Semi-Discretized Approximation of Stability of Sine-Gordon System with Average-Central Finite Difference Scheme

Abstract

In this study, the energy control and asymptotic stability of the 1D sine-Gordon equation were investigated from the viewpoint of numerical approximation. An order reduction method was employed to transform the closed-loop system into an equivalent system, and an average-central finite difference scheme was constructed. This scheme is not only energy-preserving but also possesses uniform stability. The discrete multiplier method was utilized to obtain the uniformly asymptotic stability of the discrete systems. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established. Finally, several numerical experiments based on the eigenvalue distribution of the linearized approximation systems were conducted to demonstrate the effectiveness of the numerical approximating algorithm.

1. Introduction

The sine-Gordon equation, a type of semi-linear wave equation, is utilized to model a range of physical phenomena. This includes the behavior of Josephson junctions, seismic activity such as earthquakes and slow slip and after-slip events, and dislocation within materials. The sine-Gordon equation has recently grown in importance as a class of classical infinite-dimensional nonlinear dynamical systems. The controlled system exhibits high nonlinearity, making finding global or roughly global solutions challenging. Nevertheless, numerous studies have been conducted in recent years to explore the numerical solution of this equation. For instance, Zhou provided a more precise estimate of the dimension of the global attractor for discretization of the damped sine-Gordon equation with the periodic boundary condition [1]. Three effective energy-conserving compact finite difference schemes for numerically solving the sine-Gordon equation with homogeneous Dirichlet boundary conditions were developed [2]. Radial basis functions (RBFs) and a finite difference technique were used to derive the approximate solution of the two-dimensional (2D) time-fractional stochastic sine-Gordon equation in non-rectangular domains in [3].

Energy control problems arise in many disciplines, such as engineering and physics, among others. Examples of these issues have been addressed in previous studies [4,5,6]. While the regulation or tracking of linear models as control objectives has been the focus of the majority of previous efforts, the control of oscillatory modes is one of the main areas of control theory and is inspired by real-world issues such as vibration separation and suppression. Some of the developed techniques can be used to stabilize wave motion and/or other distributed systems. Over the last few decades, control methods addressing regulation and tracking issues for systems described by partial differential equations (PDEs), including optimum control, resilient control, adaptive control, etc., have been proposed. Additionally, during the 1990s, a growing trend that focused on oscillation shaping and synchronization, as opposed to suppression, emerged, bringing forth new research areas. However, until recently, distributed systems and wave control were included in an energy control strategy that was first proposed in [7,8] based on the speed-gradient method. The primary reason is that the necessary set energy level might be a complex, unbounded set. However, there has not been much research on the sine-Gordon model of energy regulation in an infinite-dimensional environment. In [9], the problem of observer-based boundary control of sine-Gordon model energy was investigated. Specifically, a Luenberger-type observer for the sine-Gordon equation was analyzed. The authors not only addressed the observer-based boundary control of sine-Gordon model energy but also provided explicit bounds on the system parameters. These bounds are crucial, as they ensure the exponential decay of the estimation error. This means that in the above study, the authors established specific limits within which the system parameters must lie to guarantee that the error in estimating the system state decreases exponentially over time. Furthermore, speed-gradient output feedback for the energy control of the sine-Gordon model was proposed. The problem was numerically studied in [10], further supporting the validity of this strategy. The simplest speed-gradient-proportional algorithm performed the best across all three performance criteria (limit error, transient time, and threshold of stability for the sampling interval). The “relay” speed-gradient algorithm was devised and adaptively improved in [11] by employing state feedback and in-domain actuators. However, these studies were based on continuous systems, and there are not many discrete systems with which to advance research in this field.

Discretization is a crucial process for PDE systems in terms of enabling the application of theoretical results in engineering technology. In most scenarios, a PDE is first discretized into a sequence of finite-dimensional systems represented by ordinary differential equations (ODEs), which are familiar to most engineers. This discretization procedure primarily focuses on the spatial variable while keeping the time variable continuous. In a sense, this process approximates an infinite-dimensional system by using finite-dimensional systems and is referred to as semi-discretization. Secondly, preserving the essential control or geometric properties inherent in the original PDEs is a primary concern during the semi-discretization process.

Therefore, building structure-preserving algorithms and energy-preserving or stability-preserving algorithms is a research hot spot in computational mathematics [5] and control theory [12,13]. For example, the uniform exponential stability of semi-discrete schemes for linear hyperbolic PDEs has been discussed extensively in the past decade (see [14,15,16,17,18,19,20]). However, there are few results on the uniform asymptotic stability of nonlinear systems. Up until now, research on uniform approximation of nonlinear systems has been primarily concentrated on systems with nonlinear boundary conditions [12]. The task of approximating the uniformly asymptotic stability of the sine-Gordon system is particularly challenging due to the nonlinear term appearing in system equations.

In our work, the aims are to fill the gap and to tackle this problem for the first time by establishing suitable discrete time-domain energy multipliers. More precisely, the sine-Gordon system was investigated from the viewpoint of numerical approximation, in line with previous studies [15,16]. A new finite difference scheme that is both energy- and stability-preserving was designed for the sine-Gordon equation. For this purpose, the PDE component was first numerically discretized, and it was turned into a family of ODEs. Subsequently, the uniform stability of this family of discrete systems was studied, i.e., whether or not the discrete energies uniformly converge concerning the mesh size. To validate uniform stability, the common approach is to follow each step of the continuous case (see, for instance, [15,16,18,20]). The discrete multiplier method was utilized to obtain the uniform stability of the discrete system. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established.

Based on the above, the novelty of this work is two-fold. First, an appropriate numerical approximating scheme for a nonlinear PDE that is both energy-preserving and stability-preserving was designed. Secondly, we generalized the conclusions reported in [15,16,20] from the exponential stability of linear models to the asymptotic stability of nonlinear systems.

