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Existence and qualitative behavior of solutions of abstract differential-algebraic equations

Maria Filipkovska

TL;DR

This work develops a comprehensive framework for abstract differential-algebraic equations (ADAEs) in Banach spaces, allowing ADAEs of arbitrarily high index to be analyzed via a reduction to explicit differential and algebraic subsystems using regular pencils $P(\lambda)=\lambda A+B$ and associated space decompositions. Central to the approach are nonlocal implicit function theorems that guarantee existence, uniqueness, and regularity of implicit components $\eta(t,x)$, enabling global solvability and precise blow-up/boundedness characterizations through Lyapunov-type functionals. The authors derive two complementary strategies (a first approach leveraging an implicit definition of the algebraic block, and a second approach handling more general right-hand sides) and tailor the results to index-1 and index-2 pencils, including detailed canonical decompositions and specialized theorems. The findings provide rigorous criteria for maximal intervals of existence, global solvability, and Lagrange stability/instability, with broad applicability to control, mechanics, robotics, kinetics, and network models that generate ADAEs or PDAEs. Overall, the paper advances the theory of semilinear ADAEs by extending solvability and qualitative behavior analysis to high-index pencils and by offering constructive decomposition tools for converting ADAEs into tractable explicit systems.

Abstract

Abstract differential-algebraic equations (ADAEs) of a semilinear type are studied. Theorems on the existence and uniqueness of solutions and the maximal interval of existence, on the global solvability of the ADAEs, the boundedness of solutions and the blow-up of solutions are presented. Previously, an ADAE is reduced to a system of explicit differential equations and algebraic equations by using projectors. The number of equations of the system depends on the index of the characteristic pencil of the ADAE. We consider the pencil of an arbitrarily high index.

Existence and qualitative behavior of solutions of abstract differential-algebraic equations

TL;DR

This work develops a comprehensive framework for abstract differential-algebraic equations (ADAEs) in Banach spaces, allowing ADAEs of arbitrarily high index to be analyzed via a reduction to explicit differential and algebraic subsystems using regular pencils and associated space decompositions. Central to the approach are nonlocal implicit function theorems that guarantee existence, uniqueness, and regularity of implicit components , enabling global solvability and precise blow-up/boundedness characterizations through Lyapunov-type functionals. The authors derive two complementary strategies (a first approach leveraging an implicit definition of the algebraic block, and a second approach handling more general right-hand sides) and tailor the results to index-1 and index-2 pencils, including detailed canonical decompositions and specialized theorems. The findings provide rigorous criteria for maximal intervals of existence, global solvability, and Lagrange stability/instability, with broad applicability to control, mechanics, robotics, kinetics, and network models that generate ADAEs or PDAEs. Overall, the paper advances the theory of semilinear ADAEs by extending solvability and qualitative behavior analysis to high-index pencils and by offering constructive decomposition tools for converting ADAEs into tractable explicit systems.

Abstract

Abstract differential-algebraic equations (ADAEs) of a semilinear type are studied. Theorems on the existence and uniqueness of solutions and the maximal interval of existence, on the global solvability of the ADAEs, the boundedness of solutions and the blow-up of solutions are presented. Previously, an ADAE is reduced to a system of explicit differential equations and algebraic equations by using projectors. The number of equations of the system depends on the index of the characteristic pencil of the ADAE. We consider the pencil of an arbitrarily high index.

Paper Structure

This paper contains 14 sections, 29 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. The paper has the following structure.
  3. Notation used throughout the paper:
  4. Definitions used throughout the paper.
  5. Index of a regular pencil of operators, direct decompositions of spaces and the associated projectors
  6. Index of the pencil
  7. Direct decompositions of the operator domains and spaces and the associated projectors
  8. Nonlocal (global) implicit function theorems
  9. Global solvability, Lagrange stability and instability of the ADAEs
  10. First approach
  11. Second approach
  12. Cases of the characteristic pencil of index 1 and 2
  13. The index-2 case.
  14. The index-1 case.

Key Result

Theorem 3.1

Let $F\in C({\mathcal{T}}\times D_X\times D_Y,Z)$ and let there exist an open set $M_X\subseteq D_X$ and a set $M_Y\subseteq D_Y$ such that the following holds: Then equation F has a unique solution $y=\eta(t,x)$, i.e., $F(t,x,\eta(t,x))=0$, for each $(t,x)\in {\mathcal{T}}\times M_X$ and the function $\eta\in C({\mathcal{T}}\times M_X,M_Y)$ satisfies locally a Lipschitz condition with respect to

Theorems & Definitions (62)

  • Definition 2.1
  • Theorem 3.1: existence and uniqueness of a nonlocal implicit function
  • proof
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['Lem-ImplicitFunc']}
  • Theorem 3.2: existence and uniqueness of a nonlocal implicit function
  • proof
  • Corollary 3.1
  • proof
  • Remark 3.1
  • ...and 52 more