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Forward completeness implies bounded reachable sets for time-delay systems on the state space of essentially bounded measurable functions

Lucas Brivadis, Antoine Chaillet, Andrii Mironchenko, Fabian Wirth

Abstract

We consider time-delay systems with a finite number of delays in the state space $L^\infty\times\mathbb{R}^n$. In this framework, we show that forward completeness implies the bounded reachability sets property, while this implication was recently shown by J.L. Mancilla-Aguilar and H. Haimovich to fail in the state space of continuous functions. As a consequence, we show that global asymptotic stability is always uniform in the state space $L^\infty\times\mathbb{R}^n$.

Forward completeness implies bounded reachable sets for time-delay systems on the state space of essentially bounded measurable functions

Abstract

We consider time-delay systems with a finite number of delays in the state space . In this framework, we show that forward completeness implies the bounded reachability sets property, while this implication was recently shown by J.L. Mancilla-Aguilar and H. Haimovich to fail in the state space of continuous functions. As a consequence, we show that global asymptotic stability is always uniform in the state space .
Paper Structure (5 sections, 5 theorems, 24 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Time-delay systems in Linf x Rn
  3. Main result
  4. Relations between GAS and UGAS
  5. Conclusion

Key Result

Theorem II.3

Let $\mathcal{X}$ be either $\mathcal{X}^0$ or $\mathcal{X}^\infty$. Given any initial conditions $x_0\in\mathcal{X}$ and any $u\in\mathcal{U}$, there exists the unique maximal $\mathcal{X}$-solution $x(\cdot,x_0,u)$ of the Cauchy problem syst initialized at $x_0$ corresponding to an input $u$. Deno

Figures (2)

  • Figure 1: Proof architecture of Theorem \ref{['th:main']}. Implications proved in the present article are indicated in blue. Implications known from the literature and trivial implications are indicated in black, as well as an implication known to be false ($\not\Rightarrow$). Together, these implications are sufficient to establish Theorem \ref{['th:main']}.
  • Figure 2: Proof architecture of Theorem \ref{['th:gas']}. Implications proved in the present article are depicted in blue. Implications known from the literature, trivial implications, as well as an implication known to be false ($\not\Rightarrow$) are indicated in black. Together, these implications are sufficient to establish Theorem \ref{['th:gas']}.

Theorems & Definitions (18)

  • Definition II.1: $\mathcal{X}$-solutions
  • Remark II.2
  • Theorem II.3: Existence, uniqueness
  • proof
  • Remark II.4
  • Definition II.5
  • Definition II.6
  • Remark II.7
  • Lemma II.8
  • proof
  • ...and 8 more