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Zbl 1138.93361
Haak, Bernhard H.; Kunstmann, Peer Christian
Admissibility of unbounded operators and well-posedness of linear systems in Banach spaces. (English)
[J] Integral Equations Oper. Theory 55, No. 4, 497-533 (2006). ISSN 0378-620X; ISSN 1420-8989/e

Summary: 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 $X = L^{p}(\Omega)$, $1 < p < \infty$, with boundary observation and control and prove its wellposedness for several function spaces $Y$ and $U$ on the boundary $\Omega$.

MSC 2000:: *93C25 Control systems in abstract spaces
34G10 Linear ODE in abstract spaces
47A60 Functional calculus of operators
47D06 One-parameter semigroups and linear evolution equations
47N70 Appl. of operator theory in systems theory, circuits, etc.
93B36 $H^\infty$-control

Keywords: linear systems; admissibility; square function estimates

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