This remainder of this paper is organized as follows: In Section 2, the problem formulation is presented, and relevant definitions and the control algorithm are introduced. In Section 3, an order reduction method is utilized to transform the sine-Gordon equation into an equivalent system, and a suitable semi-discretization system is designed. The main result of this study is then presented. In Section 4, the several numerical experiments conducted to demonstrate the effectiveness of the numerical approximating scheme are presented. In Section 5, some conclusions and remarks are presented.

2. Problem Formulation and Speed-Gradient Control Law

Consider the one-dimensional sine-Gordon equation incorporating initial conditions and boundary control mechanisms.

$$\begin{array}{cc}\hfill & z_{tt}(t,x)-k{z}_{xx}(t,x)+\beta sinz(t,x)=0,t>0,x\in (0,1),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & z(t,0)=0,z_x(t,1)=u\left(t\right),t>0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & z(0,x)=z^0\left(x\right),z_t(0,x)=z^1\left(x\right),x\in (0,1),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & y\left(t\right)=z_t(t,1),t\ge 0,\hfill \end{array}$$

(4)

where $t\ge 0$ is the time variable, $x\in (0,1)$ is a spatial variable, $z_t(t,x)$ represents the first-order derivative with respect to the time variable, $z_x(t,x)$ represents the first-order derivative with respect to the spatial variable, $\beta >0$ and $k>0$, $z^0\in H^1(0,1),z^1\in L^2(0,1)$ are given functions, $u\left(t\right)$ is a control input, and $y\left(t\right)$ is the output.

$$H\left(t\right)={\int }_0^1\left[{\displaystyle \frac{z_t^2(t,x)}{2}}+k{\displaystyle \frac{z_x^2(t,x)}{2}}+\beta \left(1-cosz(t,x)\right)\right]\mathrm{d}x$$

is the Hamiltonian that corresponds to the sine-Gordon equation. The quantity ($H\left(t\right)$) is conserved along solutions of the unforced system (see (19)). In addition, $H\left(t\right)$ is greater than or equal to zero, and $H\left(t\right)$ equals zero only when $z(t,x)$ is equal to zero. As a result, $H\left(t\right)$ can be thought of as the energy at time t, and $H\left(t\right)$ is replaced with $E\left(t\right)$.

A control law ($u\left(t\right)$) must be identified that will make the energy of the system (1)–(4) conform to the expected value ($E^{*}$) (see also [21]), taking the following form:

$$\underset{t\to +\infty }{lim}E\left(t\right)=E^{*}.$$

(5)

Let us design a control law that addresses the energy control issue using the Speed-Gradient (SG) approach ([22,23]). First, we introduce the goal function, i.e.,

$$Q\left(t\right)={\displaystyle \frac{1}{2}}{(E\left(t\right)-E^{*})}^2,$$

(6)

which is the difference between the present and required energy. The derivative of this function is

$$Q_t=(E\left(t\right)-E^{*}){\int }_0^1\left(z_t{z}_{tt}+k{z}_x{z}_{xt}+\beta sinz{z}_t\right)\mathrm{d}x.$$

(7)

Substituting $k{z}_{xx}(t,x)-\beta sinz(t,x)$ for $z_{tt}$ and integrating the term $z_x{z}_{xt}$ by parts, one obtains

$$Q_t=(E\left(t\right)-E^{*})[k{z}_x(t,1)z_t(t,1)-k{z}_x(t,0)z_t(t,0)].$$

(8)

Thus, the boundary condition ($z(t,0)=0$) yields

$$Q_t=k(E\left(t\right)-E^{*})u\left(t\right)y\left(t\right).$$

(9)

In the second step, the gradient (${\nabla }_u{Q}_t$) of $Q_t$ with respect to the component ($u\left(t\right)$) is derived as

$${\nabla }_u{Q}_t=k(E\left(t\right)-E^{*})y\left(t\right).$$

(10)

In the third step, the vector function ($\eta (z,z_t)$) is selected to satisfy the acute angle condition (${\eta }^T{\nabla }_u{Q}_t\ge 0$). The following control law results from this:

$$u_0\left(t\right)=-k\gamma (E\left(t\right)-E^{*})y\left(t\right),$$

(11)

where the scalar gain is defined as $\gamma >0$. A more generic control algorithm is expressed in the following form:

$$u\left(t\right)=-\gamma \psi (E\left(t\right)-E^{*})y\left(t\right),$$

(12)

where $\psi :\mathbb{R}\to \mathbb{R}$ is a continuous function such that $\psi \left(0\right)=0$ and $\psi \left(s\right)s>0$ for $s\ne 0$.

3. The Semi-Discretized System

In this section, the original system (1)–(4) is first transformed into an equivalent system. A spatial semi-discretized finite difference approach is then built for the order-reduced system. Furthermore, uniform stability is investigated for these semi-discretized systems.

3.1. Order Reduction and Semi-Discretization

The introduction of auxiliary functions $v(t,x)=z_t(t,x)$ and $w(t,x)=z_x(t,x)$, (1)–(4) with feedback control (12) can be rewritten as the following equivalent system:

$$\begin{array}{cc}\hfill & v_t(t,x)-k{w}_x(t,x)+\beta sinz(t,x)=0,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & w_t(t,x)-v_x(t,x)=0,t\ge 0,0<x<1,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & z_t(t,x)=v(t,x),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & v(t,0)=0,w(t,1)=-\gamma \psi (E\left(t\right)-E^{*})v(t,1),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & v(0,x)=v^0\left(x\right),w(0,x)=w^0\left(x\right),\hfill \end{array}$$

(17)

where $(v^0,w^0)=(z^1,z_x^0)\in L^2(0,1)\times L^2(0,1)$. The energy ($E\left(t\right)$) of (13)–(17) now equivalently reads

$$E\left(t\right)={\int }_0^1\left[{\displaystyle \frac{v^2(t,x)}{2}}+k{\displaystyle \frac{w^2(t,x)}{2}}+\beta (1-cosz(t,x))\right]\mathrm{d}x,$$

