Stability Analysis of Second-Order Linear PDEs on Time Scales

Liu, Ruihong; Zhang, Chuan; Zhang, Xianfu; Meng, Fanwei; Zhang, Hao

doi:https://doi.org/10.1155/2023/8226450

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8226450 | https://doi.org/10.1155/2023/8226450

Stability Analysis of Second-Order Linear PDEs on Time Scales

Ruihong Liu,¹Chuan Zhang,¹Xianfu Zhang,²Fanwei Meng,¹and Hao Zhang³

Academic Editor: Chunrui Zhang

Received27 Oct 2022

Revised28 Feb 2023

Accepted27 Mar 2023

Published21 Apr 2023

Abstract

This paper presents the stability analysis of a class of second-order linear partial differential equations (PDEs) on time scales with diffusion operator and first-order partial derivative. According to the time scale theory, the Lyapunov functional method, and some inequality techniques, sufficient conditions for exponential stability are strictly obtained, and the results are generalized for that where both the discrete-time and continuous-time cases are considered jointly. In addition, the theoretical results are applied to exponential synchronization of reaction-diffusion neural networks (RDNNs). Simulation examples are given to verify the feasibility of our results.

1. Introduction

Many real systems can be modeled as partial differential equations (PDEs), which generally express differential relationships among multiple objective things. The PDEs have shown practical applications in many fields including the electricity, light, and heat. Based on physics principles and laws, the PDEs can be roughly divided into three categories: hyperbolic, parabolic, and elliptic PDEs. Studies on various PDEs have gained wide attention among researchers in recent years [1–5].

Stability is a fundamental problem of equations or systems, and the stability of PDEs has been extensively studied in literature. For example, Taniguchi investigated the exponential stability of stochastic delayed PDEs in [6]. Practical exponential stability of nonlinear monotonic stochastic PDEs was discussed in [7]. Sufficient conditions for exponential stability of the second-order stochastic functional PDEs were obtained in [8]. Gahlawat and Valmorbida performed the Lyapunov analysis to determine the exponential stability of linear PDEs with one spatial dimension in [9]. With the aid of finite difference technique, Esmailzadeh et al. presented a numerical solution scheme to solve hyperbolic PDEs in [10]. In addition, as special parabolic PDEs, the stability of reaction-diffusion systems or reaction-diffusion neural networks (RDNNs) has also led to many achievements [11–13]. However, the above PDEs under consideration are all independent continuous-time and discrete-time cases. It is meaningful to introduce time scales to unify the continuous-time and discrete-time cases.

The time scales calculus theory was pioneered by Hilger [14], then improved and refined by Bohner and Peterson [15, 16]. Time scales theory unifies difference equation and differential equation so that they can be studied under a same framework. Recent related studies have shown that time scales are not only a theoretical territory of mathematics, but also an effective tool to handle many practical matters [17–20]. Although extensive research has been conducted on the stability analysis of dynamic equations on time scales, most of them were limited to ordinary differential equations, and few studies on the stability of PDEs on time scales were made [21–23]. To overcome this shortcoming, this study examines the stability of the second-order linear PDEs on time scales based on the Lyapunov functional method. In addition, there were many achievements on synchronization problems of RDNNs [24, 25]. We also attempt to deal with the exponential stability of RDNNs on time scales.

Inspired by the previous works, this paper adopts the Lyapunov functional method to study the global exponential stability of the second-order linear PDEs on time scales. The main contributions can be summarized as follows:(1)Compared to the existing works [21–23], the PDEs under consideration is more general for that where both the diffusion operator and second-order partial differential terms are taking into account.(2)Based on the Lyapunov functional and inequality methods, the sufficient conditions of stability are presented, and the results are generalized that they can unify the continuous-time and discrete-time cases. In addition, we extend the results of exponential stability to the application of RDNNs.

The rest of this paper is organized as follows. In Section 2, the relevant definitions and hypothetical lemmas are introduced. In Section 3, a model of second-order linear PDEs on time scales is defined, and its stability is analyzed. Section 4 gives an application on synchronization of RDNNs on time scales. The results are verified in two examples in Section 5. Conclusions are drawn in Section 6.

Notations: throughout the paper, is a time scale and ; is an open bounded domain in with a smooth boundary ; is a set consisting of vector functions , which is -continuous with respect to and ; is a Banach space with the norm of , where and ; , where .

2. Preliminaries

Before giving the stability analysis, it is necessary to introduce some basic concepts and lemmas of time scales theory [15, 16].

Definition 1. [15] For , the forward and backward jump operators are, respectively, defined by and . If , then is called right-scattered, and is called left-scattered if . Also, if and , then is called right-dense; is called left-dense if and . The graininess function is defined by . Set is obtained from the time scale as follows: if has a left-scattered maximum , then ; otherwise, .

Definition 2. [15] Let and define by for all , namely, .

Definition 3. [15] Let and , then can be defined as a number with a property that for any , and there exists , for all , such that:then, is called a delta derivative of on . is called rd-continuous if it is continuous at right dense points in and its left-sided limits exist at left-dense points in . A set of all rd-continuous functions is denoted by . is called regulated if it is rd-continuous and its right-side limits exist at all right-side points in and its left-side limits exist at all left-side points in .

