Abstract
As in the previous chapter, we shall denote by H, U, and Y the Hilbert spaces of states, controls, and observations, respectively. We consider a dynamical system, whose state x(t) is subject to the following equation:
where u ∈ L2(0, T;U) and A: D(A) ⊂ H → H generates an analytic semigroup in H. However, in the current case, the linear operator B is not supposed to be bounded from U into H. This situation has been discussed at length in Chapters 1 and 2 (Part II). However some key constructions will be repeated here as needed. In that case many possibilities could be considered. However, in practice, it will be natural to consider situations where B maps U into the dual space (D(A*)′ of D(A*). This will be apparent in the following Examples 1.1 and 1.2. Equivalently, B is supposed to be of the form B = (λ0−A)D, where \( D \in \mathcal{L}{\text{(}}U;H{\text{)}} \) and λ0 is an element in ρ(A). Under these assumptions we write the state equation as
or in the mild form as
Remark that formula (1.1) is meaningful and x ∈ L2(0, T;H); see Chapters 1 to 3 of Part II. This formula will represent the state of our system.
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© 2007 Birkhäuser Boston
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(2007). Unbounded Control Operators: Parabolic Equations With Control on the Boundary. In: Representation and Control of Infinite Dimensional Systems. Systems & Control: Foundations & Applications. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4581-6_10
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DOI: https://doi.org/10.1007/978-0-8176-4581-6_10
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4461-1
Online ISBN: 978-0-8176-4581-6
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