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Control and Observation Operators

Part of the Birkhäuser Advanced Texts / Basler Lehrbücher book series (BAT)

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 ω0(\( \mathbb{T} \)). Recall from Section 2.10 that X1 is D(A) with the norm ‖z1 = ‖(βI - A)z‖, where β∋ρ(A) is fixed, while X−1 is the completion of X with respect to the norm ‖z−1 = ‖ (βI - A)−1z‖. 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2009

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