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The control of physical systems which are described by partial differential equations is a very active research field in the intersection of many engineering disciplines and natural sciences, not least mathematics. This book aims to make several contributions to this field with a focus on fluid control.As core of this book a new method for the reduction of the complexity of linear instationary systems with distributed parameters is proposed. Motivated by the idea, that the input/output map of a system contains the crucial information for controlling the system, the i/o map of the infinite-dimensional system is directly discretized in space and time. This discretization can then be represented by means of an i/o matrix, which is especially suited for the fast approximate solution of optimization problems. Analytical error estimates are established, and numerical examples are shown for the heat equation and for the Stokes equations.Moreover, this book deals with the controllabilityof so-called Couette flows, with some aspects of adaptive finite element discretizations and with i/o software interfaces for CFD codes.