# 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