Tensor finite-time convergent gradient neural network for periodic Sylvester tensor equations with robotic applications

Mengyan Xie; Yang Zhang; Mengyan Xie; Yang Zhang

doi:10.3934/era.2026317

Electronic Research Archive

2026, Volume 34, Issue 10: 7330-7361. doi: 10.3934/era.2026317

Previous Article Next Article

Research article

Tensor finite-time convergent gradient neural network for periodic Sylvester tensor equations with robotic applications

Mengyan Xie ¹,
Yang Zhang ^{2
,
,}

1.
College of Information Technology, Shanghai Ocean University, Shanghai 201306, China
2.
Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada

Received: 18 June 2026 Revised: 06 August 2026 Accepted: 17 August 2026 Published: 26 August 2026

The periodic Sylvester tensor equation (PSTE) arises in multidimensional periodic systems. This paper proposes a tensor finite-time convergent gradient neural network (Tensor-FTCGNN) with a global adaptive gain for solving a class of PSTEs. Using the Einstein product, the periodically coupled tensor equations are formulated as a unified linear operator equation on a product tensor space, and the corresponding adjoint operator and global gradient are derived. Tensor linear operator theory and Lyapunov analysis are employed to establish the global finite-time convergence of the ideal continuous-time model and to derive an explicit upper bound for its settling time. For practical numerical implementation, a regularized Tensor-FTCGNN is introduced, which reaches an $ \epsilon $-dependent high-accuracy region in finite time and subsequently converges exponentially to the exact solution. Numerical experiments compare the proposed regularized model with representative gradient neural network (GNN) models, verify the theoretical finite-time upper bound through the observed Phase-Ⅰ entrance times, and evaluate its convergence behavior over the tested tensor sizes. Finally, the proposed framework is applied to cooperative periodic trajectory reconstruction for a multiarm robotic system, illustrating its potential for multidimensional robotic applications.
- periodic Sylvester tensor equation,
- Einstein product,
- tensor operator,
- finite-time convergence,
- gradient neural network,
- multiarm robotic systems
Citation: Mengyan Xie, Yang Zhang. Tensor finite-time convergent gradient neural network for periodic Sylvester tensor equations with robotic applications[J]. Electronic Research Archive, 2026, 34(10): 7330-7361. doi: 10.3934/era.2026317

Related Papers:

Abstract

The periodic Sylvester tensor equation (PSTE) arises in multidimensional periodic systems. This paper proposes a tensor finite-time convergent gradient neural network (Tensor-FTCGNN) with a global adaptive gain for solving a class of PSTEs. Using the Einstein product, the periodically coupled tensor equations are formulated as a unified linear operator equation on a product tensor space, and the corresponding adjoint operator and global gradient are derived. Tensor linear operator theory and Lyapunov analysis are employed to establish the global finite-time convergence of the ideal continuous-time model and to derive an explicit upper bound for its settling time. For practical numerical implementation, a regularized Tensor-FTCGNN is introduced, which reaches an $ \epsilon $-dependent high-accuracy region in finite time and subsequently converges exponentially to the exact solution. Numerical experiments compare the proposed regularized model with representative gradient neural network (GNN) models, verify the theoretical finite-time upper bound through the observed Phase-Ⅰ entrance times, and evaluate its convergence behavior over the tested tensor sizes. Finally, the proposed framework is applied to cooperative periodic trajectory reconstruction for a multiarm robotic system, illustrating its potential for multidimensional robotic applications.

