The first chapter reviews the types of finite dimensional systems theory results which will be generalized to infinite dimensions in chapters 3-7 and gives several examples of how such problems may arise in distributed and delay systems. To make these notes self-contained, chapter 2 presents the known results on semigroups, which we shall need in later chapters. Chapters 3-7 form the core of the book, namely the extension of the finite dimensional systems theory results outlined in Chapter 1 for time-invariant inear systems. All these results are proved in detail and are illustrated by several examples and would be appropriate for an introductory graduate course in linear systems theory. Chapters 8 & 9 are concerned with extensions of the results of Chapters 3-7 to more complicated systems, namely time dependent systems and distributed systems allowing for boundary control and point observations. The difficulties which arise in trying to extend the results to more general systems are technical mathematical ones rather than conceptual ones. As the technical details are already available in the literature, we have chosen to motivate the approach using simple examples and have omitted proofs which are heavy technical extensions of those in earlier chapters. So although a complete presentation of the results is available in Chapters 8 and 9, many proofs are given in outline only. Again, considerable attention is given to analyzing the implications for distributed and delay systems by means of examples.

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