On local solutions to time-varying linear systems

Abstract

This paper develops an intrinsic framework for local solutions to time-varying linear systems with real meromorphic coefficients. Local solutions on compact intervals naturally form a sheaf of vector spaces. We introduce regularity triples to distinguish regularity across singularities, one-sided regularity at singularities, and regularity away from singularities. The local formulation isolates a continuation property implicit in established global frameworks and yields a simple definition of behavioural controllability. We establish an algebraically verifiable sufficient condition for continuation, prove its necessity for controllability, and derive an algebraic characterization of controllability based on the Teichmüller-Nakayama form, recovering and slightly generalizing established results.

arXiv comment28 pages, 1 figure; substantially revised exposition and updated references