Controllability and invariance properties of time-periodic systems

In this paper, control sets, i.e. maximal subsets of approximate controllability, are introduced for time-periodic parameter dependent systems with open loop control. It is shown that the time-periodicity causes some special topological properties of control sets that influence their merging and bifurcation process. These properties allow an enhancement of the available set-oriented numerical methods for the approximation of control sets. For the application of these methods, the so-called inner-pair condition must be satisfied. It is shown that this condition holds for a wide class of time-periodic control-affine systems that includes, for instance, coupled excited oscillators if the number of controls equals the degrees of freedom. The application of the methods to the controlled escape equation with periodic driving term where the parameter acts on the control range gives insight into the changing dynamics and different types of mergers of control sets.