Singular Trajectories and Their Role in Control Theory

By: Bernard Bonnard, Monique Chyba

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Paperback | 12 May 2003

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The role of singular trajectories in control theory is analysed in this volume that contains about 60 exercieses and problems. A section is devoted to the applications of singular trajectories to the optimisation of batch reactors. The theoretical paart based on the Martinet case concerns the singulatrity analysis of singular trajectories in sub-Riemannian geometry. An algorithm is gibven to evaluate conjugate points and a final chapter discusses open problems. The volume will interest mathematicians and engineers.

Industry Reviews

From the reviews:

"This impressive book discusses the general theory of singular trajectories, and presents in detail a large number of important results ... . The book is, however much broader and comprehensive than the title may suggest, and the authors have gone out of their way to make it accessible ... . would be useful for a two-semester graduate course ... . those who appreciate really deep mathematics, written with passion ... will find this book rewarding and will read it ... with enormous pleasure." (Hector J. Sussmann, IEEE Transactions on Automatic Control, Vol. 50 (2), February, 2005)

"This book presents a detailed and comprehensive account of singular trajectories and their ramifications in control theory and related areas of mathematics. ... it may be suitable as a textbook in an advanced course on nonlinear control theory. ... Along the way the authors review a considerable amount of standard material from nonlinear control theory, thereby making the book accessible and useful to neophytes as well as experts." (Kevin A. Grasse, Mathematical Reviews, 2004 f)

"The book brings together a wealth of results on the topic of singular trajectories and its application, which for a large part was scattered up to now throughout the literature. Hence the book serves as a very valuable source of information for every researcher interested in these and related areas. Generally speaking, the book is well-accessible. ... Summarizing, the book is a very welcome and useful addition to the literature, especially as a research monograph and source of information and inspiration." (A. J. Van der Schaft, Nieuw Archief voor Wiskunde, Vol. 8 (2), 2007)

