Topological properties of prolongations and stable sets for semigroup actions (Q1998814)

The authors investigate topological properties of Lyapunov stable sets for semigroup actions on Tychonoff spaces. Under certain conditions, they show that: (1) A compact set is stable if and only if its components are stable; (2) A compact set is asymptotically stable if and only if it has a finite number of components, each of which is asymptotically stable. Moreover, they characterize the stability of a compact set by means of \(\omega\)-limit sets, prolongations, prolongational limit sets and the existence of a fundamental system of neighborhoods of it. Finally, they present some applications to continuous flows and affine control systems.