Control sets and their boundaries under parameter variation

Control sets, i.e., maximal sets of approximate controllability, are described for systems with parameter-dependent control range. If the so-called inner-pair condition is satisfied, it is known that generically they change continuously under parameter variation while mergers of control sets produce discontinuous transitions. The inner-pair condition is proved for a class of control-affine systems on RN. Furthermore, continuity results for exit and entrance boundaries of control sets are given both for one control set that changes continuously and for two merging control sets. The results are illustrated by means of the controlled escape equation.