Controllability and Observability for a Class of Infinite Dimensional Systems

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 z₀ ∈ H at time t₀ to the state z₁ ∈ H at time t₁ using minimum energy controls. These results were obtained by considering the controllability operator

$$ \begin{gathered} L_T :L^2 (0,T;U) \to H \hfill \\ {\text{ }}:u \mapsto \int_0^T {e^{(T - s)A} Bu(s)ds,} \hfill \\ \end{gathered} $$

and its adjoint

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

and studying the relation between the ranges and null spaces of these two operators and by showing that controllability is equivalent to invertibility of L_TL ^* _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.