Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let be arbitrary K matrices, where K and d both 2. For any 0<Δ<∞, we denote by the set of all switching sequences satisfying tj−tj−1 Δ and Differently from the classical weak- topology and L1-norm, we equip with the topology so that the “one-sided Markov-type shift” , defined by is continuous, which is different from and simpler than the classical continuous-time “translation”. We study the stability of the linear switched dynamics (A): where if tj−10, then for any +-ergodic probability on , either or Some applications are presented, including: (i) equivalence of various stabilities; (ii) almost sure exponential stability of periodically switched stable systems; (iii) partial stability; and (iv) how to approach arbitrarily the stable manifold by that of periodically switched signals and how to select a stable switching signal for any initial data.