UNIFORMLY EXPONENTIALLY STABLE APPROXIMATION FOR THE TRANSMISSION LINE WITH VARIABLE COEFFICIENTS AND ITS APPLICATION

Bingfeng Zhang; Fu Zheng; Yong He

doi:10.11948/20230361

2024 Volume 14 Issue 4

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Bingfeng Zhang, Fu Zheng, Yong He. UNIFORMLY EXPONENTIALLY STABLE APPROXIMATION FOR THE TRANSMISSION LINE WITH VARIABLE COEFFICIENTS AND ITS APPLICATION[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2228-2256. doi: 10.11948/20230361

Citation:

Bingfeng Zhang, Fu Zheng, Yong He. UNIFORMLY EXPONENTIALLY STABLE APPROXIMATION FOR THE TRANSMISSION LINE WITH VARIABLE COEFFICIENTS AND ITS APPLICATION[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2228-2256. doi: 10.11948/20230361

UNIFORMLY EXPONENTIALLY STABLE APPROXIMATION FOR THE TRANSMISSION LINE WITH VARIABLE COEFFICIENTS AND ITS APPLICATION

1.
School of Mathematical Science, Bohai University, Jinzhou, Liaoning 121013, China
2.
School of Science, Hainan University, Haikou, Hainan 570228, China

Author Bio: Email: 1248264438@qq.com(B. Zhang)
Corresponding authors: Email: fuzheng@hainanu.edu.cn(F. Zheng); Email: heyong1980@tju.edu.cn(Y. He)
Fund Project: This paper was carried out under the support of the National Natural Science Foundation of China under grants no. 11871117, 12371446; and Hainan Provincial National Natural Science Foundation of China under grant No. 123MS004; and Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province; and the Specific Research Fund of the Innovation Platform for Academicians of Hainan Province; and Scientific Research Initiation Fund of Hainan University (RZ2200001240)

Abstract

Abstract

We analyze an ideal transmission line, which is defined by the telegraph equation with variable coefficients, from the perspectives of numerical analysis and control theory in this note. Because the spatially semi-discrete scheme of the original system is insufficient for discussing uniform exponential stability, we apply a similar transform to the continuous system and produce an intermediate system that may be easily analyzed. To begin, we discuss uniform exponential stability for the intermediate system using an so called average central-difference semi-discrete scheme and the direct Lyapunov function approach. The proof is the same as in the continuous case. The Trotter-Kato Theorem is used to demonstrate the stability and consistency of numerical approximation scheme. Finally, we propose a semi-discrete strategy for the original system through an inverse transform. All results on intermediate system are then translated into the original system. The numerical state reconstruction problem is addressed as an essential application of the main results. Furthermore, several numerical simulations are used to validate the effectiveness of the numerical approximating algorithms.
- Transmission line /
- exponential stability /
- state reconstruction /
- semi-discretization /
- average central-difference
MSC: 65M06, 37L15, 93D05

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References

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