Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions

Mofarreh, Fatemah; Alruwaili, Faridah; Wang, Xingtao; Elshenhab, Ahmed M.

doi:10.3390/math13223581

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Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions

¹

Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia

²

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

³

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(22), 3581; https://doi.org/10.3390/math13223581

Submission received: 3 October 2025 / Revised: 5 November 2025 / Accepted: 5 November 2025 / Published: 7 November 2025

(This article belongs to the Special Issue Recent Advances in Theory and Practice of Time-Delay Systems Analysis and Control)

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Abstract

This work introduces a unified framework for analyzing linear delay differential Sylvester matrix equations with noncommuting coefficients. The methodology employs a Kronecker product-based vectorization to transform the system, yielding explicit closed-form solutions via a novel delayed perturbation matrix function. Additionally, a delay-adapted Gramian matrix is formulated to derive necessary and sufficient controllability criteria. The approach’s efficacy is confirmed through a numerical example, demonstrating its capability in complex, noncommutative scenarios where classical methods are inapplicable.

Keywords: exact solutions; delay differential matrix equation; delayed matrix function; controllability; Kronecker product; vector operator

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MDPI and ACS Style

Mofarreh, F.; Alruwaili, F.; Wang, X.; Elshenhab, A.M. Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions. Mathematics 2025, 13, 3581. https://doi.org/10.3390/math13223581

AMA Style

Mofarreh F, Alruwaili F, Wang X, Elshenhab AM. Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions. Mathematics. 2025; 13(22):3581. https://doi.org/10.3390/math13223581

Chicago/Turabian Style

Mofarreh, Fatemah, Faridah Alruwaili, Xingtao Wang, and Ahmed M. Elshenhab. 2025. "Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions" Mathematics 13, no. 22: 3581. https://doi.org/10.3390/math13223581

APA Style

Mofarreh, F., Alruwaili, F., Wang, X., & Elshenhab, A. M. (2025). Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions. Mathematics, 13(22), 3581. https://doi.org/10.3390/math13223581

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions

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