References

[1]	Z. H. He, A. Dmytryshyn, Q. W. Wang, A new system of Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra, Linear Multilinear Algebra, 73 (2025), 1269–1309. https://doi.org/10.1080/03081087.2024.2413635 doi: 10.1080/03081087.2024.2413635
[2]	Z. H. He, L. W. Ye, Y. Z. Xu, M. L. Deng, QSVD method for arbitrary multiple Sylvester-type coupled quaternion matrix equations with applications, Comput. Appl. Math., 45 (2026), 155. https://doi.org/10.1007/s40314-025-03488-1 doi: 10.1007/s40314-025-03488-1
[3]	T. Li, Q. W. Wang, X. F. Duan, Numerical algorithms for solving discrete Lyapunov tensor equation, J. Comput. Appl. Math., 370 (2020), 112676. https://doi.org/10.1016/j.cam.2019.112676 doi: 10.1016/j.cam.2019.112676
[4]	Q. W. Wang, X. Xu, X. F. Duan, Least squares solution of the quaternion Sylvester tensor equation, Linear Multilinear Algebra, 69 (2021), 104–130. https://doi.org/10.1080/03081087.2019.1588848 doi: 10.1080/03081087.2019.1588848
[5]	M. Xie, Q. W. Wang, Y. Zhang, The BiCG algorithm for solving the minimal Frobenius norm solution of generalized Sylvester tensor equation over the quaternions, Symmetry, 16 (2024), 1167. https://doi.org/10.3390/sym16091167 doi: 10.3390/sym16091167
[6]	L. Lv, J. Chen, L. Zhang, F. Zhang, Gradient-based neural networks for solving periodic Sylvester matrix equations, J. Franklin Inst., 359 (2022), 10849–10866. https://doi.org/10.1016/j.jfranklin.2022.05.023 doi: 10.1016/j.jfranklin.2022.05.023
[7]	S. Li, C. Ma, An improved gradient neural network for solving periodic Sylvester matrix equations, J. Franklin Inst., 360 (2023), 4056–4070. https://doi.org/10.1016/j.jfranklin.2023.02.019 doi: 10.1016/j.jfranklin.2023.02.019
[8]	A. G. Wu, W. X. Zhang, Y. Zhang, An iterative algorithm for discrete periodic Lyapunov matrix equations, Automatica, 87 (2018), 395–403. https://doi.org/10.1016/j.automatica.2017.06.012 doi: 10.1016/j.automatica.2017.06.012
[9]	S. Bittanti, P. Colaneri, Periodic Systems: Filtering and Control, Springer, London, 2009. https://doi.org/10.1007/978-1-84800-911-0
[10]	S. Hara, Y. Yamamoto, T. Omata, M. Nakano, Repetitive control system: A new type servo system for periodic exogenous signals, IEEE Trans. Autom. Control, 33 (1988), 659–668. https://doi.org/10.1109/9.1274 doi: 10.1109/9.1274
[11]	T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455–500. https://doi.org/10.1137/07070111X doi: 10.1137/07070111X
[12]	T. Li, Q. W. Wang, Structure preserving quaternion biconjugate gradient method, SIAM J. Matrix Anal. Appl., 45 (2024), 306–326. https://doi.org/10.1137/23M1547299 doi: 10.1137/23M1547299
[13]	M. Xie, Q. W. Wang, J. Chen, Fixed-time TGNN model with the nonlinear activation function for online solution of Sylvester tensor equation, Numer. Algorithms, 101 (2026), 1023–1047. https://doi.org/10.1007/s11075-025-02031-x doi: 10.1007/s11075-025-02031-x
[14]	J. Xue, Y. Q. Zhao, T. Wu, J. C. W. Chan, Tensor convolution-like low-rank dictionary for high-dimensional image representation, IEEE Trans. Circuits Syst. Video Technol., 34 (2024), 13257–13270. https://doi.org/10.1109/TCSVT.2024.3442295 doi: 10.1109/TCSVT.2024.3442295
[15]	Z. Qi, Y. Ning, L. Xiao, J. Luo, X. Li, Finite-time zeroing neural networks with novel activation function and variable parameter for solving time-varying Lyapunov tensor equation, Appl. Math. Comput., 452 (2023), 128072. https://doi.org/10.1016/j.amc.2023.128072 doi: 10.1016/j.amc.2023.128072
[16]	C. Mo, D. Gerontitis, P. S. Stanimirović, Solving the time-varying tensor square root equation by varying-parameters finite-time Zhang neural network, Neurocomputing, 445 (2021), 309–325. https://doi.org/10.1016/j.neucom.2021.03.011 doi: 10.1016/j.neucom.2021.03.011
[17]	J. Li, X. Wang, K. Wang, Y. Wei, Neural networks for solving least squares solution of polynomial systems with time-varying tensors, J. Comput. Appl. Math., 474 (2026), 116952. https://doi.org/10.1016/j.cam.2025.116952 doi: 10.1016/j.cam.2025.116952
[18]	L. Xiao, J. Hu, Q. Zuo, W. Kuang, L. Li, Nonlinear zeroing neural networks with lower upper bounds for time-varying algebraic tensor Riccati equation based on improper integral, Neurocomputing, 685 (2026), 133628. https://doi.org/10.1016/j.neucom.2026.133628 doi: 10.1016/j.neucom.2026.133628
[19]	M. Xie, Q. W. Wang, Y. Zhang, An AF-CRP-enhanced zeroing neural network for time-varying generalized Sylvester matrix equations, Neurocomputing, 681 (2026), 133368. https://doi.org/10.1016/j.neucom.2026.133368 doi: 10.1016/j.neucom.2026.133368
[20]	Y. Zhang, B. Liao, G. Geng, GNN model with robust finite-time convergence for time-varying systems of linear equations, IEEE Trans. Syst. Man Cybern. Syst., 54 (2024), 4786–4797. https://doi.org/10.1109/TSMC.2024.3387023 doi: 10.1109/TSMC.2024.3387023
[21]	M. D. Petković, M. Kostadinov, J. Dakić, Gradient neural dynamics for linear matrix equations and their applications, Math. Comput. Simul., 249 (2026), 560–582. https://doi.org/10.1016/j.matcom.2026.05.031 doi: 10.1016/j.matcom.2026.05.031
[22]	J. Dakić, M. D. Petković, Gradient neural network model for the system of two linear matrix equations and applications, Appl. Math. Comput., 481 (2024), 128930. https://doi.org/10.1016/j.amc.2024.128930 doi: 10.1016/j.amc.2024.128930
[23]	P. S. Stanimirović, M. D. Petković, Gradient neural dynamics for solving matrix equations and their applications, Neurocomputing, 306 (2018), 200–212. https://doi.org/10.1016/j.neucom.2018.03.058 doi: 10.1016/j.neucom.2018.03.058
[24]	A. Einstein, Die formale Grundlage der allgemeinen Relativitätstheorie, in Albert Einstein: Akademie-Vorträge, (1914), 1030–1085.
[25]	S. Li, S. Chen, B. Liu, Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function, Neural Process. Lett., 37 (2013), 189–205. https://doi.org/10.1007/s11063-012-9241-1 doi: 10.1007/s11063-012-9241-1
[26]	S. P. Bhat, D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751–766. https://doi.org/10.1137/S0363012997321358 doi: 10.1137/S0363012997321358

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)