On local solutions to time-varying linear DAEs

TL;DR

This paper develops a framework for local solutions to time-varying linear DAEs with meromorphic coefficients, linking local and global controllability through algebraic characterizations.

Contribution

It introduces a sheaf-based approach for local solutions, establishes their equivalence to global controllability, and analyzes extension conditions for local solutions.

Findings

Local solutions form a sheaf on compact intervals.

Controllability is characterized algebraically via Teichmüller-Nakayama form.

Conditions for extending local solutions are identified.

Abstract

This paper presents a framework for local solutions to time-varying linear differential-algebraic equations (DAEs) with real meromorphic coefficients. The local solutions on compact intervals form a sheaf. This permits a simple definition of controllability in the sense of Jan C. Willems. We prove that this notion is equivalent to the established global notion by giving an algebraic characterization based on the Teichm\"uller-Nakayama form. Finally, we study conditions under which local solutions admit extension, which is necessary for controllability.