  This paper considers discontinuous dynamical systems, i.e., systems whose
associated vector field is a discontinuous function of the state. Discontinuous
dynamical systems arise in a large number of applications, including optimal
control, nonsmooth mechanics, and robotic manipulation. Independently of the
particular application, one always faces similar questions when dealing with
discontinuous dynamical systems. The most basic one is the notion of solution.
We begin by introducing the notions of Caratheodory, Filippov and
sample-and-hold solutions, discuss existence and uniqueness results for them,
and examine various examples. We also give specific pointers to other notions
of solution defined in the literature. Once the notion of solution has been
settled, we turn our attention to the analysis of stability of discontinuous
systems. We introduce the concepts of generalized gradient of locally Lipschitz
functions and proximal subdifferential of lower semicontinuous functions.
Building on these notions, we establish monotonic properties of candidate
Lyapunov functions along the solutions. These results are key in providing
suitable generalizations of Lyapunov stability theorems and the LaSalle
Invariance Principle. We illustrate the applicability of these results in a
class of nonsmooth gradient flows.
