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Criterion of the global solvability of regular and singular differential-algebraic equations

Maria Filipkovska

TL;DR

This work develops a comprehensive criterion for the global solvability of semilinear DAEs, addressing both regular and singular pencils via a block-operator pencil and direct-space decompositions. By reducing singular and regular DAEs to coupled ODE/AE systems and imposing consistency conditions on initial data, the authors establish conditions for global existence, Lagrange stability, and boundedness, and they formulate blow-up criteria through Lyapunov-type functions and differential inequalities. The results are unified in a global solvability framework that covers both regular and singular cases and is supported by concrete examples that illustrate global solvability versus finite-time blow-up. The practical impact lies in providing rigorous, checkable criteria for the long-term behavior of DAEs in engineering and physics models, enabling reliable prediction and control of complex dynamical systems.

Abstract

For regular and nonregular (singular) semilinear differential-algebraic equations (DAEs), we prove theorems on the existence and uniqueness of global solutions and on the blow-up of solutions, which allow one to identify the sets of initial values for which the initial value problem has global solutions and/or for which solutions is blow-up in finite time, as well as the regions that the solutions cannot leave. Together these theorems provide a criterion of the global solvability of semilinear DAEs. As a consequence, we obtain conditions for the global boundedness of solutions.

Criterion of the global solvability of regular and singular differential-algebraic equations

TL;DR

This work develops a comprehensive criterion for the global solvability of semilinear DAEs, addressing both regular and singular pencils via a block-operator pencil and direct-space decompositions. By reducing singular and regular DAEs to coupled ODE/AE systems and imposing consistency conditions on initial data, the authors establish conditions for global existence, Lagrange stability, and boundedness, and they formulate blow-up criteria through Lyapunov-type functions and differential inequalities. The results are unified in a global solvability framework that covers both regular and singular cases and is supported by concrete examples that illustrate global solvability versus finite-time blow-up. The practical impact lies in providing rigorous, checkable criteria for the long-term behavior of DAEs in engineering and physics models, enabling reliable prediction and control of complex dynamical systems.

Abstract

For regular and nonregular (singular) semilinear differential-algebraic equations (DAEs), we prove theorems on the existence and uniqueness of global solutions and on the blow-up of solutions, which allow one to identify the sets of initial values for which the initial value problem has global solutions and/or for which solutions is blow-up in finite time, as well as the regions that the solutions cannot leave. Together these theorems provide a criterion of the global solvability of semilinear DAEs. As a consequence, we obtain conditions for the global boundedness of solutions.
Paper Structure (23 sections, 21 theorems, 113 equations)

This paper contains 23 sections, 21 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Problem statement. Preliminaries
  3. Reduction of a singular (nonregular) DAE to a system of ordinary differential and algebraic equations
  4. Reduction of a regular semilinear DAE to a system of ordinary differential and algebraic equations
  5. Consistency condition
  6. Global solvability
  7. Global solvability of regular semilinear DAEs
  8. Global solvability of singular (nonregular) semilinear DAEs
  9. The blow-up of solutions
  10. The blow-up of solutions (Lagrange instability) of regular semilinear DAEs
  11. The blow-up of solutions (Lagrange instability) of singular semilinear DAEs
  12. The criterion of global solvability
  13. The criterion of global solvability of regular semilinear DAEs
  14. The criterion of global solvability of singular semilinear DAEs
  15. Examples
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $f\in C({\mathcal{T}}\times D,{{\mathbb R}^n})$, where $D\subseteq {{\mathbb R}^n}$ is some open set and ${\mathcal{T}}=[t_+,\infty)\subseteq [0,\infty)$, and let the operator pencil $\lambda A+B$ be a regular pencil of index not higher than 1. Assume that there exists an open set $M_1\subseteq Then there exists a unique global (i.e., on $[t_0,\infty)$) solution of the IVP DAE, ini for each i

Theorems & Definitions (42)

  • Theorem 3.1
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 3.2
  • Remark 3.1
  • proof : The proof of Theorem \ref{['Th_GlobReg-LipschObl']}
  • Lemma 3.1
  • proof : The proof of Corollary \ref{['Coroll-GlobReg1']}
  • Lemma 3.2
  • ...and 32 more