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 Sk∈Cn×n,1≤k≤KSk∈Cn×n,1≤k≤K, be arbitrarily given KK matrices and ΣK+={(kj)j=1+∞∣1≤kj≤K for each j≥1}, where n,K≥2n,K≥2. Then we study the exponential stability of the following discrete-time switched dynamics SS: xj=Skj⋯Sk1x0,j≥1 and x0∈Cn where (kj)j=1+∞∈ΣK+ can be an arbitrary switching sequence.For a probability row-vector p=(p1,…,pK)∈RK and an irreducible Markov transition matrix P∈RK×K with pP=p, we denote by μp,P the Markovian probability on ΣK+ corresponding to (p,P). By using symbolic dynamics and ergodic-theoretic approaches, we show that, if SS possesses the periodically switched stability then, (i) it is exponentially stable μp,P-almost surely; (ii) the set of stable switching sequences (kj)j=1+∞∈ΣK+ has the same Hausdorff dimension as ΣK+. Thus, the periodically switched stability of a discrete-time linear switched dynamics implies that the system is exponentially stable for “almost” all switching sequences.