(18)

and satisfies

$${\displaystyle \frac{dE\left(t\right)}{dt}}=-k\gamma \psi (E\left(t\right)-E^{*})v^2(t,1).$$

(19)

A few notations are introduced before performing finite difference semi-discretization on the system (13)–(17). A partition of the spatial domain $[0,1]$ with equal distance of $h:={\displaystyle \frac{1}{N+1}}$ for every $N\in {\mathbb{N}}^{+}$ is given, i.e.,

$$1=x_{N+1}>\cdots >x_j=jh>\cdots >x_1>x_0=0.$$

The difference operator and the average operator are denoted as

$${\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)={\displaystyle \frac{z_{j+1}\left(t\right)-z_j\left(t\right)}{h}},z_{j+{\displaystyle \frac{1}{2}}}\left(t\right)={\displaystyle \frac{z_{j+1}\left(t\right)+z_j\left(t\right)}{2}}.$$

In the sequel, the prime symbol (^′) denotes differentiation with respect to time.

By following the difference scheme reported in [15,16,19,20] and replacing $sinz(t,x_j)$ with $sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)$, a semi-discretized finite difference scheme of (13)–(17) is arrived at as follows:

$$\begin{array}{cc}\hfill & v_{j+{\displaystyle \frac{1}{2}}}^{\prime }\left(t\right)-k{\delta }_x{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)+\beta sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)=0,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & w_{j+{\displaystyle \frac{1}{2}}}^{\prime }\left(t\right)-{\delta }_x{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)=0,j=0,\cdots ,N,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & z_j^{\prime }\left(t\right)=v_j\left(t\right),j=0,\cdots ,N+1,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & v_k\left(0\right)=v_k^0,w_k\left(0\right)=w_k^0,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & v_0\left(t\right)=0,w_{N+1}\left(t\right)=-\gamma \psi (E_h\left(t\right)-E^{*})v_{N+1}\left(t\right),\hfill \end{array}$$

(24)

in which $v_j\left(t\right),w_j\left(t\right)$, and $z_j\left(t\right)$ are approximations of $v(t,x_j)$, $w(t,x_j)$, and $z(t,x_j)$, respectively; $v_k^0$ and $w_k^0$ are the approximations of the initial values of $v^0\left(x_k\right)$ and $w^0\left(x_k\right)$, respectively; and $E_h\left(t\right)$ is defined by (27) in the next subsection.

The following lemma (see [16], Lemma 2.2) is simple but extremely helpful in demonstrating the convergence of energy.

Lemma 1.

For any grid functions (${\left\{V_j\left(t\right)\right\}}_{j=0}^{N+1},{\left\{W_j\left(t\right)\right\}}_{j=0}^{N+1},{\left\{Z_j\left(t\right)\right\}}_{j=0}^{N+1}$) at mesh grids (${\left\{x_j\right\}}_{j=0}^{N+1}$), the below two summation by parts formulas hold:

$$h\sum _{j=0}^N{\delta }_x{V}_{j+{\displaystyle \frac{1}{2}}}{W}_{j+{\displaystyle \frac{1}{2}}}+h\sum _{j=0}^N{V}_{j+{\displaystyle \frac{1}{2}}}{\delta }_x{W}_{j+{\displaystyle \frac{1}{2}}}=V_{N+1}{W}_{N+1}-V_0{W}_0,$$

(25)

and

$$\begin{array}{c}\hspace{1em}h\sum _{j=0}^N{\delta }_x{V}_{j+{\displaystyle \frac{1}{2}}}{W}_{j+{\displaystyle \frac{1}{2}}}{Z}_{j+{\displaystyle \frac{1}{2}}}+h\sum _{j=0}^N{V}_{j+{\displaystyle \frac{1}{2}}}{\delta }_x{W}_{j+{\displaystyle \frac{1}{2}}}{Z}_{j+{\displaystyle \frac{1}{2}}}+h\sum _{j=0}^N{V}_{j+{\displaystyle \frac{1}{2}}}{W}_{j+{\displaystyle \frac{1}{2}}}{\delta }_x{Z}_{j+{\displaystyle \frac{1}{2}}}\\ =V_{N+1}{W}_{N+1}{Z}_{N+1}-V_0{W}_0{Z}_0-{\displaystyle \frac{h^3}{4}}\sum _{j=0}^N{\delta }_x{V}_{j+{\displaystyle \frac{1}{2}}}{\delta }_x{W}_{j+{\displaystyle \frac{1}{2}}}{\delta }_x{Z}_{j+{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(26)

3.2. Uniform Stability of (20)–(24)

The discretized energy of the finite difference scheme (20)–(24) is calculated as

$$E_h\left(t\right)=h\sum _{j=0}^N\left[{\displaystyle \frac{1}{2}}{\left|v_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\right|}^2+{\displaystyle \frac{k}{2}}{\left|w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\right|}^2+\beta (1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))\right].$$

(27)

Here, $E_h\left(t\right)$ represents a natural discretization of the continuous energy ($E\left(t\right)$) as defined in (18). Moreover, a discrete counterpart result of (19) is easily derived.

Lemma 2.

The discrete goal function ($Q_h\left(t\right)$) defined by $Q_h\left(t\right)={\displaystyle \frac{1}{2}}{\left(E_h\left(t\right)-E^{*}\right)}^2$ satisfies

$${\displaystyle \frac{d{Q}_h\left(t\right)}{dt}}=-k\gamma (E_h\left(t\right)-E^{*})\psi (E_h\left(t\right)-E^{*})v_{N+1}^2\left(t\right)\le 0,\forall t\ge 0.$$

(28)

Proof. 

