HAAK, Bernhard H.; KUNSTMANN, Peer Christian

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HAAK, Bernhard H.
INSTITUT FUER ANALYSIS
Institut de Mathématiques de Bordeaux [IMB]

KUNSTMANN, Peer Christian
INSTITUT FUER ANALYSIS

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Integral Equations and Operator Theory. 2006, vol. 55, n° 4, p. 497--533

Springer Verlag

Résumé en anglais

We study linear systems, described by operators $A$, $B$, $C$ for which the state space $X$ is a Banach space. We suppose that $-A$ generates a bounded analytic semigroup and give conditions for admissibility of $B$ and $C$ corresponding to those in G. Weiss' conjecture. The crucial assumptions on $A$ are boundedness of an $H^\infty$--calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces $Y$ or control spaces $U$ that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le~Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in $X=L^p(\Omega)$, $p \in(1,\infty)$, with boundary observation and control and prove its wellposedness for several function spaces $Y$ and $U$ on the boundary $\partial\Omega$.< Réduire

Mots clés en anglais

control theory

linear systems

admissibility

$H^\infty$--calculus

square-function estimates

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Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces

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