Hybrid systems in linear spaces: Attracting sets and periodicity

Consider an asymptotically stable linear semigroup SS of class C0C0, acting on a Banach space XX. By shifting the action of SS we obtain an affine semigroup with an arbitrary point p∈Xp∈X as a sink. Selecting a specific set PP 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 of 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.