Qualitative Theory of Control Systems

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Translations of Mathematical Monographs 141

S Title

Qualitative Theory of Control Systems

© Copyright 1994 by the American Mathematical Society
     ISBN 0-8218-4590-X
     QA402.3.D397 1994  003'.5-dc20
     LCCN 94-30834 CIP

Table of Contents

Introduction

CHAPTER 1  Implicit First-Order Differential Equations
     § 1. Simple examples
          1.1. One-dimensional mechanical system
          1.2. The net of characteristics of a mixed equation
          1.3. The net of limiting lines of a differential inequality
          1.4. Key notions in the theory of implicit first-order differential equations
          1.5. Germ and singularity
     §2. Normal forms
          2.1. Good involutions
          2.2. Normal singular points
          2.3. More on folded and cusped singularities
          2.4. Normal folded singularities
          2.5. Elliptic and hyperbolic cusps
          2.6. The real analytical case and the case of finite order of smoothness
     §3. On partial differential equations
          3.1. Elliptic and hyperbolic types
          3.2. The Cibrario normal form.
          3.3. Normal form in a neighborhood of folded singular points
     §4. The normal form of slow motions of a relaxation type equation on the break line
     §5. On singularities of attainability boundaries of typical differential inequalities on a surface
          5.1. Definitions
          5.2. Folded singularities on the boundary of the domain of definitions
          5.3. Folded singularities inside the steep domain.
     §6. Proof of Theorems 2.1 and 2.3
          6.1. Proof of Theorem 2.1
          6.2. Proof of Theorem 2.3.
          6.3. Proof of Lemma 2.1.
          6.4. Proof of Lemma 6.2.
          6.5. Proof of Lemma 6.3
     §7. Proof of Theorems 2.5 and 2.8
          7.1. Proof of Theorem 2.5
          7.2. Proof of Theorem 2.8

CHAPTER 2  Local Controllability of a System
     §1. Definitions and examples
          1.1. The class of control systems
          1.2 Steep domain, local transitivity zone, and rest zone.
          1.3. Ship drift
     §2. Singularities of a pair of vector fields on a surface
          2.1. Definitions
          2.2. The singularities of a pair of fields.
          2.3. Proofs of the proposition and the theorem
     §3. Polydynamical systems
          3.1. The simplest case (# U = 2).
          3.2. A tridynamical system.
          3.3. The points of the steep domai
          3.4. The boundary of the steep domain and the interiors of the zones.
          3.5  Singularities on the boundary of the steep domain
     §4. Classification of singularities
          4.1. Systems in general position
          4.2. Singular controls
          4.3. The critical set of a system.
          4.4. The boundary of the steep domain
          4.5. Singularities of the family of limiting lines on the boundary of the steep domain.
          4.6. Singularities in the steep domain
          4.7. Stability of singularities.
          4.8. Generalization
          4.9. Remark
     §5. The typicality of systems determined by typical sets of vector fields
          5.1. The proof of Theorem 4.1; #U = 2.
          5.2. The proof of Theorem 4.1; # U > 2.
     §6. The singular surface of a control system
          6.1. Stratification of jets of control systems
          6.2. Proof of Theorem 4.2.
          6.3. Proof of Lemma 6.5.
          6.4. Proof of Lemm 6.6
     §7. The critical set of a control system
          7.1. Stratification of multijets
          7.2. Transversality
          7.3. The beginning of the proof of Theorem 4.4
          7.4. The classes of points
          7.5. Proof of Lemma 7.9
          7.6. Proof of Lemma 7.10
     §8. Singularities of the defining set and their stability
          8.1. Proof of Theorem 4.12.
          8.2. Proof of Theorem 4.14.
          8.3. The correspondence between classes and singularities
          8.4. The stability of the sets and their singularities
          8.5. Proof of Theorem 4.16
     §9. Singularities in the family of limiting lines in the steep domain
          9.1. Completion of the proof of Theorem 4.6.
          9.2. Completion of the proof of Theorem 4.
          9.3. Proof of Theorem 4.9
          9.4. Proof of Theorem 4.10
          9.5. Proof of Theorem 4.11
          9.6  Proof of Lemma 9.1
          9.7. Proof of Lemma 9.2
          9.8 Proof of Theorem 4.15
     §10. Transversality of multiple 3-jet extensions
          10.1. Proof of Lemma 7.4
          10.2. Proofs of Lemmas 10.1 and 10.2

CHAPTER 3  Structural Stability of Control Systems
     §1. Definitions and theorems
          1.1. The class of systems
          1.2. Stability of orbits
          1.3. Conditions for structural stability
          1.4. Nonlocal controllability
          1.5. Orbit boundary
          1.6. Remark
     §2. Examples
          2.1. Swimmer drift
          2.2. Ship drift
          2.3. A structurally stable system.
     §3. A branch of the field of limiting directions
          3.1. Differentiability of the family of limiting lines
          3.2. Proof of Theorem 1.3
     §4. The set of singular limiting lines
          4.1. The necessity of conditions (A)-(D).
          4.2. The sufficiency of conditions (A)-(D).
          4.3. Proof of Theorem 1.5.
     §5. The structure of orbit boundaries
          5.1. Proof of Lemma 1.7.
          5.2. Proof of Theorem 1.8
          5.3. Proof of Theorem 1.9
     §6. Stability
          6.1. The first stage
          6.2. The second stage
          6.3. The third stage
          6.4. The fourth stage
          6.5. Proof of Theorem 1.1
     §7. Singularities of the boundary of the zone of nonlocal transitivity
          7.1. Angular points
          7.2. Singularities of the boundary of the positive orbit in the steep domain
          7.3. Singularities of the boundary of the positive orbit on the boundary of the steep domain
          7.4. The completion of the proof of Theorem 1.6

CHAPTER 4  Attainability Boundary of a Multidimensional System
     §1. Definitions and theorems
          1.1. The class of systems
          1.2. The Lipschitz chartacter of the attainability boundary.
          1.3. The Holder character of the attainability boundary
          1.4. Regular systems.
     §2. Typicality of regular systems
          2.1. The openness of the set of regular systems
          2.2. The density of the set of regular systems
     §3. The Lipschitz character of the attainability boundary
          3.1. Proof of Theorem 1.1.
          3.2. Proof of Lemma 3.1.
     §4. The quasi-Holder character of the attainability set
          4.1. Proof of Theorem 1.3
          4.2. Proof of Lemma 4.1.
          4.3. Proof of Lemma 4.4

References

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