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 \[ \dot{X}(ξ) = A_0 X(ξ) + X(ξ)A_1 + B_0 X(ξ-σ) + X(ξ-σ)B_1 + G(ξ), \] 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 \(σ>0\), \(m \in \mathbb{N}\) 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 \(B_1G(ξ)=G(ξ)B_1\) and \(B_1Ψ(ξ)=Ψ(ξ)B_1\) imposed in \cite{diblik1, diblik2}, thus obtaining explicit solution formulas for more general classes of systems.