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Optimal Control and Geometry: Integrable Systems

Book Description

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.

Table of Contents

  1. Coverpage
  2. Halftitle page
  3. Series page
  4. Title page
  5. Copyright
  6. Acknowledgments
  7. Introduction
  8. Chapter 1 The Orbit Theorem and Lie determined systems
    1. 1.1 Vector fields and differential forms
    2. 1.2 Flows and diffeomorphisms
    3. 1.3 Families of vector fields: the Orbit theorem
    4. 1.4 Distributions and Lie determined systems
  9. Chapter 2 Control systems: accessibility and controllability
    1. 2.1 Control systems and families of vector fields
    2. 2.2 The Lie saturate
  10. Chapter 3 Lie groups and homogeneous spaces
    1. 3.1 The Lie algebra and the exponential map
    2. 3.2 Lie subgroups
    3. 3.3 Families of left-invariant vector fields and accessibility
    4. 3.4 Homogeneous spaces
  11. Chapter 4 Symplectic manifolds: Hamiltonian vector fields
    1. 4.1 Symplectic vector spaces
    2. 4.2 The cotangent bundle of a vector space
    3. 4.3 Symplectic manifolds
  12. Chapter 5 Poisson manifolds, Lie algebras, and coadjoint orbits
    1. 5.1 Poisson manifolds and Poisson vector fields
    2. 5.2 The cotangent bundle of a Lie group: coadjoint orbits
  13. Chapter 6 Hamiltonians and optimality: the Maximum Principle
    1. 6.1 Extremal trajectories
    2. 6.2 Optimal control and the calculus of variations
    3. 6.3 The Maximum Principle
    4. 6.4 The Maximum Principle in the presence of symmetries
    5. 6.5 Abnormal extremals
    6. 6.6 The Maximum Principle and Weierstrass’ excess function
  14. Chapter 7 Hamiltonian view of classic geometry
    1. 7.1 Hyperbolic geometry
    2. 7.2 Elliptic geometry
    3. 7.3 Sub-Riemannian view
    4. 7.4 Elastic curves
    5. 7.5 Complex overview and integration
  15. Chapter 8 Symmetric spaces and sub-Riemannian problems
    1. 8.1 Lie groups with an involutive automorphism
    2. 8.2 Symmetric Riemannian pairs
    3. 8.3 The sub-Riemannian problem
    4. 8.4 Sub-Riemannian and Riemannian geodesics
    5. 8.5 Jacobi curves and the curvature
    6. 8.6 Spaces of constant curvature
  16. Chapter 9 Affine-quadratic problem
    1. 9.1 Affine-quadratic Hamiltonians
    2. 9.2 Isospectral representations
    3. 9.3 Integrability
  17. Chapter 10 Cotangent bundles of homogeneous spaces as coadjoint orbits
    1. 10.1 Spheres, hyperboloids, Stiefel and Grassmannian manifolds
    2. 10.2 Canonical affine Hamiltonians on rank one orbits: Kepler and Newmann
    3. 10.3 Degenerate case and Kepler’s problem
    4. 10.4 Mechanical problem of C. Newmann
    5. 10.5 The group of upper triangular matrices and Toda lattices
  18. Chapter 11 Elliptic geodesic problem on the sphere
    1. 11.1 Elliptic Hamiltonian on semi-direct rank one orbits
    2. 11.2 The Maximum Principle in ambient coordinates
    3. 11.3 Elliptic problem on the sphere and Jacobi’s problem on the ellipsoid
    4. 11.4 Elliptic coordinates on the sphere
  19. Chapter 12 Rigid body and its generalizations
    1. 12.1 The Euler top and geodesic problems on SO[sub(n)](R)
    2. 12.2 Tops in the presence of Newtonian potentials
  20. Chapter 13 Isometry groups of space forms and affine systems: Kirchhoff’s elastic problem
    1. 13.1 Elastic curves and the pendulum
    2. 13.2 Parallel and Serret–Frenet frames and elastic curves
    3. 13.3 Serret–Frenet frames and the elastic problem
    4. 13.4 Kichhoff’s elastic problem
  21. Chapter 14 Kowalewski–Lyapunov criteria
    1. 14.1 Complex quaternions and SO[sub(4)](C)
    2. 14.2 Complex Poisson structure and left-invariant Hamiltonians
    3. 14.3 Affine Hamiltonians on SO[sub(4)](C) and meromorphic solutions
    4. 14.4 Kirchhoff–Lagrange equation and its solution
  22. Chapter 15 Kirchhoff–Kowalewski equation
    1. 15.1 Eulers’ solutions and addition formulas of A. Weil
    2. 15.2 The hyperelliptic curve
    3. 15.3 Kowalewski gyrostat in two constant fields
  23. Chapter 16 Elastic problems on symmetric spaces: the Delauney–Dubins problem
    1. 16.1 The curvature problem
    2. 16.2 Elastic problem revisited – Dubins–Delauney on space forms
    3. 16.3 Curvature problem on symmetric spaces
    4. 16.4 Elastic curves and the rolling sphere problem
  24. Chapter 17 The non-linear Schroedinger’s equation and Heisenberg’s magnetic equation–solitons
    1. 17.1 Horizontal Darboux curves
    2. 17.2 Darboux curves and symplectic Fréchet manifolds
    3. 17.3 Geometric invariants of curves and their Hamiltonian vector fields
    4. 17.4 Affine Hamiltonians and solitons Concluding remarks
  25. References
  26. Index