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.