A characterization of the class of structurally stable probabilistic automata I. Discrete-time case
Titel:
A characterization of the class of structurally stable probabilistic automata I. Discrete-time case
Auteur:
Komota, Yasuo Kimtjea, Masayuki
Verschenen in:
International journal of systems science
Paginering:
Jaargang 9 (1978) nr. 4 pagina's 369-394
Jaar:
1978-04-01
Inhoud:
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 connected in the metric space of all probabilistic automata defined on the fixed state space, input space and final state set), This open, dense, convex and connected theorem may be analogous in some sense to Peixoto's open-dense theorem (Peixoto 1962) for the class of structurally stable dynamical systems defined on a compact differentiable manifold.
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