A Fluctuation Theory of Systems by Fuzzy Mapping Concept and Its Applications

Kazuo HORIUCHI  Yasunori ENDO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.11   pp.1728-1735
Publication Date: 1994/11/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Nonlinear Theory and Its Applications)
Category: Fuzzy System--Theory and Applications--
fuzzy mapping,  fixed point theorem,  fluctuation theory,  system analysis,  system design,  

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This paper proposes a methodology for fine evaluation of the uncertain behaviors of systems affected by any fluctuation of internal structures and internal parameters, by the use of a new concept on the fuzzy mapping. For a uniformly convex real Banach space X and Y, a fuzzy mapping G is introduced as the operator by which we can define a bounded closed compact fuzzy set G(x,y) for any (x,y)∈X×Y. An original system is represented by a completely continuous operator f defined on X, for instance, in a form xλ(f(x)) by a continuous operator λ: YX. The nondeterministic fluctuations induced into the original system are represented by a generalized form of the fuzzy mapping equation xGβ (x,f(x))XG(x,f(x))(ζ)β}, in order to give a fine evaluation of the solutions with respect to an arbitrarily–specified β–level. By establishing a useful fixed point theorem, the existence and evaluation problems of the "β–level-likely" solutions are discussed for this fuzzy mapping equaion. The theory developed here for the fluctuation problems is applied to the fine estimation of not only the uncertain behaviors of system–fluctuations but also the validity of system–models and -simulations with uncertain properties.