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

Let {A1,…,AK}⊂Cd×d{A1,…,AK}⊂Cd×d be arbitrary K matrices, where K and d   both ⩾2. For any 0<Δ<∞0<Δ<∞, we denote by LΔpc(R+,K) the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R+u=(λ.,t.):N→{1,…,K}×R+ satisfying tj−tj−1⩽Δtj−tj−1⩽Δ and0=:t00 where u(t)≡λju(t)≡λj if tj−10t>0, then for any ϑ+ϑ+-ergodic probability PP on LΔpc(R+,K), eitherlimj→+∞1jlog‖Φu(tj)‖<0for P-a.s. u=(λj,tj)j=1+∞; or‖Φϑ+j(u)(tj+k−tj)‖=1∀k,j⩾0for P-a.s. u=(λj,tj)j=1+∞. 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.