Control Theory from the Geometric Viewpoint

1. Front Matter

   Pages I-XIV

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2. Vector Fields and Control Systems on Smooth Manifolds

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 1-19

3. Elements of Chronological Calculus

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 21-45

4. Linear Systems

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 47-51

5. State Linearizability of Nonlinear Systems

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 53-62

6. The Orbit Theorem and its Applications

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 63-80

7. Rotations of the Rigid Body

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 81-96

8. Control of Configurations

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 97-107

9. Attainable Sets

   • Andrei A. Agrachev, Yuri L. Sachkov

   Pages 109-119

10. Feedback and State Equivalence of Control Systems

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 121-136

11. Optimal Control Problem

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 137-144

12. Elements of Exterior Calculus and Symplectic Geometry

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 145-166

13. Pontryagin Maximum Principle

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 167-189

14. Examples of Optimal Control Problems

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 191-206

15. Hamiltonian Systems with Convex Hamiltonians

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 207-209

16. Linear Time-Optimal Problem

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 211-222

17. Linear-Quadratic Problem

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 223-234

18. Sufficient Optimality Conditions, Hamilton-Jacobi Equation, and Dynamic Programming

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 235-245

19. Hamiltonian Systems for Geometric Optimal Control Problems

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 247-264

20. Examples of Optimal Control Problems on Compact Lie Groups

    • Andrei A. Agrachev, Yuri L. Sachkov

    Pages 265-291

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.

• Lie bracket
• Nonlinear system
• Optimal control
• Pontryagin-Type
• control theory
• curvature
• diffeomorphism
• geometric differential methods
• manifold
• maximum principle
• ordinary differential equation
• symplectic geometry

• SISSA-ISAS, Trieste, Italy

  Andrei A. Agrachev

• Program Systems Institute, Pereslavl-Zalessky, Russia

  Yuri L. Sachkov

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.

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