Systems with infinite-dimensional state space: The Hilbert space approach

The control of infinite-dimensional systems has received much attention from engineers and even mathematicians. Realizability although first considered in [4] has been ignored until recently. Ironically enough while state-space systems theory was developing in the early 1960's a mathematical study of scattering and of non-self-adjoint operators produced a parallel theory which was infinite dimensional from the beginning. When the close relationship between the two subjects became known time invariant infinite-dimensional systems theory advanced quickly and at a general level it now seems reasonably complete. This paper describes the connection between mathematical scattering and systems. It then gives a thorough treatment of infinite-dimensional time invariant continuous time systems. The last section lists recent scattering results which might be of engineering interest.