A characterization of the class of structurally stable probabilistic automata II. Continuous-time case
Yasuo Komota, Masayuki Kimura · International Journal of Systems Science · 1978
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 space, input space and final state sot’. This result may be analogous in some sense to Peixoto's (1962) open-dense theorem for the class of all structurally stable dynamical systems over a compact differentiable manifold and also corresponds to the open, dense, convex and connected theorem, obtained in the previous paper (Komota and Kimura 1978), for the class of all structurally stable discrete-time probabilistic automata. Finally, we introduce and study the new concepts of structural stabilities for a continuous-time probabilistic automaton under ceaseless structural perturbations. The results, almost similar to those mentioned above, are obtained with respect to structural stabilities under ceaseless structural perturbations.