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Lectures on the Calculus of Variations and Optimal Control Theory
 
L. C. Young
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7900-8
Product Code:  CHEL/304.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7912-1
Product Code:  CHEL/304.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7900-8
eBook: ISBN:  978-1-4704-7912-1
Product Code:  CHEL/304.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
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Lectures on the Calculus of Variations and Optimal Control Theory
L. C. Young
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7900-8
Product Code:  CHEL/304.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7912-1
Product Code:  CHEL/304.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7900-8
eBook ISBN:  978-1-4704-7912-1
Product Code:  CHEL/304.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 304; 1969; 337 pp
    MSC: Primary 49

    This book is divided into two parts. The first addresses the simpler variational problems in parametric and nonparametric form. The second covers extensions to optimal control theory.

    The author opens with the study of three classical problems whose solutions led to the theory of calculus of variations. They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. He gives a detailed discussion of the Hamilton-Jacobi theory, both in the parametric and nonparametric forms. This leads to the development of sufficiency theories describing properties of minimizing extremal arcs.

    Next, the author addresses existence theorems. He first develops Hilbert's basic existence theorem for parametric problems and studies some of its consequences. Finally, he develops the theory of generalized curves and “automatic” existence theorems.

    In the second part of the book, the author discusses optimal control problems. He notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints. In the control formulation, these constraints are expressed in a more convenient form in terms of control functions. After pointing out the new phenomenon that may arise, namely, the lack of controllability, the author develops the maximum principle and illustrates this principle by standard examples that show the switching phenomena that may occur. He extends the theory of geodesic coverings to optimal control problems. Finally, he extends the problem to generalized optimal control problems and obtains the corresponding existence theorems.
    Readership

    Graduate students and research mathematicians interested in calculus of variations and optimal control.
  • Table of Contents
     
