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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.16845 |
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| _version_ | 1866915575823859712 |
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| author | Asadzade, Javad A. Mahmudov, Nazim I. |
| author_facet | Asadzade, Javad A. Mahmudov, Nazim I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation of solutions to continuous and discrete first-order linear matrix equations with delay Asadzade, Javad A. Mahmudov, Nazim I. Dynamical Systems 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. |
| title | Representation of solutions to continuous and discrete first-order linear matrix equations with delay |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.16845 |