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Linear Systems and the Time Optimal Control Problem	p. 1
Linear Systems	p. 1
Accessibility Set and Controllability	p. 1
Controllability and Feedback Classification in the Autonomous Case	p. 2
Controllability	p. 2
Linear Classification	p. 3
Feedback Classification	p. 5
Stabilization	p. 6
Controllability in the Nonautonomous Case	p. 7
Application	p. 8
Time Optimal Control for Linear Systems	p. 9
Notations	p. 9
Feedback Equivalence to a Linear System	p. 15
Problem Statement	p. 15
Time Minimal Synthesis	p. 17
Notion of Synthesis	p. 17
The General Algorithm in the Plane	p. 21
Exercises	p. 24
Optimal Control for Nonlinear Systems	p. 27
A Short Visit into the Classical Calculus of Variations	p. 27
Statement of the Problem in the Holonomic Case	p. 27
Hamiltonian Equations	p. 29
Hamilton-Jacobi-Bellman Equation	p. 30
Euler-Lagrange Equations and Characteristics of the HJB Equation	p. 31
Second Order Conditions	p. 31
The Accessory Problem and the Jacobi Equation	p. 32
Conjugate Point and Local Morse Theory	p. 33
Scalar Riccati Equation	p. 34
Local C⁰ Minimizer - Extremal Field - Hilbert Invariant Integral	p. 35
Optimal Control and the Calculus of Variations	p. 36
Problem Statement	p. 36
The Augmented System	p. 37
Related Problems	p. 38
Optimal Control and the Classical Calculus of Variations	p. 38
Singular Trajectories and the Weak Maximum Principle	p. 39
First and Second Variations of <$>E^{x_0,T}<$>	p. 39
Geometric Interpretation of the Adjoint Vector	p. 42
The Weak Maximum Principle	p. 42
Abnormality	p. 43
The Weak Maximum Principle and Euler-Lagrange Equation	p. 43
Comparison with the Calculus of Variations	p. 44
LQ-Control and the Weak Maximum Principle	p. 44
Pontryagin's Maximum Principle (PMP)	p. 45
Filippov Existence Theorem	p. 52
Comments about the Existence Theorem	p. 52
Dynamic Programming and the Maximum Principle	p. 53
Exercises	p. 56
Geometric Optimal Control	p. 65
Introduction to Symplectic Geometry	p. 65
Geometric Classification of Extremals in one Dimensional Problems of Calculus of Variations	p. 69
Problem Statement and Preliminaries	p. 69
Singularity Analysis	p. 71
Generic Classification near ¿₁	p. 72
Classification and Existence of Optimal Solutions	p. 75
Time Minimum Control Problem	p. 76
Determination of the Singular Extremals	p. 77
Hamiltonian Formalism and Singular Extremals	p. 78
Geometric Classification of Extremals near ¿_i	p. 79
Normal Switching Points	p. 79
The Fold Case	p. 80
Comparaison Between the Calculus of Variations and the Time Minimum Problem for Affine Systems. The Role of Singular Extremals	p. 82
The Fuller Phenomenon	p. 82
Fuller Example	p. 82
Geometry of the Time Optimal Control in the Plane	p. 84
Preliminaries	p. 84
Generic points	p. 85
Singular arc	p. 86
Elliptic Case - The Concept of Conjugate Point	p. 88
Exercises	p. 94
Singular Trajectories and Feedback Classification	p. 97
Classification of Affine Systems	p. 99
Computations of Singular Controls	p. 99
Singular Trajectories and Feedback Classification	p. 102
Time Optimality and Feedback Classification	p. 105
Singular Trajectories and the Problem of Classification of Distributions	p. 108
Preliminaries	p. 108
Local Classification in Dimension 3, with rank D = 2	p. 109
Feedback Classification and Analytic Geometry	p. 111
Preliminaries	p. 111
Critical Hamiltonians and Symbols	p. 112
Exercises	p. 113
Controllability - Higher Order Maximum Principle - Legendre-Clebsch and Goh Necessary Optimality Conditions	p. 117
Some Notations and Formulas from Differential Geometry	p. 117
Controllability with Piecewise Constant Controls	p. 119
Integrating Distributions	p. 120
Preliminaries	p. 120
Frobenius Theorem	p. 120
Nagano-Sussmann Theorem	p. 121
C^∞-Counter Example	p. 122
Nonlinear Controllability and Chow Theorem	p. 122
Poisson Stability and Controllability	p. 124
Application	p. 125
