Null-controllability of linear parabolic-transport systems

K. Beauchard, Armand Koenig, K. L. Balc'h
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引用次数: 9

Abstract

Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.
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线性抛物输运系统的无可控性
在过去的二十年里,人们研究了抛物线-双曲型系统的可控性问题。本文是试图找到一个包含和概括以前结果的统一框架的开始。我们考虑在一维环面上具有零阶和一阶耦合且源项具有局部分布控制的常系数热输运系统。我们证明了在最优时间(由于传输分量所期望的时间)中,当控制和方程一样多时的零可控性。当控件仅作用于传输(响应)时。我们证明了在耦合项上零可控性的一个代数充要条件。整个研究依赖于基于微扰理论的细致谱分析。在求解纯输运方程直至紧化项的高双曲频率定域解上,证明了小时间条件下的负可控性结果。在大时间内证明正结果依赖于谱分解成低的、渐近的抛物线或双曲频率。
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