Descriptor Systems
For many technical systems, the modeling leads in a very natural way to combinations of ordinary differential equations and algebraic constraints. The differential equations arise, for example, from the current-voltage relations of capacitances and inductances or from the equations of motion in mechanical systems. Algebraic constraints can be, among others, Kirchhoff’s laws or holonomic or non-holonomic constraints. These are called descriptor systems or differential-algebraic equations (DAEs). From a mathematical point of view, these are implicit ordinary differential equations.
Linear time-invariant descriptor systems can be written in the form
\[E\dot{x}=Ax+Bu\]with suitable matrices $E$,$A$,$B$.
Röbenack, K.: Beitrag zur Analyse von Deskriptorsystemen.
Shaker-Verlag, Aachen, 1999, ISBN 978-3-8265-6795-7.