Backward Invariance for Linear Differential Algebraic Equations

Differential-algebraic equations (DAEs) are a widespread dynamical model that describes continuously evolving quantities defined with differential equations, subject to constraints expressed through algebraic relationships. As such, DAEs arise in many fields ranging from physics, chemistry, and engineering. In this paper we focus on linear DAEs, and develop a theory for their minimization up to an equivalence relation. We present backward invariance, which relates DAE variables that have equal solutions at all time points (thus requiring them to start with equal initial conditions) and extends the line of research on backward-type bisimulations developed for Markov chains and ordinary differential equations. We apply our results to the electrical engineering domain, showing that backward invariance can explain symmetries in certain networks as well as analyze DAEs which could not be originally treated due to their size.