Bounded Control Operators: Control Inside the Domain

Abstract

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

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

where 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.