     
    • Front Cover
    • FOREWORD
    • PREFACE
    • CONTENTS
    • Volume I LECTURES ON THE CALCULUS OF VARIATIONS
    • Preamble GENERALITIES AND TYPICAL PROBLEMS
    • §1. INTRODUCTION
    • §2. THE PLACE OF THE CALCULUS OF VARIATIONS IN RELATION TO THE REST OF MATHEMATICS AND TO SPACE SCIENCE
    • §3. STATEMENT OF THE SIMPLEST PROBLEM AND SOME COGNATE MATTERS
    • §4. EXTREMALS IN SOME CLASSICAL PROBLEMS
    • §5. SOLUTIONS Of THE PRECEDING PROBLEMS (a), (b), (c)
    • §6. THE EULER-LAGRANGE LEMMA AND SCHWARTZ DISTRIBUTIONS
    • §7. ALTERNATIVE FORMS OF THE LEMMA
    • §8. PROOF OF THE MAIN FORM OF THE LEMMA
    • §9. FIRST VARIATION, EULER EQUATION, TRANSVERSALITY
    • §10. PERRON'S PARADOX
    • Chapter I The Method of Geodesic Coverings
    • §11. INTRODUCTION
    • §12. THE VARIATIONAL ALGORITHM OF HUYGENS
    • §13. A LINK WITH ELEMENTARY CONVEXITY
    • §14. REAPPEARANCE OF THE EULER EQUATION
    • §15. THE THEOREM OF MALUS
    • §16. SUFFICIENT CONDITIONS FOR INDEPENDENCE OF THE HILBERT INTEGRAL
    • §17. INVARIANCE PROPERTIES AND AN ENVELOPE THEOREM
    • §18. GENERAL COMMENTS AND THE APPLICATIONS TO PLANE PROBLEMS
    • §19. BACKGROUND ON FlX-POINTS AND ON EXISTENCE THEOREMS FOR DlFFERENTIAL EQUATIONS AND IMPLICIT FUNCTIONS
    • Chapter II Duality and Local Embedding
    • §20. INTRODUCTION
    • §21. THE LEGENDRE TRANSFORMATION
    • §22. THE HAMILTONIAN AND ITS PROPERTIES
    • §23. CAUCHY CHARACTERISTICS
    • §24. DUALITY AND THE STANDARD HAMILTONIAN IN THE PARAMETRIC CASE
    • §25. OTHER ADMISSIBLE PARAMETRIC HAMILTONIANS
    • §26. LOCAL PASSAGE FROM PARAMETRIC TO NONPARAMETRIC CASE
    • §27. THE EMBEDDING OF SMALL EXTREMALS IN SMALL TUBES
    • §28. LOCAL EXISTENCE THEORY FOR NONPARAMETRIC VARIATIONAL PROBLEMS AND FOR ORDINARY SECOND ORDER DlFFERENTIAL EQUATIONS
    • §29. LOCAL PARAMETRIC EXISTENCE THEORY FOR THE ELLIPTIC CASE
    • Chapter Ill Embedding in the Large
    • §30. INTRODUCTION
    • §31. FIRST AND SECOND VARIATIONS AND TRANSVERSALITY
    • §32. THE SECOND VARIATION FALLACY
    • §33. THE SECONDARY HAMILTONIAN
    • §34. GEOMETRICAL INTERPRETATION OF EXACTNESS
    • §35. DISTINGUISHED FAMILIES
    • §36. CANONICAL EMBEDDINGS AND FOCAL POINTS
    • §37. THIE JACOBI THEORY OF CONJUGATE POINTS
    • §38. THE INDEX OF STABILITY OF AN EXTREMAL
    • §39. THE SECOND STAGE OF THE MORSE THEORY
    • Chapter IV Hamiltonians in the large, Convexity, Inequailities and Functional Analysis
    • §40. INTRODUCTION
    • §41. CENTER OF GRAVITY AND DISPERSAL ZONE
    • §42. CONVEXITY AND THE HAHN-BANACH THEOREM
    • §43. THE CONCEPTUAL HERITAGE OF GEORG CANTOR
    • §44. DUALITY OF CONVEX FIGURES
    • §45. DUALITY OF CONVEX FUNCTIONS
    • §46. HAMILTONIANS IN THE LARGE AND REFORMULATED VARIATIONAL THEORY
    • §47. REMARKS ON CLASSICAL INEQUALITIES
    • §48. THE DUAL UNIT BALL OF A FUNCTIONAL SPACE
    • §49. THE RIESZ REPRESENTATION
    • Chapter V Existence Theory and its Consequences
    • §50. INTRODUCTION
    • §51. THE HILBERT CONSTRUCTION AND SOME OF ITS CONSEQUENCES IN THE STANDARD PARAMETRIC CASE
    • §52. THE PARAMETRIC THEORY OF CONJUGATE POINTS AND THE PARAMETRIC JACOBI CONDITION
    • §53. THE TONELLI-CARATHEODORY UNICITY THEOREM
    • §54. ABSOLUTE AND HOMOTOPIC MINIMA ON B •• i-COMPACT DOMAINS AND MANIFOLDS
    • §55. TOWARD AN AUTOMATIC EXISTENCE THEORY
    • §56. FlRST STAGE Of AN ABSTRACT APPROACH: SEMICONTINUITY IN A B●●i-COMPACT SET
    • §§57, 58, 59
    • Chapter VI Generalized Curves and Flows
    • §60. INTRODUCTION
    • §61. INTUITIVE BACKGROUND
    • §62. A QUESTION OF SEMANTICS
    • §63. PARAMETRIC CURVES IN THE CALCULUS OF VARIATIONS
    • §64. ADMISSIBLE CURVES AS ELEMENTS OF A DUAL SPACE
