutions in $N$-dimensions. Here, we consider the synchronizable system and the partially synchronizable system in the framework of classical solutions in one dimension. We mainly study the coupled system of wave equations with different wave speeds.

We consider the following coupled system with different wave speeds:

where $U = (u_{1}, \cdots, u_{N})^T$ is the state variable and $\Lambda = \text{diag}(c_1^2, \cdots, c_N^2)$ is the wave speed matrix with $c_i > 0\ (i = 1, \cdots, N)$, which are not all equal. $A = (a_{ij})$ is an $N\times N$ coupling matrix with constant elements. $U_t$ represents the partial derivative with respect to time $t$, and $U_x$ represents the partial derivative with respect to the spatial variable $x$.

The system (1.1) satisfies the following homogeneous Dirichlet boundary conditions

and it has the following initial data

For any given $m(0 < m < N, m\in\mathbb{N})$, we give the following definition

Definition 1.1. The system given by (1.1)–(1.2) is called a partially synchronizable system if there is an initial value $(U_{0}, U_{1})$ such that the solution $U = U(t, x)$ to the problem given by (1.1)–(1.3) satisfies the following partial synchronization property:

where $\tilde{\tilde{u}}$ is called the partially synchronizable state.

The initial value $(U_0, U_1)$ of a partially synchronized solution also satisfy the following partial synchronization properties:

If $(U_0, U_1)\equiv(0, 0)$, then the system admits a solution $U\equiv 0$, which is, of course, a partially synchronized solution. Thus, in the following discussion, it is natural to exclude some trivial situations.

In Section 2, we consider the case in which both $\Lambda$ and $A$ satisfy the compatibility conditions. In Section 3, we study the partially synchronizable system without the compatibility conditions for some cases.

2.   The case with compatibility

In this section, we consider the partially synchronizable system, that is the system possesses partially synchronized solutions, for a coupled system of wave equations with different wave speeds.

Let

that is, consider an $(N-m-1)\times N$ full-row rank matrix. Let $\{e_1, \cdots, e_m, e_{m+1}\}$ be $N$-dimensional column vectors defined as follows: for $1\le s\le m$,

and for $s = m+1$,

It is obvious that

Let $u = (u_1, \cdots, u_m, u_{m+1})^T$; then, the partial synchronization condition is equivalent to

The component forms of the system given by(1.1)–(1.2) are as follows

Assume that there exists the initial value $(U_0, U_1)$ such that the system (1.1) has a partially synchronized solution $U = U(t, x):u_{m+1} = \cdots = u_{N}$. From (2.1), we get

where

We denote

where $\tilde{\Lambda}_{m+1}$ and $\tilde{A}_{m+1}$ are related to $k$. Then the system (2.2) can be written as

By (2.2), we have that

is indepndent of $k$, even though $\tilde{\Lambda}_{m+1}$ and $\tilde{A}_{m+1}$ may be related to $k$.

From (2.1), for $\forall i\in\{1, \cdots, m\}$ and $\forall k, l\in\{m+1, \cdots, N\}$, we get

When $c_{m+1} = \cdots = c_N$, we have

If $u_1, \cdots, u_m, u_{m+1}$ are linearly independent, we get

Therefore, we have the following conclusions

Theorem 2.1. If the system given by (1.1)–(1.2) is a partially synchronizable system with ${u_{m+1}}_{xx}$, where $u_1, \cdots, u_{m}, u_{m+1}$ are linearly independent, then matrices $\Lambda$ and $A$ satisfy the $C_{m+1}$-compatibility conditions $c_l = c_k\ (\forall l, k\in\{m+1, \cdots, N\})$ and (2.6), respectively. Conversely, if the system given by (1.1)–(1.2) satisfies the conditions $c_l = c_k\ (\forall l, k\in\{m+1, \cdots, N\})$ and (2.6), then for any initial value $(U_0, U_1)\in(C^2[0, L])^N\times(C^1[0, L])^N$ with the partial synchronization properties

there is a corresponding partially synchronized solution $U = (u_1, \cdots, u_m, u_{m+1}, \cdots, u_{m+1})^T$.

Then consider the corresponding coupled system of wave equations with Dirichlet boundary conditions:

in which $H(t)$ is a boundary condition.