Multiply both sides of the Equation (20) by $h{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)$ and summarize j from 0 to N to acquire

$$h\sum _{j=0}^N{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)v_{j+{\displaystyle \frac{1}{2}}}^{\prime }\left(t\right)=kh\sum _{j=0}^N{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right){\delta }_x{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)-h\beta \sum _{j=0}^N{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right).$$

(29)

Similarly, multiplying both sides of Equation (21) by $kh{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)$ and summing up j from 0 to N, we obtain

$$hk\sum _{j=0}^N{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}^{\prime }\left(t\right)=kh\sum _{j=0}^N{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right){\delta }_x{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right).$$

(30)

Then, (29)–(30) give

$${\displaystyle \frac{d{E}_h\left(t\right)}{dt}}=-k\gamma \psi (E_h\left(t\right)-E^{*})v_{N+1}^2\left(t\right).$$

(31)

Now, by noting $Q_h^{\prime }\left(t\right)=(E_h\left(t\right)-E^{*})E_h^{\prime }\left(t\right)$, the proof is completed. □

Remark 1.

In Equation (31), $\psi \left(s\right)$ is a continuous function, and $\psi \left(s\right)s>0$ for $s\ne 0$ and $\psi \left(0\right)=0$ show that

$${\displaystyle \frac{d{E}_h\left(t\right)}{dt}}\left\{\begin{array}{cc}\le 0,\hfill & \mathit{if} E_h\left(t\right)>E^{*},\hfill \\ =0,\hfill & \mathit{if} E_h\left(t\right)=E^{*},\hfill \\ \ge 0,\hfill & \mathit{if} E_h\left(t\right)<E^{*}.\hfill \end{array}\right.$$

(32)

The following assumption is introduced in order to establish the uniform convergence of $E_h\left(t\right)\to E^{*}$.

Assumption 1.

The parameters $k>0$ and $\beta >0$ in (1)–(4) satisfy the following inequalities: $0<\beta <k/4.$

A Lyapunov function is built as follows:

$$G_h\left(t\right)=E_h\left(t\right)+\epsilon \mathrm{sign}(E_h\left(t\right)-E^{*}){\mathsf{\Phi }}_h\left(t\right),$$

(33)

where $\epsilon$ is a parameter determined later, $\mathrm{sign}(·)$ is the sign function, and the auxiliary function ${\mathsf{\Phi }}_h\left(t\right)$ is defined by

$${\mathsf{\Phi }}_h\left(t\right)=h\sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right).$$

(34)

Lemma 3.

The Lyapunov function ($G_h\left(t\right)$) defined in (33) is equivalent to the discretized energy ($E_h\left(t\right)$).

$$(1-\epsilon k_0)E_h\left(t\right)\le G_h\left(t\right)\le (1+\epsilon k_0)E_h\left(t\right),$$

(35)

where $k_0=max\{1,1/k\}$ and $0<\epsilon <min\{1,k\}$.

The proof is omitted, since it only involves simple calculation. The discrete Wirtinger inequality for our approximating scheme is established as follows.

Lemma 4.

Suppose that $h(N+1)=1$ holds; then, for a sequence (${\left\{z_j\right\}}_{j=0}^{N+1}$) with $z_0=0$, we have

$$h\sum _{j=0}^N{|z_{j+{\displaystyle \frac{1}{2}}}|}^2\le h\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{|}^2.$$

(36)

Proof. 

It is easy to see that $z_{j+1}={\displaystyle \sum _{k=0}^j}[z_{k+1}-z_k]$ and

$$\begin{array}{c}\hfill |z_{j+1}{|}^2\le \sum _{j=0}^N{|1|}^2\sum _{j=0}^N|z_{j+1}-z_j{|}^2=(N+1)h^2\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{|}^2=h\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{|}^2.\end{array}$$

(37)

Thus, using inequality (37) one obtains

$$\begin{array}{c}\hfill h\sum _{j=0}^N{|z_{j+{\displaystyle \frac{1}{2}}}|}^2\le {\displaystyle \frac{h}{2}}\sum _{j=0}^N\left(|z_{j+1}{|}^2+{|z_j|}^2\right)\le (N+1)h^2\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{|}^2=h\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{|}^2.\end{array}$$

(38)

□

Lemma 5.

The auxiliary function (${\mathsf{\Phi }}_h\left(t\right)$) defined by (34) satisfies

$${\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}\le -C_0{E}_h\left(t\right)+{\displaystyle \frac{1+k{\gamma }^2{\psi }^2(E_h\left(t\right)-E^{*})}{2}}{v}_{N+1}^2\left(t\right),$$

(39)

where $C_0=1-4\beta /k>0$ according to Assumption 1.

Proof. 

Taking the derivative of ${\mathsf{\Phi }}_h\left(t\right)$ with respect to the solution of (20)–(24) yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}=& kh\sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}\left(t\right){\delta }_x{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)+h\sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)v_{j+{\displaystyle \frac{1}{2}}}\left(t\right){\delta }_x{v}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\hfill \\ \hfill -& h\beta \sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\hfill \end{array}$$

(40)

The second formula in Lemma 1 further shows

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}=-{\displaystyle \frac{kh}{2}}\sum _{j=0}^N{w}_{j+{\displaystyle \frac{1}{2}}}^2\left(t\right)+{\displaystyle \frac{k}{2}}{w}_{N+1}^2\left(t\right)-{\displaystyle \frac{hk}{8}}\sum _{j=0}^N{(w_{j+1}-w_j)}^2-{\displaystyle \frac{h}{2}}\sum _{j=0}^N{v}_{j+{\displaystyle \frac{1}{2}}}^2\left(t\right)\hfill \\ \hfill & -{\displaystyle \frac{h}{8}}\sum _{j=0}^N{(v_{j+1}-v_j)}^2+{\displaystyle \frac{1}{2}}{v}_{N+1}^2\left(t\right)-h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))\hfill \\ \hfill & +h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))-h\beta \sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}{w}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right).\hfill \end{array}$$

(41)

