  We study asymptotic stability of continuous-time systems with mode-dependent
guaranteed dwell time. These systems are reformulated as special cases of a
general class of mixed (discrete-continuous) linear switching systems on
graphs, in which some modes correspond to discrete actions and some others
correspond to continuous-time evolutions. Each discrete action has its own
positive weight which accounts for its time-duration. We develop a theory of
stability for the mixed systems; in particular, we prove the existence of an
invariant Lyapunov norm for mixed systems on graphs and study its structure in
various cases, including discrete-time systems for which discrete actions have
inhomogeneous time durations. This allows us to adapt recent methods for the
joint spectral radius computation (Gripenberg's algorithm and the Invariant
Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on
graphs.
