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

Fatemah Mofarreh, Faridah Alruwaili, Xingtao Wang and Ahmed M. Elshenhab ()
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Fatemah Mofarreh: Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia
Faridah Alruwaili: Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia
Xingtao Wang: School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Ahmed M. Elshenhab: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Mathematics, 2025, vol. 13, issue 22, 1-18

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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