Using Lemma 4 for ${\left\{z_j\right\}}_{j=0}^{N+1}$, we obtain

$$\begin{array}{cc}\hfill & -h\beta \sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\le h\beta \left|\sum _{j=0}^N{x}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)\right|\hfill \\ \hfill \le & h\beta \sum _{j=0}^N|sin{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)|\le {\displaystyle \frac{h\beta }{2}}\left(\sum _{j=0}^N|sin{z}_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2+\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\right)\hfill \\ \hfill \le & {\displaystyle \frac{h\beta }{2}}\left(\sum _{j=0}^N|z_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2+\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\right)\le {\displaystyle \frac{h\beta }{2}}\sum _{j=0}^N|{\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2+{\displaystyle \frac{h\beta }{2}}\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\hfill \\ \hfill =& h\beta \sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2,\hfill \end{array}$$

(42)

where $w_{j+{\displaystyle \frac{1}{2}}}\left(t\right)={\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right)$, which comes from $w(t,x)=z_x(t,x)$, is utilized in the last step. Then, the term $h\beta {\displaystyle \sum _{j=0}^N}(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))$ is calculated using Lemma 4 once more as follows:

$$\begin{array}{cc}\hfill & h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))\hfill \\ \hfill =& 2h\beta \sum _{j=0}^{N}sin{\displaystyle \frac{1}{2}}{|z_{j+{\displaystyle \frac{1}{2}}}\left(t\right)|}^2\le h\beta \sum _{j=0}^N|z_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\le h\beta \sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2.\hfill \end{array}$$

(43)

From the above (42) and (43), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}& \le -{\displaystyle \frac{kh}{2}}\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2+{\displaystyle \frac{k}{2}}{w}_{N+1}^2\left(t\right)-{\displaystyle \frac{kh}{8}}\sum _{j=0}^N{(w_{j+1}-w_j)}^2-{\displaystyle \frac{h}{2}}\sum _{j=0}^N|v_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\hfill \\ \hfill & -{\displaystyle \frac{h}{8}}\sum _{j=0}^N{(v_{j+1}-v_j)}^2+{\displaystyle \frac{1}{2}}{v}_{N+1}^2\left(t\right)-h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))+2h\beta \sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2\hfill \\ \hfill & \le -{\displaystyle \frac{h}{2}}\sum _{j=0}^N|v_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2+\left(2h\beta -{\displaystyle \frac{kh}{2}}\right)\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2-h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))\hfill \\ \hfill & +{\displaystyle \frac{k{\gamma }^2{\psi }^2(E_h\left(t\right)-E^{*})}{2}}{v}_{N+1}^2\left(t\right)+{\displaystyle \frac{1}{2}}{v}_{N+1}^2\left(t\right)\hfill \\ \hfill & =-{\displaystyle \frac{h}{2}}\sum _{j=0}^N|v_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2-{\displaystyle \frac{hk}{2}}\left(1-{\displaystyle \frac{4\beta }{k}}\right)\sum _{j=0}^N|w_{j+{\displaystyle \frac{1}{2}}}{\left(t\right)|}^2-h\beta \sum _{j=0}^N(1-cos{z}_{j+{\displaystyle \frac{1}{2}}}\left(t\right))\hfill \\ \hfill & +{\displaystyle \frac{1+k{\gamma }^2{\psi }^2(E_h\left(t\right)-E^{*})}{2}}{v}_{N+1}^2\left(t\right)\hfill \\ \hfill & \le -C_0{E}_h\left(t\right)+{\displaystyle \frac{1+k{\gamma }^2{\psi }^2(E_h\left(t\right)-E^{*})}{2}}{v}_{N+1}^2\left(t\right),\hfill \end{array}$$

(44)

where mesh sizes of $h=1/(N+1)$ and $C_0=1-4\beta /k>0$ are utilized. This yields the required result. □

Theorem 1.

If Assumption 1 holds, then the energies ($E_h\left(t\right)$) of the discrete systems ((20)–(24)) are uniformly convergent to $E^{*}$ with respect to the step size h, i.e.,

$$E_h\left(t\right)\to E^{*}\mathit{as}t\to \infty ,\mathit{for}\mathit{all}h\in (0,1).$$

(45)

Proof. 

By combining the derivative of function $G_h\left(t\right)$ as defined in (33) with Equation (31), one obtains

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{G}_h\left(t\right)}{dt}}& ={\displaystyle \frac{d{E}_h\left(t\right)}{dt}}+\epsilon \mathrm{sign}(E_h\left(t\right)-E^{*}){\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}\hfill \\ \hfill & =-k\gamma \psi (E_h\left(t\right)-E^{*})v_{N+1}^2\left(t\right)+\epsilon \mathrm{sign}(E_h\left(t\right)-E^{*}){\displaystyle \frac{d{\mathsf{\Phi }}_h\left(t\right)}{dt}}\hfill \end{array}$$

(46)

Let ${\mathsf{\Delta }}_h>0$ be arbitrary and suppose that $E_h\left(0\right)>E^{*}$. Therefore, taking into account (32), $E_h\left(t\right)$ decreases for a time interval, then increases for the remaining part. This means that there exists $T_{h,\mathsf{\Delta }}\in [0,+\infty )$ such that $E_h\left(t\right)\ge E^{*}+{\mathsf{\Delta }}_h$ for any $t\in [0,T_{h,\mathsf{\Delta }})$ and $E_h\left(t\right)<E^{*}+{\mathsf{\Delta }}_h$ for any $t\in (T_{h,\mathsf{\Delta }},+\infty )$. The aim is to show that $T_{h,\mathsf{\Delta }}<+\infty$ for any ${\mathsf{\Delta }}_h>0$.

