Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

TL;DR

This paper introduces a novel numerical approach using real algebraic geometry to analyze and solve differential-algebraic equations that are structurally unamenable for certain parameters, overcoming limitations of existing methods.

Contribution

It presents an embedding method and a witness point technique to perform global structural analysis on polynomial DAE systems with parameter-dependent unamenability.

Findings

Successfully constructs equivalent systems with full-rank Jacobians.

Detects degeneration across all constraint components.

Provides a comprehensive numerical structural analysis framework.

Abstract

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for polynomial systems of differential-algebraic equations, numerical methods are given to solve such cases using numerical real algebraic geometry. First, we propose an embedding method that for a given real analytic system constructs an equivalent system with a full-rank Jacobian matrix. Secondly, we introduce a witness point method, which can help to detect degeneration on all components of constraints of such systems. Thirdly, the two methods above lead to a numerical global structural analysis method for structurally unamenable differential-algebraic equations on all components of constraints.