Two-scale difference equations II. local regularity, infinite products of matrices and fractals
SIAM Journal on Mathematical Analysis
An Elementary Counterexample to the Finiteness Conjecture
SIAM Journal on Matrix Analysis and Applications
A Converse Lyapunov Theorem for Linear Parameter-Varying and Linear Switching Systems
SIAM Journal on Control and Optimization
Structure of extremal trajectories of discrete linear systems and the finiteness conjecture
Automation and Remote Control
Optimal Disturbance Attenuation for Discrete-Time Switched and Markovian Jump Linear Systems
SIAM Journal on Control and Optimization
Almost Sure Stability of Discrete-Time Switched Linear Systems: A Topological Point of View
SIAM Journal on Control and Optimization
Counterexamples to the Complex Polytope Extremality Conjecture
SIAM Journal on Matrix Analysis and Applications
Uniform stabilization of discrete-time switched and Markovian jump linear systems
Automatica (Journal of IFAC)
Analysis and synthesis of switched linear control systems
Automatica (Journal of IFAC)
On codes that avoid specified differences
IEEE Transactions on Information Theory
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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@?C^n^x^n,1@?k@?K, be arbitrarily given K matrices and @S"K^+={(k"j)"j"="1^+^~|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"k"""j...S"k"""1x"0,j=1 and x"0@?C^n where (k"j)"j"="1^+^~@?@S"K^+ can be an arbitrary switching sequence. For a probability row-vector p=(p"1,...,p"K)@?R^K and an irreducible Markov transition matrix P@?R^K^x^K with pP=p, we denote by @m"p","P the Markovian probability on @S"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 @m"p","P-almost surely; (ii) the set of stable switching sequences (k"j)"j"="1^+^~@?@S"K^+ has the same Hausdorff dimension as @S"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.