Hybrid systems in linear spaces: Attracting sets and periodicity

Consider an asymptotically stable linear semigroup S of class C0, acting on a Banach space X. By shifting the action of S we obtain an affine semigroup with an arbitrary point p∈X as a sink. Selecting a specific set P of sinks and combining the action of the corresponding semigroups gives rise to a multimodal control system which, after constraining the action of each mode, becomes a hybrid system with switching. We prove the existence of a globally attracting compact set and describe its structure. In the case of the constrained system we use this structure to prove the convergence of ergodic averages-such as the average time between switches-for a certain generic set of solutions. Next we turn to the bimodal case, typical to thermostatic control. We review a series of results where the existence of a periodic solution (with two periodic switches) has been shown. On the other hand, we announce a finite-dimensional example where no such solution exists.