  This study aims at characterizing a reachable set of a hybrid dynamical
system with a lag constraint in the switch control. The setting does not
consider any controllability assumptions and uses a level-set approach. The
approach consists in the introduction of on adequate hybrid optimal control
problem with lag constraints on the switch control whose value function allows
a characterization of the reachable set. The value function is in turn
characterized by a system of quasi-variational inequalities (SQVI). We prove a
comparison principle for the SQVI which shows uniqueness of its solution. A
class of numerical finite differences schemes for solving the system of
inequalities is proposed and the convergence of the numerical solution towards
the value function is studied using the comparison principle. Some numerical
examples illustrating the method are presented. Our study is motivated by an
industrial application, namely, that of range extender electric vehicles. This
class of electric vehicles uses an additional module -- the range extender --
as an extra source of energy in addition to its main source -- a high voltage
battery. The reachability study of this system is used to establish the maximum
range of a simple vehicle model.
