Observation and Control for Operator Semigroups pp 111-138 | Cite as

# Control and Observation Operators

## 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}(\(
\mathbb{T}
\)). Recall from Section 2.10 that *X*_{1} is *D*(*A*) with the norm ‖*z*‖_{1} = ‖(β*I* - *A*)*z*‖, where β∋ρ(*A*) is fixed, while *X*_{−1} is the completion of *X* with respect to the norm ‖*z*‖_{−1} = ‖ (β*I* - *A*)^{−1}*z*‖. 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* _{ d } ^{1} is *D*(*A**) with the norm \(
||z||_1^d = ||(\bar \beta I - A*)z||
\)
and *X* _{ d } ^{−1} 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* _{ d } ^{1} with respect to the pivot space *X*.

## Keywords

Control Operator Representation Theorem Mild Solution Continuous Semigroup Observation Operator## Preview

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