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Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces
Language
en
Article de revue
This item was published in
Integral Equations and Operator Theory. 2006, vol. 55, n° 4, p. 497--533
Springer Verlag
English Abstract
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 ...Read more >
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 $\partial\Omega$.Read less <
English Keywords
control theory
linear systems
admissibility
$H^\infty$--calculus
square-function estimates
Origin
Hal imported