Representation of solutions to continuous and discrete first-order linear matrix equations with delay

In this paper, we study continuous and discrete linear delay systems given respectively by $$\dot{X}(\xi )=A_{0}X(\xi )+X(\xi )A_1+B_{0}X(\xi -\sigma )+X(\xi -\sigma )B_1+G(\xi ),$$ and its discrete analogue $$X(u+1)=A_{0}X(u)+X(u)A_1+B_{0}X(u-m)+X(u-m)B_1+G(u),$$ where $A_0,A_1,B_0,B_1\in {\mathbb{R}}^{d\times d}$ are constant noncommuting matrices, and $\sigma >0$, $m\in \mathbb{N}$ denote the delay parameters. The main objective is to generalize the classical results of \cite{diblik1, diblik2} and to provide explicit representations of the solutions. For this purpose, we present generalized delayed exponential-type systems for both continuous and discrete cases. This approach allows us to remove the restrictive commutativity conditions $B_{1}G(\xi )=G(\xi )B_1$ and $B_1\mathrm{\Psi }(\xi )=\mathrm{\Psi }(\xi )B_1$ imposed in \cite{diblik1, diblik2}, thus obtaining explicit solution formulas for more general classes of systems.