Impulse-free jump solutions of nonlinear differential-algebraic equations
type: Item
instance of: scholarly article
title: Impulse-free jump solutions of nonlinear differential-algebraic equations
author: Yahao Chen
author: Stephan Trenn
published in: Nonlinear Analysis. Hybrid Systems
publication date: 2022-10-14T00:00:00Z
review text: Real nonlinear DAEs like \[ E(x)\dot{x} = F(x) \] \(E(x)\) being a matrix), are known to feature impasse points, when \(\det E(x_0)=0\). Here the authors focus on how define jump solutions with no Dirac impulsion, from a non-consistent initial value. It allows to model transient situations, e.g. for nonlinear electrical circuits. A consistent initial value \(x_0\) is such that the DAE has a \(C^1\) solution at its location. A jump solution is a trajectory \(\tau \mapsto J(\tau)\) defined on a bounded interval \([0,a]\) from an inconsistent value \(J(0)=x_1\) to a consistent one \(J(a)=x_0\) and it is said impulse-free when for any \(\tau\): \[ E(J(\tau))\frac{dJ(\tau)}{d\tau} = 0. \] This way of defining jump solutions is also coordinate-free. The main tools used by the authors are first the geometric approach of nonlinear DAEs (see 90's works of Rabier and Rheinboldt or Reich), then the use of a singularly perturbed associated system, to obtain the impulse-free jump solution as a limit. The main theorems are obtained for geometric index-1 DAEs: \begin{itemize} \item Theorem \(4.6\): existence and uniqueness of an impulse-free solution, in the neighborhood of a consistent point, upon specific regularity conditions. \item Theorem \(5.3\): existence and stability of an impulse-free jump solution as a limit of the solutions of the singularly perturbed system. \end{itemize} The paper is clear with non trivial examples from electricial circuits, and interesting considerations on the numerical validation of the results (in particular, Matlab is used to compute consistent initial values).
reviewed by: Gabriel Thomas
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 7601765
zbMATH Keywords: nonlinear differential-algebraic equations
zbMATH Keywords: discontinue solutions
zbMATH Keywords: inconsistent initial values
zbMATH Keywords: impulse-free jumps
zbMATH Keywords: geometric methods
zbMATH Keywords: singular perturbed systems
zbMATH Keywords: Matlab
describes a project that uses: Ode15s
describes a project that uses: Simulink
MaRDI profile type: Publication
full work available at URL: j.nahs.2022.101238
OpenAlex ID: W4284969524
Wikidata QID: Q115342771
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DOI: 10.1016/J.NAHS.2022.101238
description: scientific article; zbMATH DE number 7601765