    • §65. A HUMAN ANALOGY
    • §66. GENERALIZIED CURVES AND FLOWS, AND THEIR BOUNDARIES
    • §67. PARAMETRIC REPRESENTATION OF GENERALIZED CURVES
    • §68. EXISTENCE OF A MINIMUM
    • §69. THE NATURE OF THE GENERALIZED SOLUTIONS
    • Appendix I SOME FURTHER BASIC NOTIONS OF CONVEXITY AND INTEGRATION
    • §70. INTRODUCTION
    • §71. SEPARATION THEOREM FOR A CONVEX CONE IN 𝒞0(A)
    • §72. THE LEMMA OF THE INSUFFICIENT RADIUS
    • §73. THE DUAL SEPARATION THEOREM
    • §74. A LOCALIZATION LEMMA FOR A B··i-COMPACT SET
    • §75. RIESZ MEASURES
    • §76. EUCLIDEAN APPROXIMATION TO A BANACH VECTOR FUNCTION
    • §77. AN ELEMENTARY NORM ESTIMATE
    • §78. VECTOR INTEGRATION
    • §79. CLOSURE OF A CONVEX HULL
    • Appendix II THE VARIATIONAL SIGNIFICANCE AND STRUCTURE OF GENERALIZED FLOWS
    • §80. INTRODUCTION
    • §81. POLYGONAL FLOWS
    • §82. THE BASIS OF MODERN DUALITY IN THE CALCULUS OF VARIATIONS
    • §83. THE VARIATIONAL CONVEXITY PRINCIPLE IN ITS ELEMENTARY FORM
    • §84. A FIRST EXTENSION
    • §85. THE ENLARGEMENT PRINCIPLE AND THE FIRST CLOSURE THEOREM FOR GENERALIZED FLOWS
    • §86. THE EXTENSION TO CONSISTENT FLOWS AND BOUNDARIES
    • §87. PRELIMINARY INFORMATION ON MIXTURES AND ON THE LAGRANGE REPRESENTATION
    • §88. FURTHER. COMMENTS ON MEASURES, MIXTURES AND CONSISTENT FLOWS
    • §89. THE LAGRANGE REPRESENTATION OF A CONSISTENT FLOW
    • Volume II OPTIMAL CONTROL THEORY
    • Preamble THE NATURE OF CONTROL PROBLEMS
    • §1. INTRODUCTION
    • §2. THE MULTIPLIER RULE
    • §3. OPTIMAL CONTROL AND THE LAGRANGE PROBLEM
    • §4. THE SAD FACTS OF LIFE
    • §5. A FIRST REVISION OF THE EULER EQUATION AND OF THE MULTIPLIER RULE
    • §6. THE WEIERSTRASS CONDITION, TRANSVERSALITY, HAMILTONIANS AND A STRONG REVISED EULER RECIPE
    • §7. THE CLASSICAL CONSTRAINED HAMILTONIANS
    • §8. CONTROLS AND THE MAXIMUM PRINCIPLE
    • §9. THE MAXIMUM PRINCIPLE AND ITS SPECIAL CASES AS DEFINITIONS
    • §10. SOLUTIONS OF TWO ELEMENTARY TIME-OPTIMAL PROBLEMS
    • Chapter I Naive Optimal Control Theory
    • §11. INTRODUCTION
    • §12. DISCRETE TIME AND PROGRAMMING
    • §13. SOME BASIC REMARKS ON LINEAR DIFFERENTIAL EQUATIONS
    • §14. SUSPECTED SOLUTIONS OF THE SIMPLEST TIME-OPTIMAL PROBLEMS
    • §15. UNICITY AND OPTIMALITY
    • §16. TWO DIMENSIONAL PROBLEMS: SWITCHING TIMES AND BASIC CONSTRUCTIONS
    • §17. DISCUSSION OF CASE (a)
    • §18. DISCUSSION OF CASE (b1)
    • §19. DISCUSSION OF CASE {b2)
    • Chapter II The Application of Standard Variational Methods to Optimal Control
    • §20. INTRODUCTION
    • §21. TRAJECTORIES AND LINES OF FLIGHT
    • §22. THE SYNCHRONIZATION CONDITION AND THE NOTIONS OF STANDARD PROJECTION AND DESCRIPTIVE MAP
    • §23. THE NOTION OF A SPRAY OF FLIGHTS
    • §24. THE HILBERT INDEPENDENCE INTEGRAL
    • §25. PRELIMINARY LEMMAS
    • §26. THE THEOREM OF MALUS
    • §28. PIECING TOGETHER FRAGMENTS OF CURVES
    • §29. THE FUNDAMENTAL THEOREM AND ITS CONSEQUENCES
    • Chapter Ill Generalized Optimal Control
    • §30. INTRODUCTION
    • §31. THE PREPROBLEM
    • §32. MORE SEMANTICS
    • §33. CONVENTIONAL AND CHATTERING CONTROLS IN DIFFERENTIAL EQUATIONS
    • §34. THE HALFWAY PRINCIPLE AND THE FlLIPPOV LEMMA
    • §35. UNICITY AND A KEY LEMMA FOR APPROXIMATIONS
    • §36. CONTROL MEASURES
    • §37. A PROPER SETTING FOR OPTIMAL CONTROL PROBLEMS
    • §38. HILBERT'S PRINCIPLE OF MINIMUM
    • §39. PONTRJAGIN'S MAXIMUM PRINCIPLE
    • §39A. THE PERTURBATION
    • §39B. REDUCTION TO A SEPARATION THEOREM
    • §39C. AN EQUIVALENT FORM OF THE SEPARATION
    • §39D. PROOF OF THE MAXIMUM PRINCIPLE
    • §39E. EPILOGUE
    • References
    • Additional References
    • INDEX
    • Back Cover
  • Additional Material
     