The system is a partially synchronizable system if there exists $T > 0$ such that, for any given initial value $(U_0, U_1)\in (C^2[0, L])^N\times (C^1[0, L])^N$, there exists a boundary conditions $H(t)\in (C^2[0, T])^N$ such that the solution to the problem (2.8) is $U = U(t, x)$ and (1.3) satisfies the following partial synchronization property:

From Theorem 2.1, we have the following:

Corollary 2.2. If matrices $\Lambda$ and $A$ satisfy the conditions $c_l = c_k\ (\forall l, k\in\{m+1, \cdots, N\})$ and (2.6), then the partial exact boundary synchronization (2.9) of system (2.8) is equivalent to that for any given initial value $(U_0, U_1)\in (C^2[0, L])^N\times (C^1[0, L])^N$; also, there exists a boundary condition $H$ such that the corresponding solution $U = U(t, x)$ attains a partially synchronized state at time $t = T$:

3.   The case without compatibility

When

from (2.5), we get

Denote

In (3.2), $-\dfrac{\partial^2 u_{m+1}}{\partial x^2}$ is uniquely expressed by $u_1, \cdots, u_m, u_{m+1}$; then, we know that $\alpha_{1}, \cdots, \alpha_{m}, \alpha_{m+1}$ are constants that are independent of $k, l\ (k, l\in\{m+1, \cdots, N\})$. This gives the relationship of the matrices $\Lambda$ and $A$: $\forall k, l\in\{m+1, \cdots, N\}$:

and (3.2) becomes

Obviously, (3.5) is a necessary condition for the system given by (1.1)–(1.2) to have a partially synchronized solution. However, (3.5) is not a closed system, and it is not easy to obtain information about the partially synchrinizable state from it.

3.1.   A special case

An important special case is that (3.5) is a closed system. At this moment, $\alpha_{1} = 0, \cdots, \alpha_{m} = 0$, that is, $\forall k, l\in\{m+1, \cdots, N\}$ and $a_{k1} = a_{l1}, \cdots, a_{km} = a_{lm}$, which means that

Thus, (3.2) is reduced to a closed system

Hence, if (3.7) is ture, it must follow that

where $\lambda$ is the eigenvalue of $-\dfrac{\mathrm{d}^2}{\mathrm{d} x^2}$. We take

where $u_{\lambda}(x)$ is the eigenfunction of $-\dfrac{\mathrm{d}^2}{\mathrm{d} x^2}$ corresponding to the eigenvalue $\lambda$ and $u_{\lambda}(x)\not\equiv0$. $u_{\lambda}(x)$ and $\lambda$ satisfy the following:

From the previous discussion, we have that $\lambda = \dfrac{n^2\pi^2}{L^2}\ (n\in\mathbb{N}_{+})$. We also have the eigenfunction

where $C_2$ is a non-zero constant. Here, (3.3) becomes

Substituting (3.10) into (2.2), we get that $b(t)$ satisfies

If $u_1, \cdots, u_m, u_{\lambda}(x)$ are linearly independent, we get

Additionally,

where $\tilde{a}_{k, m+1}+\lambda c_{k}^2$ is a constant that is independent of $k$; it is denoted by

Therefore, for any given initial conditions

$b(t)$ satisfies the following:

For any given $k\ (k = m+1, \cdots, N)$, $u_{m+1}$ satisfies

Conversely, if (3.8) and (3.13) holds, then the system is a partially synchoronizable system and it has at least one partially synchronized solution.

In fact, consider (3.17); its solution $u_{m+1}$ is given by (3.10) and (3.16).

Let $U = (u_1, \cdots, u_m, u_{m+1}, \cdots, u_{m+1})$; we get

Then, $U$ is a required partially synchronized solution.

If the system given by (1.1)–(1.2) satisfies the condition $c_l\neq c_k(\forall l, k\in\{m+1, \cdots, N\})$, and (3.13), then not all of the initial values with the partial synchronization property can have partially synchronized solutions. Therefore, when $c_l\neq c_k\ (\forall l, k\in\{m+1, \cdots, N\})$, even if (1.1)–(1.2) represents a partially synchronizable system, it does not mean that all of the partially synchronized initial values of the system can have a partial synchronization solutions.

From the above, we can get the following theorem.

