  In this paper, we study the existence and the stability in the sense of
Lyapunov of solutions for\ differential inclusions governed by the normal cone
to a prox-regular set and subject to a Lipschitzian perturbation. We prove that
such, apparently, more general nonsmooth dynamics can be indeed remodelled into
the classical theory of differential inclusions involving maximal monotone
operators. This result is new in the literature and permits us to make use of
the rich and abundant achievements in this class of monotone operators to
derive the desired existence result and stability analysis, as well as the
continuity and differentiability properties of the solutions. This going back
and forth between these two models of differential inclusions is made possible
thanks to a viability result for maximal monotone operators. As an application,
we study a Luenberger-like observer, which is shown to converge exponentially
to the actual state when the initial value of the state's estimation remains in
a neighborhood of the initial value of the original system.