     
  • Reviews
     
     

    • The appearance of this book is one of the most exciting events for friends of the Calculus of Variations since the publication of Carathéodory's classic in 1935 on the calculus of variations and partial differential equations of first order. The author ... gives here a very lively, greatly stimulating, and highly personalized account of the calculus of variations and optimal control theory ... In his many refreshing asides, the author not only puts ideas and techniques into their historic perspective, but also succeeds in making men, who for many of us are merely revered names, come alive through skillful selection of quotes and descriptions of their interaction with each other and the subject matter at hand ... A beautiful book ... that is bound to stimulate many mathematicians and students of mathematics.
      MAA Monthly

    • A considerable number of heretofore unpublished results developed by the author are found ... The book is an important contribution to the calculus of variations and optimal control theory. It is most appropriate that the theory of generalized curves should be presented ... by its founder. The book is well written with an unusual and lively style. It is filled with historical remarks and with comments which enlarge one's outlook on the role of mathematics and mathematicians in our society ... This book should be mastered by anyone who wishes to become an expert in this field.
      Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 304; 1969; 337 pp
MSC: Primary 49

This book is divided into two parts. The first addresses the simpler variational problems in parametric and nonparametric form. The second covers extensions to optimal control theory.

The author opens with the study of three classical problems whose solutions led to the theory of calculus of variations. They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. He gives a detailed discussion of the Hamilton-Jacobi theory, both in the parametric and nonparametric forms. This leads to the development of sufficiency theories describing properties of minimizing extremal arcs.

Next, the author addresses existence theorems. He first develops Hilbert's basic existence theorem for parametric problems and studies some of its consequences. Finally, he develops the theory of generalized curves and “automatic” existence theorems.

In the second part of the book, the author discusses optimal control problems. He notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints. In the control formulation, these constraints are expressed in a more convenient form in terms of control functions. After pointing out the new phenomenon that may arise, namely, the lack of controllability, the author develops the maximum principle and illustrates this principle by standard examples that show the switching phenomena that may occur. He extends the theory of geodesic coverings to optimal control problems. Finally, he extends the problem to generalized optimal control problems and obtains the corresponding existence theorems.
Readership

