Chain control sets for semigroup actions
Let S be a semigroup of self-maps of a state space M. The authors' purpose is to study the subsets of M which correspond, under the action of S, to the chain control sets for control systems. We consider only the specific case where S is a subsemigroup of a Lie group G which acts transitively on M. The formal definition of the chain control sets for S requires a family F of subsets of S, and when specialized to a control semigroup recovers its original form. It is shown, under broad conditions, that a chain control set is the intersection of control sets for the semigroups generated by the neighbourhoods of the subsets in F. The analysis of these control sets turns out to be a useful tool for understanding specific properties of the semigroups in semisimple Lie groups. For instance, a finite group W(S) can be built in connection with a semigroup S \(\in\) G with nonempty interior in G. The group W(S) reflects geometrical properties of the control sets on the boundaries of G. Also, W(S) describes exactly the type of diagonalizable elements which are inside the interior of the semigroup.
scientific article; zbMATH DE number 988142