Controllability and Observability for a Class of Infinite Dimensional Systems

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


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 z0H at time t0 to the state z1H 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 LTL 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.


Hilbert Space Exact Controllability Approximate Controllability Continuous Injection Plate Equation 
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© Birkhäuser Boston 2007

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