Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities

The conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices, has recently been disproved under the guise of the finiteness conjecture. In this paper, we show that this conjecture holds in terms of Markovian probabilities. More specifically, let S-k is an element of C-nxn, 1 <= k <= K, be arbitrarily given K matrices and Sigma(+)(k) = {(k(j))(j=1)(+infinity) vertical bar 1 <= k(j) <= K for each j >= 1}, where n, K >= 2. Then we study the exponential stability of the following discrete-time switched dynamics S: x(j) = S-kj ... S(k1)x(0), j >= 1 and x(0) is an element of C-n where (k(j))(j=1)(+infinity) is an element of Sigma(+)(K) can be an arbitrary switching sequence. For a probability row-vector p = (p(1), ... p(K)) is an element of R-K and an irreducible Markov transition matrix P is an element of R-KxK with pP = p, we denote by mu(p,p) the Markovian probability on Sigma(+)(K) corresponding to (p, P). By using symbolic dynamics and ergodic-theoretic approaches, we show that, if S possesses the periodically switched stability then, (i) it is exponentially stable mu(p),P-almost surely; (ii) the set of stable switching sequences (k(j))(j=1)(+infinity) is an element of Sigma(+)(K) has the same Hausdorff dimension as Sigma(+)(K). Thus, the periodically switched stability of a discrete-time linear switched dynamics implies that the system is exponentially stable for almost all switching sequences. (C) 2011 Elsevier Ltd. All rights reserved.