Local and global canonical forms for differential-algebraic equations with symmetries

TL;DR

This paper develops local and global canonical forms for linear time-varying differential-algebraic equations with symmetries, aiding in the classification of their geometric flow properties and applications to physical systems.

Contribution

It introduces new canonical forms for symmetric DAE systems and applies them to analyze the geometric nature of their flows, including symplectic and orthogonal properties.

Findings

Canonical forms classify flow geometry as symplectic or orthogonal.

Applications to circuit simulation and incompressible flow demonstrate practical relevance.

Results facilitate analysis of dissipative Hamiltonian systems.

Abstract

Linear time-varying differential-algebraic equations with symmetries are studied. The structures that we address are self-adjoint and skew-adjoint systems. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the analysis of dissipative Hamiltonian systems arising from circuit simulation and incompressible flow.