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A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems

 

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Opened Access A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems
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Author: Ammar-Khodja, Farid
Benabdallah, Assia
Dupaix, Cédric
González Burgos, Manuel
Department: Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico
Date: 2009-08
Published in: Differential Equations & Application, 1 (3), 427-457.
Document type: Article
Abstract: In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n, with t0 ∈ [0,T], is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A|B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem.
Cite: Ammar-Khodja, F., Benabdallah, A., Dupaix, C. y González Burgos, M. (2009). A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems. Differential Equations & Application, 1 (3), 427-457.
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URI: http://hdl.handle.net/11441/48942

DOI: 10.7153/dea-01-24

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