Theorem 3.1. If the system given by (1.1)–(1.2) is a partially synchronizable system with $c_l\neq c_k\ (\forall l, k\in\{m+1, \cdots, N\})$ and (3.13), then the matrices $\Lambda$ and $A$ satisfy (3.11). And, for the following initial conditions with the partial synchronization property

where $u_{\lambda}$ is given by (3.10) and $(b_0, b_1)\neq0$, the system given by (1.1)–(1.2) has a solution with the partial synchronization property

where $b(t)$ is given by the following equation

Proof. By (3.6), (3.2) reduces to (3.7), then condition (3.3) becomes

If (3.7) is ture, $\alpha_{m+1}$ can only take the eigenvalue $\lambda$ of $-\dfrac{\mathrm{d}^2}{\mathrm{d} x^2}$; then, $\Lambda$ and $A$ satisfy (3.11). Moreover, by (3.7), $u_{m+1}$ has the form of (3.9). Substituting (3.9) into (2.2), we get

Then, since (3.13) holds and $u_{\lambda}(x)\not\equiv 0$, we get (3.14).

Conversely, if the system given by (1.1)–(1.2) has the partially synchronized initial condition (3.18), it is easy to verify that the solution (3.19) satisfies the conditions of the system given by (1.1)–(1.2) and the initial condition (3.18); thus, it is the partially synchronized solution.

3.2.   A model

We consider a sample system when $N = 3, m = 1$, as follows:

with the boundary conditions

as well as the partial synchronization initial conditions

Since $u_2 = u_3$, we get

The system is equivalent to

If $c_2 = c_3$, we suppose that $u_1$ and $u_2$ are linearly indepedent; then, there exist $a_{21} = a_{31}$ and $a_{22}+a_{23} = a_{32}+a_{33}$. We have discussed this situation in Theorem 2.1.

If $c_2\neq c_3$, we get

Substituting this formula into (3.23), we have

From (3.24) and (3.25), we can get

and

Hence, from (3.23),

Then, we get

Assume that $u_2$ is smooth enough; then, ${u_2}_{ttxx} = {u_2}_{xxtt}$. Thus, we have

that is,

This means that matrices $\Lambda$ and $A$ satisfy the compatibility condition (3.26). Obviously, this is a necessary condition.

Conversely, we consider whether the system (3.20) is a partially synchronizable system when condition (3.26) holds. Let $w = u_2-u_3$; from (3.20), we get

From the boundary condition (3.21) and initial condition (3.22), we have

and

If

i.e., if (3.24) holds, we get that $w\equiv0$; then, the system (3.20) is a partially synchronizable system. Therefore, (3.24) is a necessary and sufficient condition for system (3.20) to realize partial synchronization.

Let $W = w_{tt}-c_3^2w_{xx}$; from (3.27), we have

and

Then, we get

From (3.20), we have

then

Hence,

According to the condition (3.26), it is easy to see that

Hence,

If the initial and boundary conditions of $W$ satisfy

and

then $W\equiv0$; we immediately get $w\equiv0$. Hence, (3.20) is a partially synchronizable system.

From (3.27), $W = -[(c_3^2-c_2^2){u_2}_{xx}-(a_{21}-a_{31})u_1]$; then, $W_t = -[(c_3^2-c_2^2){u_2}_{xxt}-(a_{21}-a_{31}){u_1}_{t}]$; we need the following:

and

Then, we have

and

Regarding the boundary conditions

From (3.20), we have

By (3.21), ${u_2}_{tt}(t, 0) = 0$ and ${u_2}_{tt}(t, L) = 0$, using (3.21) again, we have

After the above discussion, we have the following conclusions.

Theorem 3.2. If the system given by (3.20)–(3.21) is a partially synchronizable system, then matrices $\Lambda$ and $A$ satisfy the compatibility condition (3.26). Conversely, if (3.26) is satisfied for the initial value $(U_0, U_1)$ with the partial synchronization property (3.22) which also satisfies the conditions (3.29)–(3.30), then the system given by (3.20)–(3.21) has a solution $U = U(t, x)$ satisfying the partial synchronazition condition

3.3.   A general case of N=3

When $N = 3$, $m = 1$, the system (1.1) can be written as

with the boundary conditions

as well as the initial conditions with the partial synchronization property:

And, $m = 1$ means that $u_2 = u_3$; then, the system (3.31) becomes

Therefore, we get a system that is equivalent to (3.34), as follows:

When $c_2 = c_3$, we require that $u_1$ and $u_2$ be linearly independent. From system (3.34), we can get

We have discussed this situation in Theorem 2.1.

