Representation of solutions to continuous and discrete first-order linear matrix equations with delay
Javad A. Asadzade, Nazim I. Mahmudov
TL;DR
The paper addresses explicit representations of solutions for both continuous-time and discrete-time linear matrix delay systems with noncommuting coefficient matrices. It develops a unified framework based on recursive auxiliary sequences $\{Q_u(\cdot)\}$ and delayed exponential-type fundamental functions $Z_D(\cdot)$ to produce exact representations for homogeneous and nonhomogeneous problems, without requiring commutativity such as $B_1 G=G B_1$ or $B_1 \Psi=\Psi B_1$. The main contributions include a generalization of the classical Diblík results to noncommutative, two-sided matrices, explicit solution formulas, and a concrete two-dimensional example that demonstrates the method's practicality. This framework advances constructive analysis of delayed matrix systems and provides tools potentially valuable for control, signal processing, and stability assessments in settings with delays and noncommutativity.
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.