  This paper studies the contraction property of time-varying
differential-algebraic equation (DAE) systems by embedding them to
higher-dimension ordinary differential equation (ODE) systems. The first result
pertains to the equivalence of the contraction of a DAE system and the uniform
global exponential stability (UGES) of its variational DAE system. Such
equivalence inherits the well-known property of contracting ODE systems on a
specific manifold. Subsequently, we construct an auxiliary ODE system from a
DAE system whose trajectories encapsulate those of the corresponding
variational DAE system. Using the auxiliary ODE system, a sufficient condition
for contraction of the time-varying DAE system is established by using matrix
measure which allows us to estimate an lower bound on the parameters of the
auxiliary system. Finally, we apply the results to analyze the stability of
time-invariant DAE systems, and to design observers for time-varying ODE
systems.
