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Well-posedness and stability of boundary delay equations

Yassine El Gantouh, Yang Liu

TL;DR

This work introduces boundary delay equations within a unified framework for linear time-invariant systems with delayed boundary conditions, proving existence, uniqueness, and positivity of solutions and deriving spectral conditions for exponential stability. By reformulating the problem on a product space and applying a positive Weiss-Staffans perturbation theorem, the authors establish well-posedness and positivity of the augmented generator and provide explicit stability criteria. The results extend to necessary and sufficient conditions for exponential stability in positive hyperbolic systems with time-delayed boundary feedback, highlighting the role of positivity in simplifying spectral stability analysis. This approach broadens the applicability of domain-perturbation and semigroup methods to boundary-delayed PDEs and related systems.

Abstract

In this paper, we introduce the notion of boundary delay equations, establishing a unified framework for analyzing linear time-invariant systems with pure time-delayed boundary conditions. We establish mild sufficient conditions for the existence, uniqueness, and positivity of solutions. Furthermore, we derive spectral criteria for exponential stability. The conditions on the perturbation generalize well-known criteria for the generation of domain perturbations of positive semigroup generators. As an application, we present necessary and sufficient conditions for the exponential stability of positive hyperbolic systems with time-delayed boundary conditions.

Well-posedness and stability of boundary delay equations

TL;DR

This work introduces boundary delay equations within a unified framework for linear time-invariant systems with delayed boundary conditions, proving existence, uniqueness, and positivity of solutions and deriving spectral conditions for exponential stability. By reformulating the problem on a product space and applying a positive Weiss-Staffans perturbation theorem, the authors establish well-posedness and positivity of the augmented generator and provide explicit stability criteria. The results extend to necessary and sufficient conditions for exponential stability in positive hyperbolic systems with time-delayed boundary feedback, highlighting the role of positivity in simplifying spectral stability analysis. This approach broadens the applicability of domain-perturbation and semigroup methods to boundary-delayed PDEs and related systems.

Abstract

In this paper, we introduce the notion of boundary delay equations, establishing a unified framework for analyzing linear time-invariant systems with pure time-delayed boundary conditions. We establish mild sufficient conditions for the existence, uniqueness, and positivity of solutions. Furthermore, we derive spectral criteria for exponential stability. The conditions on the perturbation generalize well-known criteria for the generation of domain perturbations of positive semigroup generators. As an application, we present necessary and sufficient conditions for the exponential stability of positive hyperbolic systems with time-delayed boundary conditions.

Paper Structure

This paper contains 15 sections, 19 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Notation and Problem Statement
  3. Notation and Terminology
  4. Positive C$_0$-semigroups and Extrapolation Spaces
  5. Problem Statement
  6. Preliminary results on linear time-invariant infinite-dimensional positive systems
  7. Admissible positive observation operators
  8. Admissible positive control operators
  9. A Weiss-Staffans perturbation theorem for positive systems
  10. Main results
  11. Well-posedness
  12. Stability
  13. Application
  14. Conclusion
  15. Appendix

Key Result

Proposition 3.1

Let $\mathsf{C}\in\mathcal{L}_+(D(\mathsf{A}),\mathsf{U})$ and $p\ge 1$ be a real number. If the estimate S2.7 holds for all $x\in D_+(\mathsf{A})$ and for some $\alpha > 0$, then $(\mathsf{C},\mathsf{A})$ is $L^p$-admissible.

Theorems & Definitions (57)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.1
  • Remark 3.2
  • Definition 3.2
  • Proposition 3.2
  • Definition 3.3
  • ...and 47 more