Bounded Control Operators: Control Inside the Domain

Part of the Systems & Control: Foundations & Applications book series (SCFA)


As in Chapter 1 (Part IV) we consider a dynamical system governed by the following state equation:
$$ \left\{ \begin{gathered} x'(t) = Ax(t) + Bu(t),{\text{ }}t \geqslant 0, \hfill \\ x(0) = x_0 \in H, \hfill \\ \end{gathered} \right. $$
and we use the notation introduced in §1 of that chapter. We assume that
$$ (\mathcal{H})\infty \left\{ \begin{gathered} (i){\text{ }}A{\text{ generates a }}C_0 {\text{ semigroup }}e^{tA} {\text{ in }}H{\text{,}} \hfill \\ (ii){\text{ }}B \in \mathcal{L}(U;H), \hfill \\ (iii){\text{ }}C \in \mathcal{L}(U;H). \hfill \\ \end{gathered} \right. $$
Clearly, if the assumptions \( (\mathcal{H})_\infty \) hold, then the assumptions \( (\mathcal{H}) \) of §1 in Chapter 1 (Part IV) are verified with P0 = 0.


Periodic Solution Riccati Equation Mild Solution Minimal Solution Admissible Control 
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© Birkhäuser Boston 2007

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