Control Theory from the Geometric Viewpoint

By: Andrei A. Agrachev, Yuri Sachkov

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Hardcover | 15 April 2004

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432 Pages

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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.

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Aus den Rezensionen: "Der Band ist aus Graduiertenkursen an der International School for Advanced Studies in Triest entstanden ... Mathematisch werden gute Kenntnisse der Analysis, der linearen Algebra und der Funktionalanalysis vorausgesetzt. ... Bekannte und neue Beispiele ... illustrieren hier die Fulle an Aussagen in sehr anschaulicher Weise. Insgesamt ist so ein Band entstanden, der Mathematikern und mathematisch interessierten Anwendern wertvolle Anregungen bei der Auseinandersetzung mit gesteuerten bzw. geregelten nichtlinearen Systemen und deren Optimierung bietet." (l. Troch, in: IMN - Internationale Mathematische Nachrichten, 2006, Issue 202, S. 44 f.)

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You Can Find This Book In

Non-Fiction Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Mathematics Applied Mathematics

Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Calculus & Mathematical Analysis Calculus of Variations

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Paperback

Published: 5th December 2010

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Vector fields and control systems on smooth manifolds	p. 1
Elements of chronological calculus	p. 21
Linear systems	p. 47
State linearizability of nonlinear systems	p. 53
The orbit theorem and its applications	p. 63
Rotations of the rigid body	p. 81
Control of configurations	p. 97
Attainable sets	p. 109
Feedback and state equivalence of control systems	p. 121
Optimal control problem	p. 137
Elements of exterior calculus and symplectic geometry	p. 145
Pontryagin maximum principle	p. 167
Examples of optimal control problems	p. 191
Hamiltonian systems with convex Hamiltonians	p. 207
Linear time-optimal problem	p. 211
Linear-quadratic problem	p. 223
Sufficient optimality conditions, Hamilton-Jocobi equation, and dynamic programming	p. 235
Hamiltonian systems for geometric optimal control problems	p. 247
Examples of optimal control problems on compact Lie groups	p. 265
Second order optimality conditions	p. 293
Jacobi equation	p. 333
Reduction	p. 355
Curvature	p. 363
Rolling bodies	p. 377
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Control Theory from the Geometric Viewpoint

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