This book provides, in a unified framework, an updated andrather comprehensive treatment contered on the theory of ot-pimal control with quadratic cost functional for abstractlinear systems with application to boundary/point controlproblems for partial differential equations (distributed pa-rameter systems).The book culminates with the analysisof differential andalgebraic Riccati equations which arise in the pointwisefe-edback synthesis of the optimal pair. It incorporates thecritical topics of optimal irregularity of solutions to mi-xed problems for partial differential equations, exact con-trollability, and uniform feedback stabilization. It coversthe main results of the theory - which has reached a consi-derable degree of maturity over the last few years - as wellasthe authors' basic philosophy behind it.Moreover, it provides numerous illustrative examples ofboundary/point control problems for partial differentialequations, where the abstract theory applies. However, inline with the purpose of the manuscript, many technical pro-ofs are referred to in the literature. Thus, the manuscriptshould prove useful not only to mathematicians and theoreti-cal scientists with expertise in partial differential equa-tions, operator theory, numerical analysis, control theory,etc., but also to those who simple wish to orient themselveswith the scope and status of the theory presently available.Both continuous theory and numerical approximation theorythereof are included.