Observation and Control for Operator Semigroups pp 355-384 | Cite as

# Controllability

## Abstract

**Notation.** Throughout this chapter, *U*, *X* and *Y* are complex Hilbert spaces which are identified with their duals. \(
\mathbb{T}
\) is a strongly continuous semigroup on *X*, with generator *A* : *D*(*A*)→*X* and growth bound ω_{0} (\(
\mathbb{T}
\)). Remember that we use the notation *A* and \(
\mathbb{T}_t
\) also for the extension of the original generator to *X* and for the extension of the original semigroup to *X*_{−1}. Recall also that *X* _{1} ^{ d } is *D*(*A*^{*}) with the norm \(
||z||_1^d = ||(\bar \beta I - A*)z||
\) and *X* _{−1} ^{d} is the completion of *X* with respect to the norm \(
||z||_{ - 1}^d = ||(\bar \beta I - A*)^{ - 1} z||
\). Recall that *X*_{−1} is the dual of *X* _{1} ^{ d } with respect to the pivot space *X*.

## Keywords

Weak Solution Control Operator Input Function Input Space Continuous Semigroup## Preview

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