关于一个无限线性微分方程组的稳定性和零能控性。

Pub Date : 2023-01-01 Epub Date: 2021-12-23 DOI:10.1007/s10883-021-09587-6

Abdulla Azamov, Gafurjan Ibragimov, Khudoyor Mamayusupov, Marks Ruziboev

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引用次数: 2

摘要

在这项工作中，一个线性系统的零可控性问题ℓ2，其中描述系统的线性算子的矩阵是λ∈2的无穷大矩阵ℝ 在主对角线上及其上的1s。我们证明了系统是渐近稳定的当且仅当λ≤- 1，它显示了有限维系统和无限维系统之间的细微差别。当λ≤- 1我们还证明了该系统在很大程度上是零可控的。此外，我们还展示了稳定性对范数的依赖性，即所考虑的同一系统ℓ∞ 不是渐近稳定的，如果λ=- 1.

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On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations.

In this work, the null controllability problem for a linear system in ℓ² is considered, where the matrix of a linear operator describing the system is an infinite matrix with $λ \in ℝ$ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $ℓ^{\infty}$ is not asymptotically stable if λ = - 1.

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