We present the mathematical theory of general over- and underdetermined
hybrid (switched) systems of differential-algebraic equations
(HDAEs). We give a systematic formulation of HDAEs and discuss existence
and uniqueness of solutions, the numerical computation of the switch
points and how to perform consistent initialization at switch points. We
show how numerical solution methods for DAEs can be adapted for HDAEs
and present a comparison of these methods for the real world example of
simulating an automatic gearbox.