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Frequency theorem and inertial manifolds for neutral delay equations

Mikhail Anikushin

Abstract

We study the quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by $δ$-functionals, which are unbounded in $L_{2}$. Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem (following the works of V.A. Yakubovich and A.L. Likhtarnikov), which guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; Yu. A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

Frequency theorem and inertial manifolds for neutral delay equations

Abstract

We study the quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by -functionals, which are unbounded in . Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem (following the works of V.A. Yakubovich and A.L. Likhtarnikov), which guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; Yu. A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

Paper Structure

This paper contains 15 sections, 38 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Infinite-horizon quadratic regulator problem and quadratic Lyapunov-like functionals
  3. Proof of Theorem \ref{['TH: OptimalProcessDelay']}
  4. Proof of Theorem \ref{['TH: FreqThRelaxedDelay']}
  5. Realification of the operator $P$ and its sign properties
  6. Pointwise measurement operators on embracing spaces and spaces of agalmanated functions
  7. Embracing spaces and pointwise measurement operators
  8. Spaces of adorned functions
  9. Spaces of twisted functions
  10. Spaces of agalmanated functions
  11. Applications to delay equations of neutral type in $\mathbb{R}^{n}$
  12. Properties of linear problems: structural Cauchy formula and resolvent estimates
  13. Nonlinear problems and linear inhomogeneous systems with quadratic constraints
  14. Equations with small delays
  15. Discussion

Key Result

theorem 1.1

Let the pair $(A,B)$ be $(L_{2},\mathbb{W})$-controllable in $\mathbb{E}_{0}$ and suppose that DESC:QF and DESC:FT are satisfied for the Hermitian form $\mathcal{F}$. We have the following: 1). If $\alpha_{1}>0$, then for every $v_{0} \in \mathbb{E}_{0}$ the quadratic functional $\mathcal{J}_{\mathc is a bounded linear operator. Moreover, there exists $P \in \mathcal{L}(\mathbb{E}_{0};\mathbb{E}^{

Theorems & Definitions (77)

  • remark 1.1
  • remark 1.2
  • theorem 1.1
  • theorem 1.2
  • lemma 2.1
  • proof
  • lemma 2.2
  • proof
  • lemma 2.3
  • proof
  • ...and 67 more