ABSTRACT
Many of the functional analysis ideas and results presented in this book provide the essential foundations for development of modern control theory for so-called Distributed Parameter Systems (DPS). Control of DPS is the engineering terminology used primarily for control of partial differential equations and delay differential equations, examples of which we have discussed in previous chapters of this book. The research literature contains a plethora of control theory presentations in a Hilbert space framework which rely in a fundamental way on some of the topics we have discussed including strongly continuous and analytic semigroups, unbounded operator inputs, weak formulations via sesquilinear forms, adjoint operators, and finite element approximations via the Trotter-Kato theorems in Hilbert and Banach spaces. Our formulations in previous chapters of partial differential and delay differential equations as abstract infinite dimensional differential equations in either weak form or with semigroup representations are precisely the setting for much of this research. As a final application in this book we summarize some of these control theory results. However, unlike our earlier presentations, we give only a brief sample of results with no proofs, with the sole purpose being exposure of readers to another area of science which relies on applied functional analysis in a fundamental way.
