# Bounded Control Operators: Control Inside the Domain

Chapter

## Abstract

In this chapter we consider the dynamical system governed by the equation
where

$$
\left\{ \begin{gathered}
x'(t) = Ax(t) + Bu(t),t \geqslant 0, \hfill \\
x(0) = x_0 \in H, \hfill \\
\end{gathered} \right.
$$

*A*:*D*(*A*) ⊂*H*→*H*,*B*:*U*→*H*are linear operators defined on the Hilbert spaces*H*(*state space*) and*U*(*control space*), respectively.*x*is the*state*and*u*the*control*of the system. We shall also consider another Hilbert space*Y*, the space of*observations*. The inner product and norm in*H*,*U*, and*Y*will be denoted by (·, ·) and | · |. Whenever confusion is possible a subscript*H*,*U*or*Y*will be added.## Keywords

Classical Solution Riccati Equation Mild Solution Analytic Semigroup Optimal Pair
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 Boston 2007