Andrei A. Agrachev, Yuri Sachkov

Control Theory from the Geometric Viewpoint

• Broschiertes Buch

Produktbeschreibung

Devoted to finite-dimensional deterministic control systems governed by smooth ordinary differential equations, this book treats Mathematical Control Theory from a geometric viewpoint. Covers controllability, state and feedback equivalence and optimal control.

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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.

Produktdetails

• Encyclopaedia of Mathematical Sciences 87
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-05907-0
• Softcover reprint of hardcover 1st ed. 2004
• Seitenzahl: 432
• Erscheinungstermin: 5. Dezember 2010
• Englisch
• Abmessung: 235mm x 155mm x 24mm
• Gewicht: 652g
• ISBN-13: 9783642059070
• ISBN-10: 3642059074
• Artikelnr.: 32055705

Autorenporträt

Andrei A. Agrachev Born in Moscow, Russia. Graduated: Moscow State Univ., Applied Math. Dept., 1974. Ph.D.: Moscow State Univ., 1977. Doctor of Sciences (habilitation): Steklov Inst. for Mathematics, Moscow, 1989. Invited speaker at the International Congress of Mathematicians ICM-94 in Zurich. Over 90 research papers on Control Theory, Optimization, Geometry (featured review of Amer. Math. Soc., 2002). Professional Activity: Inst. for Scientific Information, Russian Academy of Sciences, Moscow, 1977-1992; Moscow State Univ., 1989-1997; Steklov Inst. for Mathematics, Moscow, 1992-present; International School for Advanced Studies (SISSA-ISAS), Trieste, 2000-present. Current positions: Professor of SISSA-ISAS, Trieste, Italy and Leading Researcher of the Steklov Ins. for Math., Moscow, Russia   Yuri L. Sachkov Born in Dniepropetrovsk, Ukraine. Graduated: Moscow State Univ., Math. Dept., 1986. Ph.D.: Moscow State Univ., 1992. Over 20 research papers on Control Theory. Professional Activity: Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 1989-present; University of Pereslavl, 1993-present. Steklov Inst. for Mathematics, Moscow, 1998-1999; International School for Advanced Studies (SISSA-ISAS), Trieste, 1999-2001. Current positions: Senior researcher of Program Systems Institute, Pereslavl-Zalessky, Russia; Associate professor of University of Pereslavl, Russia.

Inhaltsangabe

1 Vector Fields and Control Systems on Smooth Manifolds.- 2 Elements of Chronological Calculus.- 3 Linear Systems.- 4 State Linearizability of Nonlinear Systems.- 5 The Orbit Theorem and its Applications.- 6 Rotations of the Rigid Body.- 7 Control of Configurations.- 8 Attainable Sets.- 9 Feedback and State Equivalence of Control Systems.- 10 Optimal Control Problem.- 11 Elements of Exterior Calculus and Symplectic Geometry.- 12 Pontryagin Maximum Principle.- 13 Examples of Optimal Control Problems.- 14 Hamiltonian Systems with Convex Hamiltonians.- 15 Linear Time-Optimal Problem.- 16 Linear-Quadratic Problem.- 17 Sufficient Optimality Conditions, Hamilton-Jacobi Equation, and Dynamic Programming.- 18 Hamiltonian Systems for Geometric Optimal Control Problems.- 19 Examples of Optimal Control Problems on Compact Lie Groups.- 20 Second Order Optimality Conditions.- 21 Jacobi Equation.- 22 Reduction.- 23 Curvature.- 24 Rolling Bodies.- A Appendix.- A.2 Remainder Term of the Chronological Exponential.- References.- List of Figures.

Rezensionen

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 Fülle 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.)