I THEORY.- to Differential Geometry.- to Dynamical Systems.- Controlled Systems, Controllability.- Jets of Infinite Order, Lie-B#x00E4;cklund#x2019;s Equivalence.- Differentially Flat Systems.- Flatness and Motion Planning.- Flatness and Tracking.- II APPLICATIONS.- DC Motor Starting Phase.- Displacements of a Linear Motor With Oscillating Masses.- Synchronization of a Pair of Independent Windshield Wipers.- Control of Magnetic Bearings.- Crane Control.- Automatic Flight Control Systems.

The present book is a translation and an expansion of lecture notes cor- sponding to a course of Mathematics of Control delivered during four years at the Ecole Nationale des Ponts et Chauss ees (Marne-la-Vall ee, France) to Master students. A reduced version of this course has also been given at the Master level at the University of Paris-Sud since eight years. It may the- fore serve as lecture notes for teaching at the Master or PhD level but also as a comprehensive introduction to researchers interested in atness and more generally in the mathematical theory of nite dimensional systems and c- trol. This book may be seen as an outcome of the applied research policy pi- oneered by the Ecole des Mines de Paris (now MINES-ParisTech), France, aiming not only at academic excellence, but also at collaborating with - dustries on speci c innovative projects to enhance technological innovation using the most advanced know-how. This in uence, though indirectly visible, mainly concerns the originality of some of the topics addressed here which are, in a sense, a theoretic synthesis of the author's applied contributions and viewpoints in the control eld, continuously elaborated and modi ed in contact with the industrial realities. Such a synthesis wouldn't have been made possible without the scienti c trust and nancial support of many c- panies during periods ranging from two to ten years.