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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:250

Article

Dai, Xiongping

关键词: Linear switched system;    Stability;    Recurrence;    Rotation number;    Switching indicator;    Exponential stabilizability;    Quasi-additive ergodic theorem;   

DOI  :  10.1016/j.jde.2011.01.029

来源: Elsevier

【 摘 要 】

Let {A(I), ... , A(K)) subset of C-dxd be arbitrary K matrices, where K and d both >= 2. For any 0 < Delta < infinity, we denote by L-Delta(pc)(R+. K) the set of all switching sequences u = (lambda., t.): N -> (1, ... , K} x R+ satisfying t(j) - t(j-i) <= Delta and 0=: t(0) < t(1) < ... < t(j-1) < t(j) < ... with t(j) -> +infinity Differently from the classical weak-* topology and L-1-norm, we equip L-Delta(pc)(R+. K) with the topology so that the one-sided Markov-type shift v(+) :L-Delta(pc)(R+, K) L-Delta(pc)(R+, K), defined by u = (lambda(j), t(j))(j=1)(+infinity) -> v(+)(u) = (lambda(j+1), t(j+1) - t(1))(j=1)(+infinity), is continuous, which is different from and simpler than the classical continuous-time translation. We study the stability of the linear switched dynamics (A): <(x)over dot>(t) = A(u(t))x(t), x(0) is an element of C-d and t> 0 where u(t) lambda(j) if t(j-1) < t <= t(j), for any u is an element of L-Delta(pc)(R+, K). By introducing the concept weakly Birkhoff recurrent switching signal, we show that if, under some norm parallel to . parallel to, the principal matrix Phi(u)(t) of (A) satisfies parallel to Phi(u)(t)parallel to <= 1 for all u is an element of L-Delta(pc)(R+, K) and t > 0, then for any 0+-ergodic probability P on u is an element of L-Delta(pc)(R+, K). either lim 1/j log parallel to Phi(u)(t(j))parallel to < 0 for P-a.s. u = (lambda(j), t(j))(j=1)(+infinity); j ->infinity or parallel to Phi(vj+(u)) (t(j+k) - t(j))parallel to = 1 (sic), j >= 0 for P-a.s. u = (lambda(j), t(j))(j=1)(+infinity). 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. (C) 2011 Elsevier Inc. All rights reserved.

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