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Abstract boundary delay systems and application to network flow

András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

TL;DR

The paper addresses well posedness and positivity for delayed transport on networks modeled as metric graphs. It reformulates the problem as an abstract boundary delay equation and uses Staffans-Weiss perturbation to obtain a strongly continuous semigroup generated by $\mathcal{A}_{P,L}$, together with resolvent descriptions and positivity criteria. Spectral analysis introduces the condition $1\in\rho(\Delta(\lambda))$ that characterizes the spectrum, and positivity is established under positivity of the delay operators and suitable large parameter limits. The theory is then applied to a network flow with delays, proving well posedness and positivity and providing explicit state-trajectory representations. These results advance the mathematical understanding of delay effects in network transport and support applications to material flow in interconnected systems.

Abstract

This paper investigates the well-posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well-posedness is proved by using the Staffans-Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

Abstract boundary delay systems and application to network flow

TL;DR

The paper addresses well posedness and positivity for delayed transport on networks modeled as metric graphs. It reformulates the problem as an abstract boundary delay equation and uses Staffans-Weiss perturbation to obtain a strongly continuous semigroup generated by , together with resolvent descriptions and positivity criteria. Spectral analysis introduces the condition that characterizes the spectrum, and positivity is established under positivity of the delay operators and suitable large parameter limits. The theory is then applied to a network flow with delays, proving well posedness and positivity and providing explicit state-trajectory representations. These results advance the mathematical understanding of delay effects in network transport and support applications to material flow in interconnected systems.

Abstract

This paper investigates the well-posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well-posedness is proved by using the Staffans-Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

Paper Structure

This paper contains 5 sections, 6 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Infinite-dimensional closed-loop systems
  3. Abstract boundary delay equations
  4. Spectral theory and positivity
  5. Application: Flow in network with delays

Key Result

Theorem 2.6

A well-posed triple $(A,B,C)$ is regular if and only if $\mathnormal{D}_\lambda v=R(\lambda,\mathnormal{A}_{-1})\mathnormal{B} v\in D(\mathnormal{C}_\Lambda)$ for all $v\in \mathnormal{U}$ and some $\lambda\in\rho(\mathnormal{A})$.

Theorems & Definitions (16)

  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.8
  • proof
  • Example 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 6 more