Definition 4. [15] If , then for any , the integral is defined as follows:

Definition 5. [16] If , then the exponential function can be defined by for all . The cylinder transformation is given by the following expression:

Lemma 1. [16] Assume that is a cube and is a real-valued function, which vanish on the boundary , then it holds:

Lemma 2. [16] If are differentiable at , then it holds:

Lemma 3. [16] If is differentiable at , then it holds:

Lemma 4. [16] Let be two regressive functions and denote ; then the following expressions hold:(i)(ii)(iii)(iv)(v)(vi)

Definition 6. [16] The equilibrium solution of PDEs (10) is called global exponential stability if there exist positive constants and such that:

Definition 7. [16] For each , let be a neighborhood of . Then, for , define as a mean such that, for , and there exists a right neighborhood of such that:For each , where , if is right-scattered and is continuous at , it holds:

3. Main Results

In this work, discrete and continuous domains are considered directly and jointly using the equations on time scales, and the second-order linear PDEs on the time scales under investigation is given as follows:where ; is the time variable; is the spacial variable; is the state variable; , , and are constant coefficients; is the nonlinear function; is the initial value function.

Assumption 1. [16] Assume that is Lipschitz continuous, and there exists a constant such that:

Theorem 1. Under the Assumption 1, the PDEs on time scales (10) is global exponential stability if the following condition is satisfied as follows:

Proof. Let , and the error dynamics equation is obtained as follows:where . Calculating the delta derivation of yields toIn view of integration by parts and Lemma 1, one hasConsidering Assumption 1, we can getBased on the above inequalities (15–17), it can be obtainedwhere , , .
If formula (18) holds, there exists an appropriate positive number , perhaps small, such that for any :Then, select a function defined as follows:where . By equation (20), it can be obtained that and is continuous. Next, ensure that at some point. For any , when , , there exists a constant such that and for . Setting , , one can getTaking the Lyapunov function as follows:It should be noted that . Then, the delta derivatives of can be expressed as follows:Simplifying the above formula gives as follows:In view of Lemma 4, it can be obtained thatwhich implieswhere . According to Definition 6, the PDEs on time scales (10) is global exponential stability under the condition (12).

Remark 1. At present, few research studies have been conducted to distinguish the continuous-time and discrete-time cases of PDEs. However, the time domain of PDEs is not always strictly continuous-time or discrete-time in practice. In this paper, the time scale theory is utilized to unify the continuous-time or discrete-time cases.

Remark 2. If , we have , and equation (10) can be expressed as a continuous-time PDEs:If , then , and equation (10) can be given as a discrete-time PDEs:

Remark 3. By choosing appropriate equation parameter, the equation considered in this paper can be the hyperbolic, the elliptic and the parabolic equations on time scales, the stability conditions can be easily obtained. For example, if , the equation is a semilinear parabolic equation on time scales:then sufficient condition for exponential stability of the above equation is given as follows:

4. Application

The RDNNs on time scales is a special case of the second-order linear PDEs on time scales. In this section, the exponential stability problem of driven-response RDNNs on time scales is considered. Consider the following RDNNs model as follows:where , and is the number of neurons. is the space variable. is the diffusion parameter. is the self-feedback connection weight. is the connection weight coefficient.

Take the system (31) as the drive system, then the corresponding response system can be given as follows:

Define the error , and the error equation is obtained as follows:

We can easily obtain the following proposition.

Proposition 1. The RDNNs on time scales (31) is exponential stability if the following sufficient condition is satisfied:

5. Illustrative Examples

In this section, two examples are given to verify the validity of our results.

Example 1. Consider a PDE on time scales defined as follows:where , and satisfies Lipschitz condition, and . Let , and we consider the system with the following parameters:and the initial conditions:Let ; we can haveBy calculating the inequality (12), we can haveIn this case, system (35) does not satisfy sufficient condition (2), so the system is not exponentially stable under this numerical condition (See Figure 1).

Example 2. Consider a PDE with the following parameters, and the other parameters are the same as in Example 1:and the initial conditions:Let . By calculating the inequality (12), we haveObviously, the conditions obtained are satisfied. Figure 2 shows that the equation is the global exponential stability. According to Theorem 1, the equation on time scales satisfies the global exponential stability condition.
From Figure 2, we can see that the error state of PDEs (35) is gradually reduced to the corresponding stable point. From the simulation results, it can be seen that the stable points of continuous-time and discrete-time are different, but they tend to be stable in the end.

Remark 4. The reason why equation (35) fails to converge to zero is that the numerical stability point of the equation under this condition is not zero. In the same way, when , the stable point is .

Remark 5. Examples 1 and 2 give two simulation examples of stable and unstable PDE systems. It can be seen that the selection of system parameters and initial values determine the steady state of the system. The obtained condition of exponential stability in Theorem 1 is a sufficient condition, so the system that does not meet this condition may also be stable, which will be the goal of the PDE system in our next study.

6. Conclusion

This study considers a common second-order linear PDEs and obtains the corresponding equations on time scales. Based on the time scales theory, the Lyapunov functional method, and some inequality techniques, the exponential stability of second-order linear PDEs on time scales is analyzed, and sufficient conditions for the global exponential stability are strictly presented. In addition, the stability result is applied to the synchronization of RDNNs on time scales. Finally, two examples are given to verify the correctness of our results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (grant nos. 62003189, 11671227, 61973189, and 61702356), China Postdoctoral Science Foundation (grant no. 2020M672024), Natural Science Foundation of Shandong Province (grant no. ZR2021MA016), and Youth Creative Team Sci-Tech Program of Shandong Universities (grant no. 2019KJI007).

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Copyright © 2023 Ruihong Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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