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Mathematics
Volume 13
Issue 22
10.3390/math13223631
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems

by
Fatemah Mofarreh
1 and
Ahmed M. Elshenhab
2,*
1
Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia
2
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(22), 3631; https://doi.org/10.3390/math13223631 (registering DOI)
Submission received: 26 September 2025 / Revised: 5 November 2025 / Accepted: 10 November 2025 / Published: 12 November 2025
(This article belongs to the Special Issue New Trends in Fractional Differential Equations with Applications)
Versions Notes

Abstract

This paper addresses the solvability and controllability of fractional delay differential Sylvester matrix equations with non-permutable coefficient matrices. By applying a vectorization approach and Kronecker product algebra, we transform the matrix-valued problem into an equivalent vector system, enabling the derivation of explicit solution representations using a delayed perturbation of two-parameter Mittag-Leffler-type matrix functions. We establish necessary and sufficient conditions for controllability via a fractional delay Gramian matrix, providing a computationally verifiable criterion that requires no commutativity assumptions. The theoretical results are validated through numerical examples, demonstrating effectiveness in noncommutative scenarios where classical methods fail.
Keywords: representation of solutions; fractional delay differential Sylvester matrix equation; delayed Perturbation Matrix Function; controllability; Kronecker product; vector operator representation of solutions; fractional delay differential Sylvester matrix equation; delayed Perturbation Matrix Function; controllability; Kronecker product; vector operator

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MDPI and ACS Style

Mofarreh, F.; Elshenhab, A.M. A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems. Mathematics 2025, 13, 3631. https://doi.org/10.3390/math13223631

AMA Style

Mofarreh F, Elshenhab AM. A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems. Mathematics. 2025; 13(22):3631. https://doi.org/10.3390/math13223631

Chicago/Turabian Style

Mofarreh, Fatemah, and Ahmed M. Elshenhab. 2025. "A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems" Mathematics 13, no. 22: 3631. https://doi.org/10.3390/math13223631

APA Style

Mofarreh, F., & Elshenhab, A. M. (2025). A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems. Mathematics, 13(22), 3631. https://doi.org/10.3390/math13223631

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