Control Theory from the Geometric Viewpoint / Edition 1

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Overview

This book presents some facts and methods of Mathematical Control Theory treated from the geometric viewpoint. It is devoted to finite-dimensional deterministic control systems governed by smooth ordinary differential equations. The problems of controllability, state and feedback equivalence, and optimal control are studied.

Some of the topics treated by the authors are covered in monographic or textbook literature for the first time while others are presented in a more general and flexible setting than elsewhere.

Although being fundamentally written for mathematicians, the authors make an attempt to reach both the practitioner and the theoretician by blending the theory with applications. They maintain a good balance between the mathematical integrity of the text and the conceptual simplicity that might be required by engineers. It can be used as a text for graduate courses and will become most valuable as a reference work for graduate students and researchers.

Editorial Reviews

From the Publisher

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.)

Product Details

• ISBN-13: 9783642059070
• Publisher: Springer Berlin Heidelberg
• Publication date: 12/6/2010
• Series: Encyclopaedia of Mathematical Sciences Series , #87
• Edition description: Softcover reprint of hardcover 1st ed. 2004
• Edition number: 1
• Pages: 412
• Product dimensions: 0.87 (w) x 9.21 (h) x 6.14 (d)

Meet the Author

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.

1 Vector Fields and Control Systems on Smooth Manifolds 1

1.1 Smooth Manifolds 1

1.2 Vector Fields on Smooth Manifolds 4

1.3 Smooth Differential Equations and Flows on Manifolds 8

1.4 Control Systems 12

2 Elements of Chronological Calculus 21

2.1 Points, Diffeomorphisms, and Vector Fields 21

2.2 Seminorms and $C^{\infty }(M)$-Topology 25

2.3 Families of Functionals and Operators 26

2.4 Chronological Exponential 28

2.5 Action of Diffeomorphisms on Vector Fields 37

2.6 Commutation of Flows 40

2.7 Variations Formula 41

2.8 Derivative of Flow with Respect to Parameter 43

3 Linear Systems 47

3.1 Cauchy's Formula for Linear Systems 47

3.2 Controllability of Linear Systems 49

4 State Linearizability of Nonlinear Systems 53

4.1 Local Linearizability 53

4.2 Global Linearizability 57

5 The Orbit Theorem and its Applications 63

5.1 Formulation of the Orbit Theorem 63

5.2 Immersed Submanifolds 64

5.3 Corollaries of the Orbit Theorem 66

5.4 Proof of the Orbit Theorem 67

5.5 Analytic Case 72

5.6 Frobenius Theorem 74

5.7 State Equivalence of Control Systems 76

6 Rotations of the Rigid Body 81

6.1 State Space 81

6.2 Euler Equations 84

6.3 Phase Portrait 88

6.4 Controlled Rigid Body: Orbits 90

7 Control of Configurations 97

7.1 Model 97

7.2 Two Free Points 100

7.3 Three Free Points 101

7.4 Broken Line 104

8 Attainable Sets 109

8.1 Attainable Sets of Full-Rank Systems 109

8.2 Compatible Vector Fields and Relaxations 113

8.3 Poisson Stability 116

8.4 Controlled Rigid Body: Attainable Sets 118

9 Feedback and State Equivalence of Control Systems 121

9.1 Feedback Equivalence 121

9.2 Linear Systems 123

9.3 State-Feedback Linearizability 131

10 Optimal Control Problem 137

10.1 Problem Statement 137

10.2 Reduction to Study of Attainable Sets 138

10.3 Compactness of Attainable Sets 140

10.4 Time-Optimal Problem 143

10.5 Relaxations 143

11 Elements of Exterior Calculus and Symplectic Geometry 145

11.1 Differential 1-Forms 145

11.2 Differential $k$-Forms 147

11.3 Exterior Differential 151

11.4 Lie Derivative of Differential Forms 153

11.5 Elements of Symplectic Geometry 157

12 Pontryagin Maximum Principle 167

12.1 Geometric Statement of PMP and Discussion 167

12.2 Proof of PMP 172

12.4 PMP for Optimal Control Problems 179

12.5 PMP with General Boundary Conditions 182

13 Examples of Optimal Control Problems 191

13.1 The Fastest Stop of a Train at a Station 191

13.2 Control of a Linear Oscillator 194

13.3 The Cheapest Stop of a Train 197

13.4 Control of a Linear Oscillator with Cost 199

13.5 Dubins Car 200

14 Hamiltonian Systems with Convex Hamiltonians 207

15 Linear Time-Optimal Problem 211

15.1 Problem Statement 211

15.2 Geometry of Polytopes 212

15.3 Bang-Bang Theorem 213

15.4 Uniqueness of Optimal Controls and Extremals 215

15.5 Switchings of Optimal Control 218

16.1 Problem Statement 223

16.2 Existence of Optimal Control 224

16.3 Extremals 227

16.4 Conjugate Points 229

17 Sufficient Optimality Conditions, Hamilton-Jacobi Equation,Dynamic Programming 235

17.1 Sufficient Optimality Conditions 235

17.2 Hamilton-Jacobi Equation 242

17.3 Dynamic Programming 244

18 Hamiltonian Systems for Geometric Optimal Control Problems 247

18.1 Hamiltonian Systems on Trivialized Cotangent Bundle 247

18.2 Lie Groups 255

18.3 Hamiltonian Systems on Lie Groups 260

19 Examples of Optimal Control Problems on Compact Lie Groups 265

19.1 Riemannian Problem 265

19.2 A Sub-Riemannian Problem 267

19.3 Control of Quantum Systems 271

19.4 A Time-Optimal Problem on $SO(3)$ 284

20 Second Order Optimality Conditions 293

20.1 Hessian 293

20.2 Local Openness of Mappings 297

20.3 Differentiation of the Endpoint Mapping 304

20.4 Necessary Optimality Conditions 309

20.5 Applications 318

20.6 Single-Input Case 321

21 Jacobi Equation 333

21.1 Regular Case: Derivation of Jacobi Equation 334

21.2 Singular Case: Derivation of Jacobi Equation 338

21.3 Necessary Optimality Conditions 342

21.4 Regular Case: Transformation of Jacobi Equation 343

21.5 Sufficient Optimality Conditions 346

22 Reduction 355

22.1 Reduction 355

22.2 Rigid Body Control 358

22.3 Angular Velocity Control 359

23 Curvature 363

23.1 Curvature of 2-Dimensional Systems 363

23.2 Curvature of 3-Dimensional Control-Affine Systems 373

24 Rolling Bodies 377

24.1 Geometric Model 377

24.2 Two-Dimensional Riemannian Geometry 379

24.4 Controllability 384

24.5 Length Minimization Problem 387

A Appendix 393

A.1 Homomorphisms and Operators in $C^{\infty }(M)$ 393

A.2 Remainder Term of the Chronological Exponential 395

References 399

List of Figures 407

Index 409

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