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Control Theory from the Geometric Viewpoint : Encyclopaedia of Mathematical Sciences - Andrei A. Agrachev

Control Theory from the Geometric Viewpoint

Encyclopaedia of Mathematical Sciences

Hardcover Published: 15th April 2004
ISBN: 9783540210191
Number Of Pages: 412

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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.

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.)

Vector fields and control systems on smooth manifoldsp. 1
Elements of chronological calculusp. 21
Linear systemsp. 47
State linearizability of nonlinear systemsp. 53
The orbit theorem and its applicationsp. 63
Rotations of the rigid bodyp. 81
Control of configurationsp. 97
Attainable setsp. 109
Feedback and state equivalence of control systemsp. 121
Optimal control problemp. 137
Elements of exterior calculus and symplectic geometryp. 145
Pontryagin maximum principlep. 167
Examples of optimal control problemsp. 191
Hamiltonian systems with convex Hamiltoniansp. 207
Linear time-optimal problemp. 211
Linear-quadratic problemp. 223
Sufficient optimality conditions, Hamilton-Jocobi equation, and dynamic programmingp. 235
Hamiltonian systems for geometric optimal control problemsp. 247
Examples of optimal control problems on compact Lie groupsp. 265
Second order optimality conditionsp. 293
Jacobi equationp. 333
Reductionp. 355
Curvaturep. 363
Rolling bodiesp. 377
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540210191
ISBN-10: 3540210199
Series: Encyclopaedia of Mathematical Sciences
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 412
Published: 15th April 2004
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 0.78