Graduate students and research mathematicians interested in calculus of variations and optimal control.
  • Front Cover
  • FOREWORD
  • PREFACE
  • CONTENTS
  • Volume I LECTURES ON THE CALCULUS OF VARIATIONS
  • Preamble GENERALITIES AND TYPICAL PROBLEMS
  • §1. INTRODUCTION
  • §2. THE PLACE OF THE CALCULUS OF VARIATIONS IN RELATION TO THE REST OF MATHEMATICS AND TO SPACE SCIENCE
  • §3. STATEMENT OF THE SIMPLEST PROBLEM AND SOME COGNATE MATTERS
  • §4. EXTREMALS IN SOME CLASSICAL PROBLEMS
  • §5. SOLUTIONS Of THE PRECEDING PROBLEMS (a), (b), (c)
  • §6. THE EULER-LAGRANGE LEMMA AND SCHWARTZ DISTRIBUTIONS
  • §7. ALTERNATIVE FORMS OF THE LEMMA
  • §8. PROOF OF THE MAIN FORM OF THE LEMMA
  • §9. FIRST VARIATION, EULER EQUATION, TRANSVERSALITY
  • §10. PERRON'S PARADOX
  • Chapter I The Method of Geodesic Coverings
  • §11. INTRODUCTION
  • §12. THE VARIATIONAL ALGORITHM OF HUYGENS
  • §13. A LINK WITH ELEMENTARY CONVEXITY
  • §14. REAPPEARANCE OF THE EULER EQUATION
  • §15. THE THEOREM OF MALUS
  • §16. SUFFICIENT CONDITIONS FOR INDEPENDENCE OF THE HILBERT INTEGRAL
  • §17. INVARIANCE PROPERTIES AND AN ENVELOPE THEOREM
  • §18. GENERAL COMMENTS AND THE APPLICATIONS TO PLANE PROBLEMS
  • §19. BACKGROUND ON FlX-POINTS AND ON EXISTENCE THEOREMS FOR DlFFERENTIAL EQUATIONS AND IMPLICIT FUNCTIONS
  • Chapter II Duality and Local Embedding
  • §20. INTRODUCTION
  • §21. THE LEGENDRE TRANSFORMATION
  • §22. THE HAMILTONIAN AND ITS PROPERTIES
  • §23. CAUCHY CHARACTERISTICS
  • §24. DUALITY AND THE STANDARD HAMILTONIAN IN THE PARAMETRIC CASE
  • §25. OTHER ADMISSIBLE PARAMETRIC HAMILTONIANS
  • §26. LOCAL PASSAGE FROM PARAMETRIC TO NONPARAMETRIC CASE
  • §27. THE EMBEDDING OF SMALL EXTREMALS IN SMALL TUBES
  • §28. LOCAL EXISTENCE THEORY FOR NONPARAMETRIC VARIATIONAL PROBLEMS AND FOR ORDINARY SECOND ORDER DlFFERENTIAL EQUATIONS
  • §29. LOCAL PARAMETRIC EXISTENCE THEORY FOR THE ELLIPTIC CASE
  • Chapter Ill Embedding in the Large
  • §30. INTRODUCTION
  • §31. FIRST AND SECOND VARIATIONS AND TRANSVERSALITY
  • §32. THE SECOND VARIATION FALLACY
  • §33. THE SECONDARY HAMILTONIAN
  • §34. GEOMETRICAL INTERPRETATION OF EXACTNESS
  • §35. DISTINGUISHED FAMILIES
  • §36. CANONICAL EMBEDDINGS AND FOCAL POINTS
  • §37. THIE JACOBI THEORY OF CONJUGATE POINTS
  • §38. THE INDEX OF STABILITY OF AN EXTREMAL
  • §39. THE SECOND STAGE OF THE MORSE THEORY
  • Chapter IV Hamiltonians in the large, Convexity, Inequailities and Functional Analysis
  • §40. INTRODUCTION
  • §41. CENTER OF GRAVITY AND DISPERSAL ZONE
  • §42. CONVEXITY AND THE HAHN-BANACH THEOREM
  • §43. THE CONCEPTUAL HERITAGE OF GEORG CANTOR
  • §44. DUALITY OF CONVEX FIGURES
  • §45. DUALITY OF CONVEX FUNCTIONS
  • §46. HAMILTONIANS IN THE LARGE AND REFORMULATED VARIATIONAL THEORY
  • §47. REMARKS ON CLASSICAL INEQUALITIES
  • §48. THE DUAL UNIT BALL OF A FUNCTIONAL SPACE
  • §49. THE RIESZ REPRESENTATION