Controllability and Enlargement Technique	p. 125
Problem Statement	p. 125
Evaluation of the Accessibility Set	p. 127
Legendre-Clebsch Condition	p. 129
The Multi-Inputs Case: Goh Condition	p. 131
The Concept of Rigidity - Strong Legendre Clebsch and Goh Conditions as Necessary Conditions for Rigidity	p. 132
Intrinsic Second Variation - Morse Index	p. 133
Exercises	p. 134
The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C⁰-Optimality	p. 143
Single-Input Case	p. 143
Preliminaries	p. 143
Feedback Semi-Normal Forms in the Hyperbolic and Elliptic Cases	p. 146
LQ-Model	p. 147
Approximation of the End-Point Mapping Using the LQ-Model in the Elliptic Case	p. 148
Conclusion	p. 153
The Exceptional Case	p. 153
Conclusion	p. 155
Comparison of the Elliptic-Hyperbolic Case and the Exceptional Case	p. 156
Applications	p. 157
Time Optimal Synthesis for Planar System	p. 157
Example in Dimension 3 and Connection with the Hamilton-Jacobi-Bellman Equation	p. 157
The Case in <$>{\op R}^3<$>. Intrinsic Computations of Conjugate Points- Connection with the Time Minimal Synthesis Problem. The Concept of Curvature	p. 160
Euler-Lagrange Equation	p. 160
Geometric Interpretation	p. 161
Intrinsic Computation	p. 162
The Concept of Curvature	p. 162
Conjugate Points and Time Minimal Synthesis	p. 163
Connection with the Liu-Sussmann Example	p. 164
Preliminaries	p. 164
Conclusion	p. 166
Statement and Proof of Liu-Sussmann Result	p. 166
Conclusion	p. 167
Connection with the Sub-Riemannian Geometry	p. 168
Time Minimal Control of Chemical Batch Reactors and Singular Trajectories	p. 171
Introduction	p. 171
Mathematical Model of Chemical Batch Reactors and Description of the Control Problem	p. 172
Chemical Kinetics	p. 172
Control Device	p. 174
The Optimal Control Problem	p. 175
Projected Problem	p. 175
Singular Extremals - Curvature - Conjugate Points	p. 176
Projected System	p. 177
Time Minimal Synthesis for Planar Systems in the Neighbor-hood of a Terminal Manifold of Codimension One	p. 180
Problem Statement	p. 180
Assumption C1	p. 181
The Generic C1 Case	p. 181
The Generic C1 Flat Case	p. 181
The Case C1 of Codimension One	p. 182
Generic Hyperbolic Cases	p. 185
Generic Exceptional Case	p. 188
Generic Flat Exceptional Case	p. 189
Global Time Minimal Synthesis	p. 191
Preliminaries	p. 191
Singular Arc	p. 191
Regular Arcs	p. 192
Optimal Synthesis in the Neighborhood of N	p. 193
Switching Rules	p. 193
Optimal Synthesis	p. 194
State Constraints due to the Temperature	p. 195
Preliminaries	p. 195
Optimal Synthesis	p. 198
The Problem in Dimension 3	p. 199
Preliminaries	p. 199
Stratification of N by the Optimal Feedback Synthesis	p. 199
Orientation Principle	p. 200
Switching Rules	p. 201
Local Classification near the Target	p. 203
Focal Points	p. 206
Generic Properties of Singular Trajectories	p. 209
Introduction and Notations	p. 209
Determination of the Singular Extremals with Minimal Order	p. 210
Statement of the First Generic Property	p. 211
Geometric Interpretation and the General Concept of Order	p. 211
Proof of Theorem 25	p. 213
Partially Algebraic and Semi-Algebraic Fiber Bundles	p. 217
Coordinate Systems on P(d, N)	p. 218
Evaluation of Codimension of the <$>{\cal F}(N)<$>	p. 218
End of the Proof of Theorem 25	p. 222
Genericity of Codimension One Singularity	p. 222
Proof of Theorem 26	p. 222
The "Bad" Set for Theorem 26	p. 223
Evaluation of the Codimension of B_c(N,q)	p. 224
Singularities of the Singular Flow of Minimal Order	p. 226
Preliminaries (see Sect. 4.1.3)	p. 226
Local Classification near C	p. 227
Local Classification near S\C	p. 228
The Quadratic Case	p. 229
Exercises	p. 231
Singular Trajectories in Sub-Riemannian Geometry	p. 233
Introduction	p. 233
Generalities About SR-Geometry	p. 234
Definition	p. 234
Optimal Control Theory Formulation	p. 234
Computations of the Extremals and Exponential mapping	p. 235
Research Program in SR-Geometry	p. 239
