Asymptotic Properties of Solutions to Set Dynamical Systems

Dynamical systems with trajectories given by sequences of sets are studied. For this class of generalized systems, notions of solution, invariance, and omega limit sets are defined. The structural properties of omega limit sets are revealed. In particular, it is shown that for complete and bounded solutions, the omega limit set of a bounded and complete solution is nonempty, compact, and invariant. Lyapunov-like conditions to locate omega limit sets are also derived. Tools from the theory of set convergence are conveniently used to prove the results. The findings are illustrated in several examples and applications, including the computation of reachable sets and forward invariant sets, as well as in propagation of uncertainty