It is proven by contradiction, and suppose that there exists ${\mathsf{\Delta }}_h>0$ such that $T_{h,\mathsf{\Delta }}=+\infty$. We define

$${\xi }_{h,\mathsf{\Delta }}=min\{\psi \left(s\right)|s\in [{\mathsf{\Delta }}_h,E_h\left(0\right)-E^{*}]\}>0,$$

(47)

and

$${\eta }_{h,\mathsf{\Delta }}=max\{\psi \left(s\right)|s\in [{\mathsf{\Delta }}_h,E_h\left(0\right)-E^{*}]\}>0.$$

(48)

Then, for any $t\ge 0$, one has ${\xi }_{h,\mathsf{\Delta }}\le \psi (E_h\left(t\right)-E^{*})\le {\eta }_{h,\mathsf{\Delta }}$. Therefore, in light of (35), (39) and (46), one obtains

$${\displaystyle \frac{d{G}_h\left(t\right)}{dt}}\le -{\displaystyle \frac{\epsilon C_0}{1+\epsilon k_0}}{G}_h\left(t\right)+\epsilon {\displaystyle \frac{1+k{\gamma }^2{\eta }_{h,\mathsf{\Delta }}^2}{2}}{v}_{N+1}^2\left(t\right)-k\gamma {\xi }_{h,\mathsf{\Delta }}{v}_{N+1}^2\left(t\right)$$

(49)

for any $t\ge 0$. Choosing $\epsilon >0$ sufficiently small such that $\epsilon <{\displaystyle \frac{2k\gamma {\xi }_{h,\mathsf{\Delta }}}{1+k{\gamma }^2{\eta }_{h,\mathsf{\Delta }}^2}}$, one finds that ${\displaystyle \frac{d{G}_h\left(t\right)}{dt}}\le -C_{\epsilon }{G}_h\left(t\right),\forall t\ge 0$, where $C_{\epsilon }={\displaystyle \frac{\epsilon C_0}{1+\epsilon k_0}}>0$. A direct use of Gronwall’s inequality leads to $G_{h\left(t\right)}\le G_{h\left(0\right)}{e}^{-C_{\epsilon }t}$ for any $t\ge 0$.

This means that, $\mathrm{for} \mathrm{all} h\in (0,1)$, $G_h\left(t\right)\to 0$ as $t\to \infty$, which contradicts the definition of $T_{h,\mathsf{\Delta }}$ and the fact that $E^{*}+{\mathsf{\Delta }}_h>0$. Thus, $T_{h,\mathsf{\Delta }}<+\infty$ and $E_h\left(t\right)<E^{*}+{\mathsf{\Delta }}_h$ for any $t>T_{h,\mathsf{\Delta }}$. Hence, $E_h\left(t\right)\to E^{*}$ as $t\to +\infty \mathrm{for} \mathrm{all} h\in (0,1)$ because ${\mathsf{\Delta }}_h>0$ was chosen arbitrarily.

Now, suppose that $E_h\left(0\right)<E^{*}$ and $E_h\left(t\right)\le E^{*}-{\mathsf{\Delta }}_h$ for any $t\in [0,+\infty )$, then arguing in a similar way to the case of $E_h\left(0\right)>E^{*}$, one can show that for any sufficiently small $\epsilon >0$ and some $D_{\epsilon }>0$, the following inequality holds true:

$${\displaystyle \frac{d{G}_h\left(t\right)}{dt}}\ge D_{\epsilon }{G}_h\left(t\right),\forall t\ge 0.$$

(50)

Thus, $G_h\left(t\right)\ge G_h\left(0\right)e^{D_{\epsilon }t}$ for any $t\ge 0$. In this case, we obtain$\mathrm{for} \mathrm{all} h\in (0,1)$, and $G_h\left(t\right)$ approaches positive infinity as t approaches infinity. This contradicts the assumption that $E_h\left(t\right)\le E^{*}-{\mathsf{\Delta }}_h$ for any $t\in [0,+\infty )$. Hence, $E_h\left(t\right)\to E^{*}$ as $t\to +\infty \mathrm{for} \mathrm{all} h\in (0,1)$, since ${\mathsf{\Delta }}_h>0$ is arbitrary. □

4. Numerical Analysis

In this section, we first show that the semi-discretized finite difference scheme (20)–(24) for an equivalent system (13)–(17) can also yield a semi-discretized finite difference scheme for the original system (1)–(4).

Lemma 6.

Let $z(t,x)$ be a solution of system (1)–(4). The semi-discretized average-central finite difference scheme (20)–(24) can be used to numerically solve the original system (1)–(4) as follows:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4}}\left(z_{j-1}^{″}\left(t\right)+2z_j^{″}\left(t\right)+z_{j+1}^{″}\left(t\right)\right)\hfill \\ \hfill & -k{\delta }_x^2{z}_j\left(t\right)+{\displaystyle \frac{\beta }{2}}(sin{z}_{j-{\displaystyle \frac{1}{2}}}+sin{z}_{j+{\displaystyle \frac{1}{2}}})=0,j=1,\dots ,N,\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left(z_N^{″}\left(t\right)+z_{N+1}^{″}\left(t\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}sin{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{k}{h}}{\delta }_x{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{k}{h}}\gamma \psi (E_h\left(t\right)-E^{*})z_{N+1}^{\prime }\left(t\right)=0,\hfill \\ \hfill & z_0\left(t\right)=0,z_j\left(0\right)=z^0\left(x_j\right),z_j^{\prime }\left(0\right)=z^1\left(x_j\right),j=0,1,2,\dots ,N+1,\hfill \end{array}\right.$$

(51)

where $z_j\left(t\right)$ represents approximations of $z(t,x_j)$ for $j=0,1,\cdots ,N+1$.

Proof. 