  • Chapter V Existence Theory and its Consequences
  • §50. INTRODUCTION
  • §51. THE HILBERT CONSTRUCTION AND SOME OF ITS CONSEQUENCES IN THE STANDARD PARAMETRIC CASE
  • §52. THE PARAMETRIC THEORY OF CONJUGATE POINTS AND THE PARAMETRIC JACOBI CONDITION
  • §53. THE TONELLI-CARATHEODORY UNICITY THEOREM
  • §54. ABSOLUTE AND HOMOTOPIC MINIMA ON B •• i-COMPACT DOMAINS AND MANIFOLDS
  • §55. TOWARD AN AUTOMATIC EXISTENCE THEORY
  • §56. FlRST STAGE Of AN ABSTRACT APPROACH: SEMICONTINUITY IN A B●●i-COMPACT SET
  • §§57, 58, 59
  • Chapter VI Generalized Curves and Flows
  • §60. INTRODUCTION
  • §61. INTUITIVE BACKGROUND
  • §62. A QUESTION OF SEMANTICS
  • §63. PARAMETRIC CURVES IN THE CALCULUS OF VARIATIONS
  • §64. ADMISSIBLE CURVES AS ELEMENTS OF A DUAL SPACE
  • §65. A HUMAN ANALOGY
  • §66. GENERALIZIED CURVES AND FLOWS, AND THEIR BOUNDARIES
  • §67. PARAMETRIC REPRESENTATION OF GENERALIZED CURVES
  • §68. EXISTENCE OF A MINIMUM
  • §69. THE NATURE OF THE GENERALIZED SOLUTIONS
  • Appendix I SOME FURTHER BASIC NOTIONS OF CONVEXITY AND INTEGRATION
  • §70. INTRODUCTION
  • §71. SEPARATION THEOREM FOR A CONVEX CONE IN 𝒞0(A)
  • §72. THE LEMMA OF THE INSUFFICIENT RADIUS
  • §73. THE DUAL SEPARATION THEOREM
  • §74. A LOCALIZATION LEMMA FOR A B··i-COMPACT SET
  • §75. RIESZ MEASURES
  • §76. EUCLIDEAN APPROXIMATION TO A BANACH VECTOR FUNCTION
  • §77. AN ELEMENTARY NORM ESTIMATE
  • §78. VECTOR INTEGRATION
  • §79. CLOSURE OF A CONVEX HULL
  • Appendix II THE VARIATIONAL SIGNIFICANCE AND STRUCTURE OF GENERALIZED FLOWS
  • §80. INTRODUCTION
  • §81. POLYGONAL FLOWS
  • §82. THE BASIS OF MODERN DUALITY IN THE CALCULUS OF VARIATIONS
  • §83. THE VARIATIONAL CONVEXITY PRINCIPLE IN ITS ELEMENTARY FORM
  • §84. A FIRST EXTENSION
  • §85. THE ENLARGEMENT PRINCIPLE AND THE FIRST CLOSURE THEOREM FOR GENERALIZED FLOWS
  • §86. THE EXTENSION TO CONSISTENT FLOWS AND BOUNDARIES
  • §87. PRELIMINARY INFORMATION ON MIXTURES AND ON THE LAGRANGE REPRESENTATION
  • §88. FURTHER. COMMENTS ON MEASURES, MIXTURES AND CONSISTENT FLOWS
  • §89. THE LAGRANGE REPRESENTATION OF A CONSISTENT FLOW
  • Volume II OPTIMAL CONTROL THEORY
  • Preamble THE NATURE OF CONTROL PROBLEMS
  • §1. INTRODUCTION
  • §2. THE MULTIPLIER RULE
  • §3. OPTIMAL CONTROL AND THE LAGRANGE PROBLEM
  • §4. THE SAD FACTS OF LIFE
  • §5. A FIRST REVISION OF THE EULER EQUATION AND OF THE MULTIPLIER RULE
  • §6. THE WEIERSTRASS CONDITION, TRANSVERSALITY, HAMILTONIANS AND A STRONG REVISED EULER RECIPE
  • §7. THE CLASSICAL CONSTRAINED HAMILTONIANS
  • §8. CONTROLS AND THE MAXIMUM PRINCIPLE
  • §9. THE MAXIMUM PRINCIPLE AND ITS SPECIAL CASES AS DEFINITIONS
  • §10. SOLUTIONS OF TWO ELEMENTARY TIME-OPTIMAL PROBLEMS
  • Chapter I Naive Optimal Control Theory
  • §11. INTRODUCTION
  • §12. DISCRETE TIME AND PROGRAMMING
  • §13. SOME BASIC REMARKS ON LINEAR DIFFERENTIAL EQUATIONS
  • §14. SUSPECTED SOLUTIONS OF THE SIMPLEST TIME-OPTIMAL PROBLEMS
