Parametric Smith iterative algorithms for discrete Lyapunov matrix equations

Ai-Guo Wu; Ying Zhang; Hui-Jie Sun

doi:10.3934/jimo.2019093

Article Contents

2020, Volume 16, Issue 6: 3047-3063. Doi: 10.3934/jimo.2019093

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Parametric Smith iterative algorithms for discrete Lyapunov matrix equations

School of Mechanical Engineering and Automation, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China

^* Corresponding author: zhangyinghit@126.com
^* Corresponding author: zhangyinghit@126.com

Received: October 31, 2018

Revised: February 28, 2019

Early access: July 2019

Published: October 2020

The authors are supported by Shenzhen Municipal Basic Research Project for Discipline Layout with Project No.JCYJ20170811160715620, by the National Natural Science Foundation of China under Grant No. 61822305, by Guangdong Natural Science Foundation under Grant No. 2017A030313340, and by Shenzhen Municipal Project for International Cooperation with Project No. GJHZ20180420180849805

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Abstract

An iterative algorithm is established in this paper for solving the discrete Lyapunov matrix equations. The proposed algorithm contains a tunable parameter, and includes the Smith iteration as a special case, and thus is called the parametric Smith iterative algorithm. Some convergence conditions are developed for the proposed parametric Smith iterative algorithm. Moreover, the optimal parameter for the proposed algorithm to have the fastest convergence rate is also provided for a special case. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithm.

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Mathematics Subject Classification: Primary: 34D20, 15A06; Secondary: 15A24.

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Figure 1. Convergence performance of the algorithm (3) for Example 1

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Figure 2. Spectral radius of $ G $ for Example 1

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Figure 3. Convergence curve of the algorithm (5) for Example 1

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Figure 4. Convergence performance of the algorithm (3) for Example 2

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Figure 5. Spectral radius of $ G $ for Example 2

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Figure 6. Convergence curve of the algorithm (5) for Example 2

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Figure 7. Spectral radius of $ G $ for Example 3

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Figure 8. Convergence curve of the algorithm (5) for Example 3

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