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A non-local reduction principle for cocycles in Hilbert spaces

Mikhail Anikushin

Abstract

We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for $C_{0}$-semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem.

A non-local reduction principle for cocycles in Hilbert spaces

Abstract

We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for -semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem.

Paper Structure

This paper contains 12 sections, 30 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Structure of amenable sets
  3. The map $\Phi$ and inertial manifold $\mathfrak{A}$
  4. Abstract periodic cocycles
  5. Delayed feedback systems
  6. Linear part
  7. Frequency-domain condition
  8. Parabolic problems with monotone nonlinear boundary controls
  9. Linear part
  10. Frequency-domain condition
  11. Frequency theorem of the Yakubovich-Likhtarnikov for $C_{0}$-semigroups
  12. Spectral properties of solutions to Lyapunov inequalities

Key Result

Theorem 1.1

Suppose that (H1)-(H4) and one of (COM1) or (COM2) hold. Then the map $\Pi_{q} := { \left.\nulldelimiterspace \Pi \newline \right|_{\mathfrak{A}(q)} } \colon \mathfrak{A}(q) \to \mathbb{H}^{-}$ is a homeomorphism.

Theorems & Definitions (66)

  • remark 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • remark 2
  • Theorem 1.4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 56 more