Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation

Andrii Mironchenko, Fabian Wirth, Antoine Chaillet, Lucas Brivadis

Abstract

We show that the existence of a Lyapunov-Krasovskii functional (LKF) with pointwise dissipation (i.e. dissipation in terms of the current solution norm) suffices for input-to-state stability, provided that uniform global stability can also be ensured using the same LKF. To this end, we develop a stability theory, in which the behavior of solutions is not assessed through the classical norm but rather through a specific LKF, which may provide significantly tighter estimates. We discuss the advantages of our approach by means of an example.

A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation

Abstract

We show that the existence of a Lyapunov-Krasovskii functional (LKF) with pointwise dissipation (i.e. dissipation in terms of the current solution norm) suffices for input-to-state stability, provided that uniform global stability can also be ensured using the same LKF. To this end, we develop a stability theory, in which the behavior of solutions is not assessed through the classical norm but rather through a specific LKF, which may provide significantly tighter estimates. We discuss the advantages of our approach by means of an example.
Paper Structure (19 sections, 23 theorems, 115 equations)

This paper contains 19 sections, 23 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Time-delay systems
  4. Definitions
  5. Main result
  6. Discussion
  7. Relations to existing results
  8. Criterion of uniform global asymptotic stability
  9. Example
  10. V-stability theory
  11. V-ISS superposition theorem
  12. Lyapunov-Krasovskii condition for V-UGS
  13. Bounds on solutions
  14. Proof of Theorem \ref{['thm:Krasovskii-Theorem']}
  15. Conclusion and perspectives
  16. ...and 4 more sections

Key Result

Proposition 2.5

Given an LKF candidate $V:\mathcal{X}^n\to\mathbb{R}_+$, consider the following statements: Then the following relations hold: If $V$ is coercive, then all four statements are equivalent.

Theorems & Definitions (38)

  • Definition 2.1: LKF candidate
  • Remark 2.2
  • Definition 2.3: $V$-ISS / ISS
  • Remark 2.4
  • Proposition 2.5: $V$-ISS $\Rightarrow$ ISS
  • Definition 2.6: pointwise/LKF-wise ISS LKF
  • Remark 2.7
  • Proposition 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 28 more