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A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

Sebastien Fueyo, Yacine Chitour

Abstract

This paper proves a corona theorem for the algebra of Radon measures compactly supported in $\mathbb{R}_-$ and this result is applied to provide a necessary and sufficient Hautus--type frequency criterion for the $L^1$ exact controllability of linear controlled delayed difference equations (LCDDE). Hereby, it solves an open question raised in [6].

A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

Abstract

This paper proves a corona theorem for the algebra of Radon measures compactly supported in and this result is applied to provide a necessary and sufficient Hautus--type frequency criterion for the exact controllability of linear controlled delayed difference equations (LCDDE). Hereby, it solves an open question raised in [6].
Paper Structure (8 sections, 6 theorems, 42 equations)

This paper contains 8 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and definitions
  3. Notations
  4. Radon measures framework
  5. The truncation operator
  6. Topological properties of the quotient algebra $M(\mathbb{R}_-)/(p)$
  7. A corona theorem for a subalgebra of Radon measures negatively and compactly supported
  8. $L^1$ exact controllability of linear difference delay control systems

Key Result

Lemma 2.1

The following assertions hold true:

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • Remark 4.3
  • proof
  • ...and 4 more