When $c_2\neq c_3$, let $u_2 = u_3$; from the third equation in (3.35), we get

and, substituting (3.36) into the second equation in (3.35), we obtain

From (3.36) and (3.37), we have

Assume that $u_2$ is smooth enough, then, ${u_2}_{xxtt} = {u_2}_{ttxx}$; thus, we get

Substituting (3.31), (3.36) and (3.37) into the above equation, we get

It can be assumed that ${u_1}_{xx}$, $u_1$ and $u_2$ are linearly independent. Then, we obtain

and

From the above, and under the assumption that $c_1$, $c_2$ and $c_3$ are not equal to each other, we get

and

Obviously, (3.39)–(3.41) are necessary conditions for the system (3.31) to be a partially synchronizable system.

Conversely, we consider whether the system (3.31) can realize partial synchronization when conditions (3.39)–(3.41) are satisfied. Denote $w = u_2-u_3$; then, $w(0, x) = 0$, $w_t(0, x) = 0$ and $w$ satisfies

with the boundary conditions

If

then we have that $w(t, x) = 0$; hence, (3.31) is a partially synchronizable system. Therefore, (3.42) is not only a necessary condition for the system (3.31) to realize partial synchronization, but it is also a sufficient condition. However, (3.42) is not a self-closed equation, which is difficult to solve. We want to obtain the algebraic conditions for the coupling matrix $A$ and the wave speed matrix $\Lambda$.

Denote $W = w_{tt}-c_3^2w_{xx}-(a_{23}-a_{33})w$; then,

We can get

and

From this we have

From (3.31), it follows that

and

Substituting the above formulas into (3.44), we get

i.e.,

By (3.39)–(3.41), the above equation can be simplified as follows:

Then, conditions (3.39)–(3.41) are not sufficient to close the equation (3.45). We want $W$ to satisfy a self-closed system. For this purpose, we can assume that

Combining this with (3.39)–(3.41), we get

then, $W$ satisfies the conditions for a self-closed system:

Obviously, the requirements of (3.47) are stronger than those of (3.39)–(3.41).

If the initial and boundary conditions of $W$ satisfy

and

then $W\equiv0$; we immediately get $w\equiv0$. Hence, (3.31) is a partially synchronizable system.

From (3.43), we get that $W = -(c_3^2-c_2^2){u_2}_{xx}-(a_{21}-a_{31})u_1+(a_{32}+a_{33})u_2$; then,

we must have

and

Thus,

and

Regarding the boundary conditions

From (3.31) and (3.46), we have

By (3.21), ${u_2}_{tt}(t, 0) = 0$ and ${u_2}_{tt}(t, L) = 0$, using (3.21) agian, we have

From the above discussion, we get the following theorem.

Theorem 3.3. If the system given by (3.31)–(3.32) is a partially synchronizable system in which ${u_1}_{xx}$, $u_1$ and $u_2$ are linearly independent then matrix $\Lambda$ and matrix $A$ satisfy the conditions (3.39)–(3.41). Conversely, if condition (3.47) holds for the initial value $(U_0, U_1)$ with the partial synchronization property (3.33), which satisfies the conditions (3.48)–(3.49), then the system given by (3.31)–(3.32) has a solution $U = U(t, x)$ satisfying the partial synchronazition condition

Remark 3.4. The status of variables $u_2$ and $u_3$ in (3.31) are equal. In the previous processing step, we retained $u_2$, eliminated $u_3$ and obtained the condition (3.47). In fact, if we keep $u_3$ and eliminate $u_2$, we can obtain the following conditions:

Compared with condition (3.47), it can be seen that, with the exception that

remain unchanged, the conditions

exhibit a symmetrical state. Therefore, whether the condition (3.51) is combined with the condition (3.52) or (3.53), the system (3.31) can have partially synchronized solutions.

Remark 3.5. Actually, if we do not have conditions (3.39)–(3.41) and directly require (3.45) to be a self-closed system, we have

Thus, we can directly get (3.47). Therefore, the previous steps provide us a method to realize the partial synchronization of the system given by (3.31)–(3.32).

3.4.   The general case with N variables

For the case of $N$ variables in which the first $m$ variables do not require synchronization, we rewrite the equation (1.1) as follows:

with the boundary conditions

and the initial conditions with the partial synchronization property

We want to get

For $c_{m+1} = \cdots = c_N$, we have discussed this situation in Theorem 2.1.

When $c_{m+1}, \cdots, c_N$ are different from each other, if we require (3.57) to hold from the previous discussion, we get condition (3.5).

