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