A characterization of the class of structurally stable probabilistic automata II. Continuous-time case

Abstract

Hayashi et al. (1971) have investigated the problem of strong stability for a continuous time probabilistic automaton introduced by Knast (1969) as a generalization of discrete-time probabilistic automaton to the continuous-time case. In this paper, from the viewpoint of applying the conceptual framework of structural stability theory for dynamical systems over a compact differentiable manifold to the analysis of these problems for discrete-state stochastic systems, we attempt to extend their local (structural) stability theory for continuous-time probabilistic automata to a global one in such a sense as explained later. As a result of this investigation, it is revealed that a given continuous-time probabilistic automaton is structurally stable if and only if it is ergodic. Furthermore, we demonstrate that ‘ the class of all structurally stable continuous-time probabilistic automata is open, dense, convex and connected in the metric space of all continuous-time probabilistic automata over the fixed state ...