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Geometrie differentielle classique --- Surfaces minima

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Mathematiques --- Geometrie differentielle --- Physique theorique --- Colloque --- Colloque --- Colloque

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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 optimization --- Geometry, Differential --- Differentiable dynamical systems --- Commande, théorie de la --- Optimisation mathématique --- Géométrie différentielle --- Dynamique différentiable --- Control theory --- Control theory. --- Differentiable dynamical systems. --- Geometry, Differential. --- Mathematical optimization. --- Commande, Théorie de la. --- Dynamique différentiable. --- Géométrie différentielle. --- Optimisation mathématique. --- System theory. --- Systems Theory, Control. --- Systems, Theory of --- Systems science --- Science --- Philosophy

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This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The following two chapters are devoted to fiber bundles and homotopy theory of fibrations. Vector bundles have been emphasized, although principal bundles are also discussed in detail. The last three chapters study bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres. Chapter 5, with its focus on the tangent bundle, also serves as a basic introduction to Riemannian geometry in the large. This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry. Gerard Walschap is Professor of Mathematics at the University of Oklahoma where he developed this book for a series of graduate courses he has taught over the past few years.

Geometry, Differential --- Fiber bundles (Mathematics) --- Differentiable manifolds --- Characteristic classes --- Géométrie différentielle --- Faisceaux fibrés (Mathématiques) --- Variétés différentiables --- Classes caractéristiques --- Differential geometry --- Geometry, Differential. --- Géométrie différentielle --- Faisceaux fibrés (Mathématiques) --- Variétés différentiables --- Classes caractéristiques