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

2025/09/21 by Javad A. Asadzade, Asadzade, Javad A., Nazım I. Mahmudov +1
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2509.16845

openalex publication_date 2025/09/21 · openalex created_date 2025/10/02 · openalex updated_date 2026/07/29

Abstract

In this paper, we study continuous and discrete linear delay systems given respectively by X(ξ) = A₀ X(ξ) + X(ξ)A₁ + B₀ X(ξ-σ) + X(ξ-σ)B₁ + G(ξ), and its discrete analogue X(u+1) = A₀ X(u) + X(u)A₁ + B₀ X(u-m) + X(u-m)B₁ + G(u), where \(A₀, A₁, B₀, B₁ ∈ ℝ^{d × d}\) are constant noncommuting matrices, and \(σ>0\), \(m ∈ ℕ\) denote the delay parameters. The main objective is to generalize the classical results of \citediblik1, 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 \(B₁G(ξ)=G(ξ)B₁\) and \(B₁Ψ(ξ)=Ψ(ξ)B₁\) imposed in \citediblik1, diblik2, thus obtaining explicit solution formulas for more general classes of systems.

Citations

• Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability
• Representation of solutions of discrete linear delay systems with non permutable matrices

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