It follows from $v(t,x)=z_t(t,x)$ and $w(t,x)=z_x(t,x)$ that

$$v_{j+{\displaystyle \frac{1}{2}}}=z_{j+{\displaystyle \frac{1}{2}}}^{\prime },w_{j+{\displaystyle \frac{1}{2}}}={\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}},j=0,1,\dots ,N.$$

(52)

By multiplying both sides of the Equation (20) by ${\displaystyle \frac{h}{2}}$, one obtains

$${\displaystyle \frac{k(w_{j+1}-w_j)}{2}}-{\displaystyle \frac{h\beta }{2}}sin{z}_{j+{\displaystyle \frac{1}{2}}}={\displaystyle \frac{h}{2}}{v}_{j+{\displaystyle \frac{1}{2}}}^{\prime },j=0,\dots ,N.$$

(53)

Then, combining with (52), for $j=0,\dots ,N$ we have

$$w_{j+1}={\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{h}{2k}}{z}_{j+{\displaystyle \frac{1}{2}}}^{″}+{\displaystyle \frac{h\beta }{2k}}sin{z}_{j+{\displaystyle \frac{1}{2}}},w_j={\delta }_x{z}_{j+{\displaystyle \frac{1}{2}}}-{\displaystyle \frac{h}{2k}}{z}_{j+{\displaystyle \frac{1}{2}}}^{″}-{\displaystyle \frac{h\beta }{2k}}sin{z}_{j+{\displaystyle \frac{1}{2}}}.$$

(54)

Eliminating the variable $w_j$, one obtains

$$-{\displaystyle \frac{z_{j+1}-2z_j+z_{j-1}}{h^2}}+{\displaystyle \frac{1}{4k}}(z_{j-1}^{″}+2z_j^{″}+z_{j+1}^{″})+{\displaystyle \frac{\beta }{4k}}(sin{z}_{j-{\displaystyle \frac{1}{2}}}+sin{z}_{j+{\displaystyle \frac{1}{2}}})=0,$$

(55)

which yields the first equation of (51). Setting $j=N$ in the first equation of (54) derives

$$w_{N+1}={\delta }_x{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{h}{2k}}{z}_{N+{\displaystyle \frac{1}{2}}}^{″}+{\displaystyle \frac{h\beta }{2k}}sin{z}_{N+{\displaystyle \frac{1}{2}}},$$

(56)

amd with the right boundary of (24), the second equation of (51) is derived. The left boundary condition and initial conditions are trivial. The proof of the lemma is complete. □

In order to perform numerical simulations, the first choice of the approximation of the nonlinear term ($sin{z}_{j+{\displaystyle \frac{1}{2}}}$) is $z_{j+{\displaystyle \frac{1}{2}}}$ for $j=1,2,\cdots ,N$. In this case, (51) becomes

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4}}\left(z_{j-1}^{″}\left(t\right)+2z_j^{″}\left(t\right)+z_{j+1}^{″}\left(t\right)\right)\hfill \\ \hfill & -k{\delta }_x^2{z}_j\left(t\right)+{\displaystyle \frac{\beta }{2}}(z_{j-{\displaystyle \frac{1}{2}}}+z_{j+{\displaystyle \frac{1}{2}}})=0,j=1,\dots ,N,\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left(z_N^{″}\left(t\right)+z_{N+1}^{″}\left(t\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{k}{h}}{\delta }_x{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{k}{h}}\gamma \psi (E_h\left(t\right)-E^{*})z_{N+1}^{\prime }\left(t\right)=0,\hfill \\ \hfill & z_0\left(t\right)=0,z_j\left(0\right)=z^0\left(x_j\right),z_j^{\prime }\left(0\right)=z^1\left(x_j\right),j=0,1,2,\dots ,N+1,\hfill \end{array}\right.$$

(57)

The discrete scheme (57) should be rewritten as a state space formulation. For this purpose, we define $(N+1)\times (N+1)$ matrices as follows:

$$P=\left(\begin{array}{cccccc}2& 1& & & & \\ 1& 2& 1& & & \\ & \ddots & \ddots & \ddots & \\ & & 1& 2& 1& \\ & & & 2& 2\end{array}\right),B=\left(\begin{array}{cccccc}2& -1& & & & \\ -1& 2& -1& & & \\ & \ddots & \ddots & \ddots & \\ & & -1& 2& -1& \\ & & & -1& 1\end{array}\right),$$

(58)

and $A=diag\{0,\dots ,a\}$ with $a=k\gamma \psi (E_h\left(t\right)-E^{*})$ belonging to ${\mathbb{C}}^{(N+1)\times (N+1)}$. Then, (51) can be rewritten respectively as the vectorial forms:

$$\left\{\begin{array}{c}{\displaystyle \frac{P}{4}}{Z}_h^{″}\left(t\right)+{\displaystyle \frac{\beta }{4}}P{Z}_h\left(t\right)+{\displaystyle \frac{k}{h^2}}B{Z}_h+{\displaystyle \frac{1}{h}}A{Z}_h^{\prime }=0,\hfill \\ Z_h\left(0\right)=Z_h^0,Z_h^{\prime }\left(0\right)=Z_h^1,\hfill \end{array}\right.$$

(59)

where $Z_h\left(t\right)={\left(z_1\left(t\right),\dots ,z_{N+1}\left(t\right)\right)}^{\top }$, $Z_h^0={\left(z_1^0,\dots ,z_{N+1}^0\right)}^{\top }$, $Z_h^1={\left(z_1^1,\dots ,z_{N+1}^1\right)}^{\top }$.

By setting

$$P_h=\left(\begin{array}{cc}I& 0\\ 0& {\displaystyle \frac{1}{4}}P\end{array}\right),B_h=\left(\begin{array}{cc}0& I\\ -{\displaystyle \frac{\beta }{4}}P-{\displaystyle \frac{k}{h^2}}B& -{\displaystyle \frac{1}{h}}A\end{array}\right),$$

(60)

the differential Equation (57) is then equivalent to

$$P_h{Y}_h^{\prime }\left(t\right)=B_h{Y}_h\left(t\right)\iff Y_h^{\prime }\left(t\right)=D_h{Y}_h\left(t\right),$$

(61)

with $D_h:=P_h^{-1}{B}_h$ and $Y_h^{\prime }\left(t\right)={(Z_h{\left(t\right)}^{\top },Z_h^{\prime }{\left(t\right)}^{\top })}^{\top }$. Note that $P_h$ is evidently invertible (see [24]), but it corresponds to the average operator. If $P_h$ is replaced by the identity operator (I), the system (61) would not contain the average operator, and the corresponding scheme restores the classical finite difference for the original system (1)–(4).

