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Admissible observation operators. Semigroup criteria of admissibility. (English) Zbl 0856.93021
Let \(\{S(t)\}_{t\geq0}\) be a strongly continuous semigroup of linear bounded operators over a Hilbert space \(H\), with generator \(A\). Let \(C\in{\mathcal L}(D_A,Y)\) – called observation operator – where \(D_A\) is the domain of \(A\) and \(Y\) is another Hilbert space.
This paper studies admissible observation operators, i.e., those operators \(C\) for which there exists \(\gamma>0\) such that: \[ \int^\infty_0|CS(t)u_0|^2_Y dt\leq\gamma|u_0|^2_H,\qquad \forall\;u_0\in D_A. \] Necessary and sufficient conditions for admissibility are obtained via conditions for a certain operator to generate a uniformly bounded semigroup. Moreover, this semigroup is also weakly stable. Semigroup criteria of admissibility of exact observable systems are then derived. It turns out that admissibility and exact observability are equivalent to a certain operator to be similar to the generator of an isometric semigroup.

MSC:
93B28 Operator-theoretic methods
93B07 Observability
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