This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters."

This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters."

Mathematical Control Theory is a branch of Mathematics having as one of its main aims the establishment of a sound mathematical foundation for the c- trol techniques employed in several di?erent ?elds of applications, including engineering, economy, biologyandsoforth. Thesystemsarisingfromthese- plied Sciences are modeled using di?erent types of mathematical formalism, primarily involving Ordinary Di?erential Equations, or Partial Di?erential Equations or Functional Di?erential Equations. These equations depend on oneormoreparameters thatcanbevaried, andthusconstitute thecontrol - pect of the problem. The parameters are to be chosen soas to obtain a desired behavior for the system. From the many di?erent problems arising in Control Theory, the C. I. M. E. school focused on some aspects of the control and op- mization ofnonlinear, notnecessarilysmooth, dynamical systems. Two points of view were presented: Geometric Control Theory and Nonlinear Control Theory. The C. I. M. E. session was arranged in ?ve six-hours courses delivered by Professors A. A. Agrachev (SISSA-ISAS, Trieste and Steklov Mathematical Institute, Moscow), A. S. Morse (Yale University, USA), E. D. Sontag (Rutgers University, NJ, USA), H. J. Sussmann (Rutgers University, NJ, USA) and V. I. Utkin (Ohio State University Columbus, OH, USA). We now brie?y describe the presentations. Agrachev's contribution began with the investigation of second order - formation in smooth optimal control problems as a means of explaining the variational and dynamical nature of powerful concepts and results such as Jacobi ?elds, Morse's index formula, Levi-Civita connection, Riemannian c- vature.