Representation of Solutions to Continuous and Discrete First‐Order Linear Matrix Equations With Delay

ABSTRACT In this paper, we study continuous and discrete linear delay systems given respectively by and its discrete analogue where are constant non‐commuting matrices, and , denote the delay parameters. The main objective is to generalize the classical results of [6, 7] and to provide explicit representations of the solutions. For this purpose, we present generalized delayed exponential‐type matrix functions for both continuous and discrete cases. This approach allows us to remove the restrictive commutativity conditions and imposed in [6, 7], thus obtaining explicit solution formulas for more general classes of systems. Additionally, we establish Ulam‐Hyers stability criteria for both continuous and discrete systems, confirming their robustness under small perturbations. The theoretical results are illustrated with numerical examples demonstrating the applicability of the proposed methods.