The significance of the discrete scheme (61) is discussed here. We present thee plots in Figure 1, Figure 2 and Figure 3. Figure 1 depicts the maximal real parts of $D_h$ (with the average operator) and $B_h$ (without the average operator) for $N=10:5:400$. Figure 2 depicts the distributions of the eigenvalues of $D_h$ and $B_h$ in which $N=200$. In Figure 1 and Figure 2, the feedback control (a) is set as a random number located within the interval of $[-0.5,0.5]$. It is easy to see from Figure 1 that the maximal real parts of the eigenvalues of $D_h$ are almost negative, and those of $B_h$ are positive. This means that the classical finite difference does not preserve uniform asymptotic stability. This is in accordance with the analysis reported in [24]. However, the average-central finite difference scheme (20)–(24), which corresponds to $D_h$, may be uniformly asymptotically stable. This matches the main result of Theorem 1. Moreover, the importance of the feedback control gain ($a=k\gamma \psi (E_h\left(t\right)-E^{*})$) is reflected through Figure 1 and Figure 3. If a is fixed at $a=0.5$, the stability of the average-central finite difference scheme (20)–(24) is still preserved, and the maximal real parts of $D_h$ are almost the same. The other parameters are selected as $\beta =k=\gamma =1$ in these figures.

Figure 1. Maximal real parts of eigenvalues of the semi-discrete schemes with random a in the case of the first substitution.

Figure 2. Eigenvalue distribution of the semi-discrete scheme with random a and $N=200$ in the case of the first substitution.

Figure 3. Maximal real parts of eigenvalues of the semi-discrete schemes with $a=0.5$ in the case of the first substitution.

In the next numerical simulations, the second choice of the approximation of the nonlinear term ($sin{z}_{j+{\displaystyle \frac{1}{2}}}$) is $z_{j+1}$ for $j=1,2,\cdots ,N$. In this scenario, (51) becomes

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4}}\left(z_{j-1}^{″}\left(t\right)+2z_j^{″}\left(t\right)+z_{j+1}^{″}\left(t\right)\right)\hfill \\ \hfill & -k{\delta }_x^2{z}_j\left(t\right)+{\displaystyle \frac{\beta }{2}}\left(z_j+z_{j+1}\right)=0,j=1,\dots ,N,\hfill \\ \hfill & {\displaystyle \frac{1}{2}}\left(z_N^{″}\left(t\right)+z_{N+1}^{″}\left(t\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}{z}_{N+1}+{\displaystyle \frac{k}{h}}{\delta }_x{z}_{N+{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{k}{h}}\gamma \psi (E_h\left(t\right)-E^{*})z_{N+1}^{\prime }\left(t\right)=0,\hfill \\ \hfill & z_0\left(t\right)=0,z_j\left(0\right)=z^0\left(x_j\right),z_j^{\prime }\left(0\right)=z^1\left(x_j\right),j=0,1,2,\dots ,N+1,\hfill \end{array}\right.$$

(62)

The state space formulation of the discrete scheme (62) is expressed as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{P}{4}}{Z}_h^{″}\left(t\right)+\beta D{Z}_h\left(t\right)+{\displaystyle \frac{k}{h^2}}B{Z}_h+{\displaystyle \frac{1}{h}}A{Z}_h^{\prime }=0,\hfill \\ Z_h\left(0\right)=Z_h^0,Z_h^{\prime }\left(0\right)=Z_h^1,\hfill \end{array}\right.$$

(63)

with an $(N+1)\times (N+1)$ matrix as follows:

$$D={\displaystyle \frac{1}{2}}\left(\begin{array}{cccccc}1& 1& & & & \\ & 1& 1& & & \\ & & \ddots & \ddots & \\ & & & 1& 1& \\ & & & & 1\end{array}\right).$$

(64)

Moreover, the differential Equation (62) is then equivalent to

$$P_h{Y}_h^{\prime }\left(t\right)=C_h{Y}_h\left(t\right)\iff Y_h^{\prime }\left(t\right)=G_h{Y}_h\left(t\right),$$

(65)

with $G_h:=P_h^{-1}{C}_h$ and

$$C_h=\left(\begin{array}{cc}0& I\\ -\beta D-{\displaystyle \frac{k}{h^2}}B& -{\displaystyle \frac{1}{h}}A\end{array}\right).$$

(66)

Figure 4, Figure 5 and Figure 6 are plotted for the discrete scheme (65) as above. The k, $\beta$, $\gamma$, and a parameters are the same as above. From these figures, it is easy to see that the discrete schemes ((61) and (65)) are similar. However, discrete scheme (65) is simpler than (61) and has lower computational costs. This implies that (65) is a more optimal approximation.

Figure 4. Maximal real parts of eigenvalues of the semi-discrete schemes with random a in the case of the second substitution.

Figure 5. Eigenvalue distribution of the semi-discrete scheme with random a and $N=200$ in the case of the second substitution.

Figure 6. Maximal real parts of eigenvalues of the semi-discrete schemes with $a=0.5$ in the case of the second substitution.

5. Conclusions

In this study, we investigated the asymptotic stability of the sine-Gordon system from the viewpoint of numerical approximation. It is widely recognized that the traditional finite difference scheme cannot uniformly preserve the exponential stability of a continuous system. Furthermore, given the challenge of the nonlinear term emerging in the model, we employed the order-reduction method to create a spatially semi-discretized finite difference scheme specifically designed for continuous systems. Moreover, to cope with the challenging nature of the nonlinear term in the proof of the main result, a discrete Wirtinger inequality suitable for our approximating scheme was established. More interestingly, we discovered that the proof of uniformly asymptotic stability for the semi-discretized system, using the order-reduction method, could be conducted in parallel with that of its continuous PDE counterpart. Importantly, the method proposed in this paper is not limited to the specific nonlinear PDE under investigation but can be readily applied to other partial differential equations or coupled systems where exponential stability can be established. However, we should point out that convergence and stability analyses of algorithms are not within the scope of this article. Moreover, the current numerical experiments may not necessarily reflect the nonlinearity of the system. These issues are worth investigating in further research.