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

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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 ϵ-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.

Keywords

periodic Sylvester tensor equation / Einstein product / tensor operator / finite-time convergence / gradient neural network / multiarm robotic systems

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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

[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

[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

[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

[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

[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

[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

[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

[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

[11] T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. https://doi.org/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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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