# Controllability and Observability for a Class of Infinite Dimensional Systems

Chapter

## Abstract

In §2.1 and §2.2 of Chapter 1 of Part I, we have discussed criteria for controllability, and observability for finite dimensional systems and have also shown that when the system is controllable we can transfer the state and its adjoint and studying the relation between the ranges and null spaces of these two operators and by showing that controllability is equivalent to invertibility of

*z*_{0}∈*H*at time*t*_{0}to the state*z*_{1}∈*H*at time*t*_{1}using minimum energy controls. These results were obtained by considering the controllability operator$$
\begin{gathered}
L_T :L^2 (0,T;U) \to H \hfill \\
{\text{ }}:u \mapsto \int_0^T {e^{(T - s)A} Bu(s)ds,} \hfill \\
\end{gathered}
$$

$$
\begin{gathered}
L_T^* :H \to L^2 (0,T;U) \hfill \\
{\text{ }}:y \mapsto B*e^{(T - \cdot )A*} y, \hfill \\
\end{gathered}
$$

*L*_{T}*L*_{ T }^{*}. As we have remarked (see Remark 2.1, Chapter 1 of Part I) in some sense the same ideas can be used to obtain characterizations of controllability when the spaces*U*and*X*are infinite dimensional Hilbert spaces, but at the expense of using much elaborate technical machinery. In this chapter we discuss questions of controllability for parabolic and second-order hyperbolic equations, the plate equation, and Maxwell’s equations.## Keywords

Hilbert Space Exact Controllability Approximate Controllability Continuous Injection Plate Equation
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