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Discrete dynamical systems: inverse problems and related topics

Alexander Mikhaylov, Victor Mikhaylov

TL;DR

...

Abstract

In this review, we extend the Boundary Control method\, -- \,an approach to inverse problems based on control theory for dynamical systems \, -- \,to inverse problems for discrete dynamical systems. We apply our results to classical moment problems, Toda lattices, Weyl functions, de Branges spaces, Krein-Stieltjes strings, and also to problems of numerical simulations.

Discrete dynamical systems: inverse problems and related topics

TL;DR

...

Abstract

In this review, we extend the Boundary Control method\, -- \,an approach to inverse problems based on control theory for dynamical systems \, -- \,to inverse problems for discrete dynamical systems. We apply our results to classical moment problems, Toda lattices, Weyl functions, de Branges spaces, Krein-Stieltjes strings, and also to problems of numerical simulations.
Paper Structure (29 sections, 41 theorems, 226 equations, 1 figure)

This paper contains 29 sections, 41 theorems, 226 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Inverse dynamic problems for Jacobi matrices
  3. Setup of the problem
  4. Forward problem, operators of the Boundary Control method.
  5. Krein equations
  6. Factorization method
  7. Characterization of inverse data.
  8. Discrete Schrödinger operator
  9. Inverse problem for complex Jacobi matrices
  10. Classical moment problems
  11. Setup of the problem
  12. Spectral representation of connecting and response operators
  13. Finite Jacobi matrix
  14. Semi-infinite Jacobi matrix
  15. Truncated moment problem
  16. ...and 14 more sections

Key Result

Lemma 1

The solution to equation (Jacobi_dyn) can be represented as follows where $w_{n,s}$ satisfies the Goursat problem

Figures (1)

  • Figure 1: Discrete wave equation.

Theorems & Definitions (59)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Lemma 2
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • ...and 49 more