Some Aspects of Stability for Semigroup Actions and Control Systems

Abstract

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\), and let \(\mathcal {F}\) be a family of subsets of \(S\). For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\). The point \(x\) is forward \(\mathcal {F}\)-Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \). The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\)-nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \)-prolongational limit set \(J\left( x,\mathcal {F}\right) \). The paper contains applications to control systems.