A characterization of the class of structurally stable probabilistic automata I. Discrete-time case

Abstract

The behaviour of a probabilistic automaton is essentially characterized by products of matrices selected from its finite set of transition stochastic matrices. It is of interest to know under what conditions these matrix products are structurally stable against small perturbations of the entries in the transition matricies. From the viewpoint of applying the framework of structural stability theory for dynamical systems defined on a manifold to the analysis of a structural stability problem for discrete-state stochastic systems, this paper deals with the structural stability problem for probabilistic automata that arises when one considers the effects caused by small perturbations of their transition matrices upon their ergodic properties their output functions and other behaviour, The necessary and sufficient conditions for a given probabilistic automaton to be structurally stable is derived. Furthermore, we prove that ( the class of structurally stable probabilistic automata is open, dense, convex and c...