Invariance properties and convergence of solutions of set dynamical systems are studied. Using a framework for systems with set-valued states, notions of stability and detectability, similar to the existing results for classical dynamical systems, are defined and used to obtain information about the convergence properties of solutions. In particular, it is shown that local stability, detectability, and boundedness can be combined to conclude convergence of set-valued solutions. Under the assumption of bounded solutions and outer semicontinuity of the set-valued maps that define the system's dynamics, invariance properties for set dynamical systems are also presented along with an invariance principle. The invariance principle involves the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results.

In this paper, stability properties for discrete-time dynamical systems with set-valued states are studied. We use previous results on detectability and invariance properties to present an extension of Krasovskii and Lyapunov stability results for set dynamical systems, under the assumption of outer semicontinuity of the set-valued maps that define the system's dynamics. We also propose a formulation for closed-loop control systems with state-feedback, within the framework of set dynamical systems. Examples illustrate the results.