On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

Abstract

Consider a non-autonomous continuous-time linear system in which the timedependent matrix determining the dynamics is piecewise constant and takes finitely many values A 1 , . . . , A N . This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices {A 1 , . . . , A N }, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes A 1 , . . . , A N , on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices A 1 , . . . , A N converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.

Performance estimation of switched linear systems via n×(n+2) generalized coordinate transformations

“…There are various results of spectral analysis on the discrete-time switched linear systems based on spectral radius. Interested readers may refer to [42][43][44] for details.…”

Stability of shuffled switched linear systems: A joint spectral radius approach

“…However, the stability condition based on the JSR turns out to be conservative in the case of constrained switching sequences. Some works have adapted the notion of joint spectral radius to switching signals generated by deterministic [11,18,33] or stochastic [15,10] finite state automata. However, none of these constraints allow to deal with shuffled switching signals.…”