Let X denote a locally compact metric space and ϕ: X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X,ϕ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we generalize Zadeh’s original extension principle. In this paper we study basic properties, such as the continuity of so-called g-fuzzifications. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly convergent maps on X induces a uniformly convegent sequence of continuous maps on the space of fuzzy sets, and (ii) a conjugacy (a semi-conjugacy, resp.) between two discrete dynamical systems can be extended to a conjugacy (a semi-conjugacy, resp.) between fuzzified dynamical systems. Moreover, at the end of this paper we show that there are connections between g-fuzzifications and crisp dynamical systems via set-valued dynamical systems and skew-product (triangular) maps. Throughout this paper we consider different topological structures in the space of fuzzy sets; namely, the sendograph, endograph and levelwise topologies.

We prove that the space of fuzzy compacta in a Peano continuum is homeomorphic to the Hilbert space 2. As a corollary we find that the spaces of fuzzy compacta in Rn and 2 are also homeomorphic to Hilbert space. © 2011 Elsevier B.V. All rights reserved.

Abstract: This work deals with the problems of the Weakly Structurable Continuous Dynamic System (WSCDS) optimal control and briefly discuss the results developed by G. Sirbiladze [17]. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control processes, for which we use R. Bellman’s optimality principle and take into consideration the gain-loss fuzzy process. A separate consideration is given to the case where an extremal fuzzy control process acting on the WSCDS (1) depends and (2) does not depend on an WSCDS state. Applying Bellman’s optimality principle and assuming that the gain-loss process exists for the WSCDS, a variant of the fuzzy integral representation of an optimal control is given for the WSCDS. This variant employs the instrument of extended extremal fuzzy composition measures constructed in [16]. The questions of defining a fuzzy gain relation for the WSCDS are considered, taking into account the avail-able expert knowledge on the WSCDS subject-matter. An example of constructing of the WSCDS optimal control is presented.