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Admissibility of control and observation operators for semigroups: a survey. (English) Zbl 1083.93025

Ball, Joseph A. (ed.) et al., Current trends in operator theory and its applications. Proceedings of the international workshop on operator theory and its applications (IWOTA), Virginia Tech, Blacksburg, VA, USA, August 6–9, 2002. Basel: Birkhäuser (ISBN 3-7643-7067-X/hbk). Operator Theory: Advances and Applications 149, 199-221 (2004).
Consider the abstract linear system \(\dot{x}(t) = Ax(t)\), \(y(t) = C x(t)\), where \(A\) is the infinitesimal generator of the strongly continuous semigroup \(T(t)\) on the Hilbert space \(H\), and \(C\) is a bounded operator from the domain of \(A\), \(D(A)\), to the Hilbert space \(Y\). Since \(C\) is bounded on \(D(A)\), it is clear that \(CT(t)x_0\) is well-defined for \(x_0 \in D(A)\). The operator \(C\) is defined to be (infinite-time) admissible if the mapping \(x_0 \rightarrow CT(t)x_0\) extends to a bounded mapping from \(H\) to \(L^2((0,\infty);Y)\). In the last 15 years, the characterization of admissible output operators has been a lively research topic. In this publication, the authors give an overview of these results. Depending on the semigroup the admissibility question relates to many other branches of mathematics. For instance, Carleson measures (diagonal semigroups), Hankel operators, reproducing kernel thesis, and Feffermann’s duality theorem (right-shift semigroup), \(H^{\infty}\)-calculus (analytic semigroups). The results discussed in this paper are for abstract systems. For partial differential equations there are other specific methods for checking admissibility. However, these fall outside the scope of this article.
For the entire collection see [Zbl 1050.47002].

MSC:

93C25 Control/observation systems in abstract spaces
93B28 Operator-theoretic methods
47D06 One-parameter semigroups and linear evolution equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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