On local solutions to time-varying linear DAEs

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 Teichmüller-Nakayama form. Finally, we study conditions under which local solutions admit extension, which is necessary for controllability.