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Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

  • Book
  • © 2016

Overview

Authors:
  1. Russell Johnson

    1. Università degli Studi di Firenze, Firenze, Italy

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  2. Rafael Obaya

    1. Universidad de Valladolid, Valladolid, Spain

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  3. Sylvia Novo

    1. Universidad de Valladolid, Valladolid, Spain

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  4. Carmen Núñez

    1. Universidad de Valladolid, Valladolid, Spain

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  5. Roberta Fabbri

    1. Università degli Studi di Firenze, Firenze, Italy

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  • Provides a detailed study of the concept of rotation and of the solutions
  • of linear nonautonomous Hamiltonian systems of general dimension
  • Explores the applications of linear Hamiltonian systems to
  • questions arising in classical oscillation theory of such systems, and to the
  • theory of linear control systems with time-varying coefficients
  • Applies known results of the theory of periodic Hamiltonian systems to the general nonautonomous case and presents various new properties

Part of the book series: Developments in Mathematics (DEVM, volume 36)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xxi
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  2. Nonautonomous Linear Hamiltonian Systems

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 1-75

  3. The Rotation Number and the Lyapunov Index for Real Nonautonomous Linear Hamiltonian Systems

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 77-124

  4. The Floquet Coefficient for Nonautonomous Linear Hamiltonian Systems: Atkinson Problems

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 125-179

  5. The Weyl Functions

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 181-248

  6. Weak Disconjugacy for Linear Hamiltonian Systems

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 249-328

  7. Nonautonomous Control Theory: Linear Regulator Problem and the Kalman–Bucy Filter

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 329-365

  8. Nonautonomous Control Theory: A General Version of the Yakubovich Frequency Theorem

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 367-418

  9. Nonautonomous Control Theory: Linear-Quadratic Dissipative Control Processes

    • Russell Johnson, Rafael Obaya, Sylvia Novo, Carmen Núñez, Roberta Fabbri
    Pages 419-486

  10. Back Matter

    Pages 487-497
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Keywords

About this book

This monograph contains an in-depth analysis of the dynamics given by a linear Hamiltonian system of general dimension with nonautonomous bounded and uniformly continuous coefficients, without other initial assumptions on time-recurrence. Particular attention is given to the oscillation properties of the solutions as well as to a spectral theory appropriate for such systems. The book contains extensions of results which are well known when the coefficients are autonomous or periodic, as well as in the nonautonomous two-dimensional case. However, a substantial part of the theory presented here is new even in those much simpler situations.


The authors make systematic use of basic facts concerning Lagrange planes and symplectic matrices, and apply some fundamental methods of topological dynamics and ergodic theory. Among the tools used in the analysis, which include Lyapunov exponents, Weyl matrices, exponential dichotomy, and weak disconjugacy, a fundamentalrole is played by the rotation number for linear Hamiltonian systems of general dimension. The properties of all these objects form the basis for the study of several themes concerning linear-quadratic control problems, including the linear regulator property, the Kalman-Bucy filter, the infinite-horizon optimization problem, the nonautonomous version of the Yakubovich Frequency Theorem, and dissipativity in the Willems sense.


The book will be useful for graduate students and researchers interested in nonautonomous differential equations; dynamical systems and ergodic theory; spectral theory of differential operators; and control theory.

Authors and Affiliations

  • Università degli Studi di Firenze, Firenze, Italy

    Russell Johnson, Roberta Fabbri

  • Universidad de Valladolid, Valladolid, Spain

    Rafael Obaya, Sylvia Novo, Carmen Núñez

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