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Discrete generalized-Sylvester matrix equation solved by RNN with a novel direct discretization numerical method

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Abstract

In the fields of artificial intelligence and control engineering, generalized-Sylvester matrix equation is considered as an important mathematic problem, and its solving process is usually viewed as a challenge that deserves particular attention. In this paper, a creative discrete-form recurrent neural network (RNN) model is developed, analyzed and investigated for solving discrete-form time-variant generalized-Sylvester matrix equation (DF-TV-GSME), which is derived by a direct discretization numerical method. Specifically, first of all, DF-TV-GSME, which includes the well-known Lyapunov matrix equation and Sylvester matrix equation, is presented as the target problem of this research. Secondly, for solving such problem, different from the traditional discrete-form RNN design philosophy, second-order Taylor expansion is applied to derive the discrete-form RNN model. This creative process avoids involving the continuous time-variant environment and continuous-form model. Then, theoretical properties analyses are presented, which present the convergence and precision of the discrete-form RNN model. Abundant numerical experiments are further carried out with different perspectives of DF-TV-GSME, which further confirm the excellent properties of discrete-form RNN model.

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Funding

This work was supported in part by the National Natural Science Foundation of China (with numbers 61906164 and 61972335), in part by the Natural Science Foundation of Jiangsu Province of China (with number BK20190875), in part by the Six Talent Peaks Project in Jiangsu Province (with number RJFW-053), in part by Jiangsu “333” Project, in part by Qinglan project of Yangzhou University, in part by the Cross-Disciplinary Project of the Animal Science Special Discipline of Yangzhou University, in part by the Yangzhou University Interdisciplinary Research Foundation for Animal Husbandry Discipline of Targeted Support (with number yzuxk202015), in part by the Yangzhou City-Yangzhou University Science and Technology Cooperation Fund Project (with number YZ2021157), in part by the Yangzhou University Top-level Talents Support Program (2021, 2019) by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (with numbers KYCX21_3234 and SJCX22_1709).

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Authors and Affiliations

  1. School of Information Engineering, Yangzhou University, Yangzhou, 225127, China

    Yang Shi, Chenling Ding, Bin Li & Xiaobing Sun

  2. Jiangsu Province Engineering Research Center of Knowledge Management and Intelligent Service, Yangzhou University, Yangzhou, 225127, China

    Yang Shi, Chenling Ding, Bin Li & Xiaobing Sun

  3. College of Engineering, Swansea University, Fabian Way, Swansea, UK

    Shuai Li

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  1. Yang Shi
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  2. Chenling Ding
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Correspondence to Yang Shi or Shuai Li.

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Shi, Y., Ding, C., Li, S. et al. Discrete generalized-Sylvester matrix equation solved by RNN with a novel direct discretization numerical method. Numer Algor 93, 971–992 (2023). https://doi.org/10.1007/s11075-022-01449-x

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  • DOI: https://doi.org/10.1007/s11075-022-01449-x

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