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Left-coprimeness condition for the reachability in finite time of pseudo-rational systems of order zero with an application to difference delay systems

Sébastien Fueyo

TL;DR

The paper develops a finite-time reachability theory for pseudo-rational systems of order zero and applies it to difference-delay systems with distributed delays. It proves that quasi-reachability and reachability coincide in finite time above the universal bound $T>-l( ext{det}(Q))$, and provides a constructive left-coprimeness criterion for $L^1$ targets, with a deeper analysis for the scalar case and regularity-based sufficiency results for higher dimensions. For distributed-delay systems, a Hautus-type frequency-domain test and a matrix-rank condition yield a practical reachability criterion, together with a simple bound $T_{ ext{min}}\uparrow doldsymbol{ extLambda}_N$ for minimal time. The work is underpinned by Paley–Wiener–Schwartz and corona-type distribution results, extending previous scalar findings to the matrix setting and highlighting open questions on sufficiency of the frequency-domain criterion. Overall, the framework provides actionable criteria and input-construction methods for finite-time controllability of a broad class of delay systems with potential engineering and biological applications.

Abstract

The paper investigates the reachability in finite time of pseudo-rational systems of order zero. A bound on the minimal time of reachability is obtained and the reachability property for integrable functions is characterized in terms of a left-coprime condition. The results are applied to difference delay systems with distributed delays.

Left-coprimeness condition for the reachability in finite time of pseudo-rational systems of order zero with an application to difference delay systems

TL;DR

The paper develops a finite-time reachability theory for pseudo-rational systems of order zero and applies it to difference-delay systems with distributed delays. It proves that quasi-reachability and reachability coincide in finite time above the universal bound , and provides a constructive left-coprimeness criterion for targets, with a deeper analysis for the scalar case and regularity-based sufficiency results for higher dimensions. For distributed-delay systems, a Hautus-type frequency-domain test and a matrix-rank condition yield a practical reachability criterion, together with a simple bound for minimal time. The work is underpinned by Paley–Wiener–Schwartz and corona-type distribution results, extending previous scalar findings to the matrix setting and highlighting open questions on sufficiency of the frequency-domain criterion. Overall, the framework provides actionable criteria and input-construction methods for finite-time controllability of a broad class of delay systems with potential engineering and biological applications.

Abstract

The paper investigates the reachability in finite time of pseudo-rational systems of order zero. A bound on the minimal time of reachability is obtained and the reachability property for integrable functions is characterized in terms of a left-coprime condition. The results are applied to difference delay systems with distributed delays.
Paper Structure (9 sections, 10 theorems, 42 equations)

This paper contains 9 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Pseudo-rational systems of order zero
  4. Minimal time of reachability
  5. Left-coprimeness reachability conditions
  6. Application to delay systems
  7. Invertibility of a Radon measure linked to a difference delay systems with distributed delays
  8. Paley-Wiener-Schwartz theorem
  9. A corona theorem for distributions with compact support in $\mathbb{R}_-$

Key Result

Theorem 4.1

A $\Sigma^Q$ pseudo-rational system of order zero is $X^{Q,q}$ quasi-reachable (resp. reachable) if and only if it is $X^{Q,q}$ quasi-reachable (resp. reachable) in time $\widetilde{T}$ for all $\tilde{T}>-l(\det(Q))$. In particular, an upper bound on the minimal time of quasi-reachability (resp. re

Theorems & Definitions (33)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • proof : Proof Theorem \ref{['bound_minimal_time_controllability']}
  • ...and 23 more