This report addresses a class of nonsmooth dynamical systems which model the motion of state trajectories that are constrained to evolve in time-varying non-convex sets using the framework of differential inclusions. Using Lyapunov-based analysis, sufficient conditions are proposed for local stability in such systems, while specifying the basin of attraction. If the sets governing the motion of state trajectories are moving with bounded variation, then the resulting state trajectories are also of bounded variation, and unlike the convex case, the stability conditions depend on the size of jumps allowed in the sets. Based on the stability analysis, a Luenberger-like observer is proposed which is shown to converge asymptotically to the actual state provided the initial value of the state estimation error is small enough. In addition, a semi-global practically stable observer, based on the high-gain approach, is designed to reduce the state estimation error to the desired accuracy in finite time which is then combined with the locally convergent observer to obtain semi-global asymptotically convergent state estimates.