Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory

TL;DR

This paper introduces geometric methods for analyzing nonlinear differential-algebraic equations (DAEs), connecting them with nonlinear control systems, and generalizing classical forms to better understand their structure and solutions.

Contribution

It develops a geometric reduction approach for nonlinear DAEs, proposes an explicitation procedure linking DAEs to control systems, and derives nonlinear generalizations of the Weierstrass form.

Findings

A new geometric reduction method for nonlinear DAEs

An explicitation procedure connecting DAEs with control systems

Nonlinear generalizations of the Weierstrass form

Abstract

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word "external" means that we consider a DAE (locally) everywhere and "internal" means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.