Conversely, we want to find the conditions that make the system (3.57) be a partially synchronizable system. Here, we use the same method as the case when $N = 3, m = 1$. Let

Obviously, $(w_1, \cdots, w_{N-m-1})\equiv0$ is equivalent to condition (3.57).

Next, we want to find the condition which can realize $(w_1, \cdots, w_{N-m-1})\equiv0$. Let the $(m+1)$-th equation in (3.54) be subtracted from the $N$-th equation in (3.54), and, consistent with the mark in (2.3), we denote

we get

The equations of $w_2, \cdots, w_{N-m-1}$ can be obtained in the same way. For example,

Thus, we get the system of $(w_1, \cdots, w_{N-m-1})$ with the following boundary conditions

and the initial conditions

If the conditions

hold, then the vector $(w_1, \cdots, w_{N-m-1})$ satisfies the conditions of a self-closed system. We get that $(w_1, \cdots, w_{N-m-1})\equiv0$. Hence, the system (3.54) is a partially synchronizable system. Obviously, this is a sufficient condition. However, (3.60) is not a self-closed equation, so it is different to solve. We want to get the algebraic conditions of matrices $A$ and $\Lambda$.

Let

From (3.59), we get

Then, we have

and

From the above two equations, we get

We want (3.62) be a self-closed system of $W_1$; thus, we simplify the above formula as follows. Using system (3.54), we have

and

Then, ${u_2}_{tt}-c_N^2{u_2}_{xx}, \cdots, {u_m}_{tt}-c_N^2{u_m}_{xx}$ can be obtained similarly. For example,

Substituting the above formula into (3.62), we get

After simplification, we obtain

Assume that the matrices $A$ and $\Lambda$ satisfy the following conditions:

For short

Then, (3.62) becomes ${W_1}_{tt}-c_{N}^2{W_1}_{xx} = 0$. Using the same method to calculate the equations of $W_2, \cdots, W_{N-m-1}$, we get the conditions as follows: for all $i = m+1, \cdots, N-1$,

This can be abbreviated as follows: for all $i = m+1, \cdots, N-1$,

If condition (3.65) holds, then we get the self-closed systems of $W_1, \cdots, W_{N-m-1}$:

Assume that the initial and boundary conditions of $({W_1}, \cdots, {W_{N-m-1}})$ satisfy

Then, $({W_1}, \cdots, {W_{N-m-1}})\equiv0$. We immediately get that $(w_1, \cdots, w_{N-m-1})\equiv0$. This implies that (3.54) is a partially synchronizable system.

Now, we consider the initial and boundary conditions of ${W_1}, \cdots, {W_{N-m-1}}$. For $W_1$, from (3.61), we get

then,

We must have

and

thus, we require that

and

Regarding the boundary conditions of $W_1$,

From (3.54), we have

From (3.55), we have that ${u_N}_{tt}(t, 0) = 0$ and ${u_N}_{tt}(t, L) = 0$. Using (3.55) again, we get

For $W_2, \cdots, W_{N-m-1}$, we can get similar conclusions for the initial value: for $i = m+1, \cdots, N-1$,

and

Remark 3.6. By (3.2)–(3.3), we know that conditions (3.69)–(3.70) are equivalent to

and

Hence, (3.69)–(3.70) are not $2(N-m-1)$ conditions, but only two conditions.

To sum up, we get the conclusion:

Theorem 3.7. If the system given by (3.54)–(3.55) satisfies the compatibility condition (3.65), the initial value $(U_0, U_1)$ has the partial synchronization property (3.56), which satisfy conditions (3.71)–(3.72), then the system given by (3.54)–(3.55) has a corresponding solution $U = U(t, x)$ that satisfies the partial synchronization condition (3.57).

Remark 3.8. Under the assumption of Theorem 5, not all partially synchronized initial values $(U_0, U_1)$ have corresponding partially synchronized solutions, as only the initial values satisfying (3.71)–(3.72) can have partially synchronized solutions. Therefore, even if the system with different wave speeds is a partially synchronizable system, it does not mean that, for any given initial value satisfying the partial synchronization property, there is a corresponding partially synchronized solution.

Use of AI tools declaration

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

Acknowledgments

The research of this work was supported in part by the National Natural Science Foundation of China [grant numbers 11831011, 12161141004]. This work was also partially supported by the Institute of Modern Analysis–A Shanghai Frontier Research Center.

Conflict of interest

The authors declare there is no conflict of interest.