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 \[ \dot{X}(\xi) = A_0 X(\xi) + X(\xi)A_1 + B_0 X(\xi-\sigma) + X(\xi-\sigma)B_1 + G(\xi), \] 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 \(B_1G(\xi)=G(\xi)B_1\) and \(B_1\Psi(\xi)=\Psi(\xi)B_1\) imposed in \cite{diblik1, diblik2}, thus obtaining explicit solution formulas for more general classes of systems.