Relative invariance for monoid actions

Let $G$ be a topological group and $S\subset G$ a submonoid of $G$ acting on the topological space $M$. Let $J$ be a subset of $M$. Our purpose here is to study the subsets of $M$ which correspond, under the action of $S$, to the relative (with respect to $J$) invariant control sets for control systems [4]. The relation $x\thicksim y$ if $y\in {\rm cl}(Sx)$ and $x\in {\rm cl}(Sy)$ is an equivalence relation and the classes with respect to this relation with nonempty interior in $M$ are the control sets for the action of $S$. It is given conditions for the existence and uniqueness of relative invariant classes. As it was done for the control sets, we define an order in the classes and relate it to the relative invariant classes. We also show under certain condition that the relative invariant classes are relatively closed in $J$.