Classification	p. 239
Singularity Theory of the Exponential Mapping and of the Distance Function	p. 239
Privileged Coordinates and Graded Normal Forms	p. 240
Regular and Singular Points	p. 240
Adapted and Privileged Coordinates	p. 241
Nilpotent Approximation	p. 242
Graded Approximation	p. 242
The Contact Case of Order -1 or the Heisenberg-Brockett Example	p. 242
The Contact Case in <$>{\op R}^3<$>	p. 242
Symmetry Group in the Heisenberg Case	p. 243
Heisenberg SR-Geometry and the Dido Problem	p. 243
Geodesies	p. 244
Conjugate Points	p. 245
Sphere and Wave Front	p. 245
Conclusion About the SR-Heisenberg Geometry	p. 246
The Generic Contact Case	p. 247
Normal Forms in the Contact Case	p. 247
Generic Conjugate Locus	p. 248
The Martinet Case	p. 249
Preliminaries	p. 249
Normal Form	p. 250
Orthonormal Frame	p. 253
Graded Normal Form	p. 253
Geodesics	p. 254
Riemannian Metric on the Plane (x, y)	p. 255
Asymptotic Foliation Associated to the Normal Form at Order 0	p. 256
Properties of the Asymptotic Foliations	p. 257
Integrable Case of Order 0 and Elliptic Integrals	p. 258
The Martinet Flat Case	p. 262
Estimates of the Sphere and of the Wave Front near the Abnormal Direction in the Flat Case and exp -ln Category	p. 266
Intersection of S(0, r) with the Cut Locus	p. 266
Conclusion Deduced from the Martinet SR-Flat Geometry Concerning the role of Abnormal Geodesies in SR-Geometry	p. 268
Behaviors of the Normal Geodesies near the Abnormal Direction - Geodesic C⁰-Rigidity	p. 268
Nonproperness of the First Return Mapping R₁ and Geometric Consequence	p. 269
Cut Locus in Martinet SR-Geometry	p. 271
Exercises	p. 273
Micro-Local Resolution of the Singularity near a Singular Trajectory - Lagrangian Manifolds and Symplectic Stratifications	p. 281
Introduction	p. 281
Lagrangian Manifolds	p. 281
Application to Classical Calculus of Variations	p. 282
Singularity Theory of the Generating Function - Generating Family	p. 283
Application to Optimal Control Theory and to SR-Geometry	p. 285
The SR-Normal Case	p. 285
Jacobi Fields	p. 287
Reduction Algorithm in the SR-Case	p. 288
The Singular Case	p. 289
A Geometric Remark	p. 289
Heisenberg Case	p. 289
Martinet Flat Case	p. 289
Resolution of the Singularity in the Hyperbolic Case	p. 292
Normal Form	p. 292
Intrinsic Second-Order Derivative	p. 293
Goh Transformation	p. 294
Lagrangian Manifolds in the Hyperbolic Case	p. 294
Examples	p. 296
Resolution of the Singularity in the Exceptional Case	p. 302
Normal Form - Intrinsic Derivative	p. 302
The Operators	p. 303
Isotropic Manifolds in the Exceptional Case	p. 303
Micro-Local Analysis of SR-Martinet Ball of Small Radius -Lagrangian Stratification of the Martinet Sector	p. 306
Preliminaries	p. 306
The Smooth Abnormal Sector	p. 307
The Smoothness of the Sphere in the Abnormal Direction	p. 307
The Martinet Sector	p. 309
Symplectic Stratifications	p. 312
Transcendence of the Sector	p. 312
The Micro-Local Analysis of the Whole Martinet Sphere	p. 313
Exercises	p. 314
Numerical Computations	p. 321
Introduction	p. 321
Numerical Algorithm	p. 321
Applications	p. 324
Contact Case in Dimension 3	p. 324
The Martinet Flat Case	p. 324
The Tangential Case	p. 328
The Elliptic Case	p. 329
The Hyperbolic Case	p. 330
Conclusion and Perspectives	p. 335
The Category of the Distance Function in SR-Geometry	p. 335
SR-Distances and Control of the Oscillations of Solutions of Ordinary Differential Equations	p. 335
Optimal Control Problems with Bounded State Variables	p. 338
Maximum Principle with State Constraints	p. 338
Singular Trajectories and Regularity of Stabilizing Feedback	p. 339
Computations of Singular Trajectories	p. 340
Computations of Conjugate Points	p. 340
Exercises	p. 341
References	p. 343
Index	p. 349
Table of Contents provided by Publisher. All Rights Reserved.

Singular Trajectories and Their Role in Control Theory

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