Solving quaternion nonsymmetric algebraic Riccati equations through zeroing neural networks

Houssem Jerbi; Izzat Al-Darraji; Saleh Albadran; Sondess Ben Aoun; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis; Houssem Jerbi; Izzat Al-Darraji; Saleh Albadran; Sondess Ben Aoun; Theodore E. Simos; Spyridon D. Mourtas; Vasilios N. Katsikis

doi:10.3934/math.2024281

AIMS Mathematics

2024, Volume 9, Issue 3: 5794-5809. doi: 10.3934/math.2024281

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Solving quaternion nonsymmetric algebraic Riccati equations through zeroing neural networks

1.
Department of Industrial Engineering, College of Engineering, University of Hail, Ha'il 81481, Saudi Arabia
2.
Al-Khwarizmi College of Engineering, University of Baghdad, Baghdad 10081, Iraq
3.
Department of Electrical Engineering, College of Engineering, University of Hail, Ha'il 81481, Saudi Arabia
4.
Department of Computer Engineering, College of Computer Science and Engineering, University of Hail, Ha'il 81451, Saudi Arabia
5.
Center for Applied Math. and Bioinformatics, Gulf Univ. for Science and Technology, West Mishref 32093, Kuwait
6.
Department of Medical Research, China Medical Univ. Hospital, China Medical Univ., Taichung City 40402, Taiwan
7.
Laboratory of Inter-Disciplinary Problems in Energy Production, Ulyanovsk State Technical Univ., 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
8.
Data Recovery Key Laboratory of Sichun Province, Neijing Normal Univ., Neijiang 641100, China
9.
Section of Mathematics, Dept. of Civil Engineering, Democritus Univ. of Thrace, Xanthi 67100, Greece
10.
Department of Economics, Division of Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian Univ. of Athens, Sofokleous 1 Street, 10559 Athens, Greece
11.
Laboratory "Hybrid Methods of Modelling and Optimization in Complex Systems", Siberian Federal Univ., Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia

Received: 09 December 2023 Revised: 18 January 2024 Accepted: 24 January 2024 Published: 31 January 2024
MSC : 65F20, 68T05

Many variations of the algebraic Riccati equation (ARE) have been used to study nonlinear system stability in the control domain in great detail. Taking the quaternion nonsymmetric ARE (QNARE) as a generalized version of ARE, the time-varying QNARE (TQNARE) is introduced. This brings us to the main objective of this work: finding the TQNARE solution. The zeroing neural network (ZNN) technique, which has demonstrated a high degree of effectiveness in handling time-varying problems, is used to do this. Specifically, the TQNARE can be solved using the high order ZNN (HZNN) design, which is a member of the family of ZNN models that correlate to hyperpower iterative techniques. As a result, a novel HZNN model, called HZ-QNARE, is presented for solving the TQNARE. The model functions fairly well, as demonstrated by two simulation tests. Additionally, the results demonstrated that, while both approaches function remarkably well, the HZNN architecture works better than the ZNN architecture.
- zeroing neural networks,
- nonsymmetric algebraic Riccati equations,
- quaternions,
- dynamical system
Citation: Houssem Jerbi, Izzat Al-Darraji, Saleh Albadran, Sondess Ben Aoun, Theodore E. Simos, Spyridon D. Mourtas, Vasilios N. Katsikis. Solving quaternion nonsymmetric algebraic Riccati equations through zeroing neural networks[J]. AIMS Mathematics, 2024, 9(3): 5794-5809. doi: 10.3934/math.2024281

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Abstract

Many variations of the algebraic Riccati equation (ARE) have been used to study nonlinear system stability in the control domain in great detail. Taking the quaternion nonsymmetric ARE (QNARE) as a generalized version of ARE, the time-varying QNARE (TQNARE) is introduced. This brings us to the main objective of this work: finding the TQNARE solution. The zeroing neural network (ZNN) technique, which has demonstrated a high degree of effectiveness in handling time-varying problems, is used to do this. Specifically, the TQNARE can be solved using the high order ZNN (HZNN) design, which is a member of the family of ZNN models that correlate to hyperpower iterative techniques. As a result, a novel HZNN model, called HZ-QNARE, is presented for solving the TQNARE. The model functions fairly well, as demonstrated by two simulation tests. Additionally, the results demonstrated that, while both approaches function remarkably well, the HZNN architecture works better than the ZNN architecture.

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