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Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks

Maria Filipkovska

TL;DR

The work develops a qualitative theory for nonregular semilinear DAEs by exploiting a block decomposition of singular pencils to separate singular and regular dynamics, enabling a reduction to an ODE/AE system. It provides comprehensive criteria for global solvability, Lagrange stability and instability, and dissipativity, supplemented by weaker-condition variants and constructive guidance for choosing Lyapunov-type functions. The results are applied to isothermal gas networks, deriving DAEs from isothermal Euler equations and illustrating network-level dynamics, boundary conditions, and isothermal pipeline models. The framework supports concrete isothermal network analysis and offers a pathway to analyze blow-up and ultimate boundedness in descriptor systems with singular pencils, with implications for control and safety in gas transportation systems.

Abstract

The conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and the conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations are obtained. An example demonstrating the application of the obtained results is considered. Isothermal models of gas networks are proposed as applications.

Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks

TL;DR

The work develops a qualitative theory for nonregular semilinear DAEs by exploiting a block decomposition of singular pencils to separate singular and regular dynamics, enabling a reduction to an ODE/AE system. It provides comprehensive criteria for global solvability, Lagrange stability and instability, and dissipativity, supplemented by weaker-condition variants and constructive guidance for choosing Lyapunov-type functions. The results are applied to isothermal gas networks, deriving DAEs from isothermal Euler equations and illustrating network-level dynamics, boundary conditions, and isothermal pipeline models. The framework supports concrete isothermal network analysis and offers a pathway to analyze blow-up and ultimate boundedness in descriptor systems with singular pencils, with implications for control and safety in gas transportation systems.

Abstract

The conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and the conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations are obtained. An example demonstrating the application of the obtained results is considered. Isothermal models of gas networks are proposed as applications.
Paper Structure (15 sections, 12 theorems, 138 equations)

This paper contains 15 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Problem statement, definitions and preliminary constructions
  3. Remarks on differential inequalities
  4. Block form of a singular pencil, the corresponding direct decompositions of spaces and projectors
  5. Reduction of a DAE with the singular characteristic pencil to a system of ordinary differential and algebraic equations
  6. Global solvability of singular (nonregular) semilinear DAEs
  7. Lagrange stability of singular semilinear DAEs
  8. Lagrange instability of singular semilinear DAEs (the blow-up of solutions in finite time)
  9. Dissipativity (ultimate boundedness) of singular semilinear DAEs
  10. Replacement some conditions of the theorems by weaker ones
  11. On the choice of the functions $\chi$ and $V$ when checking the conditions of proved theorems
  12. Isothermal models of gas networks in the form of DAEs
  13. A model of a gas flow for a single pipe (in the isothermal case)
  14. A model of a gas network (in the isothermal case)
  15. Analysis of a singular (nonregular) semilinear DAE with the characteristic pencil of the rank $\mathop{\mathrm{rank}}(\lambda A+B) <n,m$

Key Result

Proposition 2.1

For the operators $A,\, B\colon{{\mathbb R}^n}\to{{\mathbb R}^m}$, which form a singular pencil $\lambda A+B$, there exist the decompositions of the spaces ${{\mathbb R}^n}$, ${{\mathbb R}^m}$ into the direct sums of subspaces (which can always be construct) with respect to which $A$, $B$ have the block structure where $A_s=A|_{X_s},\, B_s=B|_{X_s}\colon X_s\to Y_s$ and $A_r=A|_{X_r},\, B_r=B|_{

Theorems & Definitions (32)

  • Definition 2.1
  • Proposition 2.1: see Fil.singFil.KNU2019
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3: Fil.KNU2019
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Theorem 3.2
  • ...and 22 more