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Optimal Control and Geometry: Integrable Systems

$84.99

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: July 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107113886

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About the Authors
  • The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.

    • Provides a cohesive view of the theory of integrable systems with conceptual clarity and originality of approach
    • Delivers a new framework for tackling problems of optimality
    • Allows readers in many fields to understand both the calculus of variations and control theory in a new light
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    Reviews & endorsements

    'The book is written in a very refreshing style that reflects the author's contagious enthusiasm for the subject. It manages this by focusing on the geometry, using the precise language of differential geometry, while not getting bogged down by analytic intricacies. Throughout, the book pays much attention to historical developments and evolving and contrasting points of view, which is also reflected in the rich bibliography of classical resources.' Matthias Kawski, MathSciNet

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    Product details

    • Date Published: July 2016
    • format: Hardback
    • isbn: 9781107113886
    • length: 423 pages
    • dimensions: 235 x 159 x 27 mm
    • weight: 0.73kg
    • contains: 4 tables
    • availability: Available
  • Table of Contents

    1. The orbit theorem and Lie determined systems
    2. Control systems. Accessibility and controllability
    3. Lie groups and homogeneous spaces
    4. Symplectic manifolds. Hamiltonian vector fields
    5. Poisson manifolds, Lie algebras and coadjoint orbits
    6. Hamiltonians and optimality: the Maximum Principle
    7. Hamiltonian view of classic geometry
    8. Symmetric spaces and sub-Riemannian problems
    9. Affine problems on symmetric spaces
    10. Cotangent bundles as coadjoint orbits
    11. Elliptic geodesic problem on the sphere
    12. Rigid body and its generalizations
    13. Affine Hamiltonians on space forms
    14. Kowalewski–Lyapunov criteria
    15. Kirchhoff–Kowalewski equation
    16. Elastic problems on symmetric spaces: Delauney–Dubins problem
    17. Non-linear Schroedinger's equation and Heisenberg's magnetic equation. Solitons.

  • Author

    Velimir Jurdjevic, University of Toronto
    Professor Velimir Jurdjevic is one of the founders of geometric control theory. His pioneering work with H. J. Sussmann was deemed to be among the most influential papers of the century and his book, Geometric Control Theory, revealed the geometric origins of the subject and uncovered important connections to physics and geometry. It remains a major reference on non-linear control. Jurdjevic's expertise also extends to differential geometry, mechanics and integrable systems. His publications cover a wide range of topics including stability theory, Hamiltonian systems on Lie groups, and integrable systems. He has spent most of his professional career at the University of Toronto.

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