  • §15. UNICITY AND OPTIMALITY
  • §16. TWO DIMENSIONAL PROBLEMS: SWITCHING TIMES AND BASIC CONSTRUCTIONS
  • §17. DISCUSSION OF CASE (a)
  • §18. DISCUSSION OF CASE (b1)
  • §19. DISCUSSION OF CASE {b2)
  • Chapter II The Application of Standard Variational Methods to Optimal Control
  • §20. INTRODUCTION
  • §21. TRAJECTORIES AND LINES OF FLIGHT
  • §22. THE SYNCHRONIZATION CONDITION AND THE NOTIONS OF STANDARD PROJECTION AND DESCRIPTIVE MAP
  • §23. THE NOTION OF A SPRAY OF FLIGHTS
  • §24. THE HILBERT INDEPENDENCE INTEGRAL
  • §25. PRELIMINARY LEMMAS
  • §26. THE THEOREM OF MALUS
  • §28. PIECING TOGETHER FRAGMENTS OF CURVES
  • §29. THE FUNDAMENTAL THEOREM AND ITS CONSEQUENCES
  • Chapter Ill Generalized Optimal Control
  • §30. INTRODUCTION
  • §31. THE PREPROBLEM
  • §32. MORE SEMANTICS
  • §33. CONVENTIONAL AND CHATTERING CONTROLS IN DIFFERENTIAL EQUATIONS
  • §34. THE HALFWAY PRINCIPLE AND THE FlLIPPOV LEMMA
  • §35. UNICITY AND A KEY LEMMA FOR APPROXIMATIONS
  • §36. CONTROL MEASURES
  • §37. A PROPER SETTING FOR OPTIMAL CONTROL PROBLEMS
  • §38. HILBERT'S PRINCIPLE OF MINIMUM
  • §39. PONTRJAGIN'S MAXIMUM PRINCIPLE
  • §39A. THE PERTURBATION
  • §39B. REDUCTION TO A SEPARATION THEOREM
  • §39C. AN EQUIVALENT FORM OF THE SEPARATION
  • §39D. PROOF OF THE MAXIMUM PRINCIPLE
  • §39E. EPILOGUE
  • References
  • Additional References
  • INDEX
  • Back Cover

  • The appearance of this book is one of the most exciting events for friends of the Calculus of Variations since the publication of Carathéodory's classic in 1935 on the calculus of variations and partial differential equations of first order. The author ... gives here a very lively, greatly stimulating, and highly personalized account of the calculus of variations and optimal control theory ... In his many refreshing asides, the author not only puts ideas and techniques into their historic perspective, but also succeeds in making men, who for many of us are merely revered names, come alive through skillful selection of quotes and descriptions of their interaction with each other and the subject matter at hand ... A beautiful book ... that is bound to stimulate many mathematicians and students of mathematics.
    MAA Monthly

  • A considerable number of heretofore unpublished results developed by the author are found ... The book is an important contribution to the calculus of variations and optimal control theory. It is most appropriate that the theory of generalized curves should be presented ... by its founder. The book is well written with an unusual and lively style. It is filled with historical remarks and with comments which enlarge one's outlook on the role of mathematics and mathematicians in our society ... This book should be mastered by anyone who wishes to become an expert in this field.
    Mathematical Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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