Theory of fuzzy systems
A systematic theory of ''fuzzy systems'' is proposed, this ''fuzzy system'' being a mathematical model for the evolution, in time, of a ''fuzzy set'' (not well defined set). For any given t, the set x(t) is a fuzzy subset of some state space X, and this subset is given by a function \(\mu_{x(t)}:X\to [0,1]\), which reminds a measure of the likelihood that a given point \(u\in X\) belongs to the set x(t). The strong deviation from the usual probabilistic approach lies in the law of composition, which is defined on the basis of a min-max property. The usual differential equation defining the dynamical system, \(\dot x(t)=f(t,x(t))\) is substituted by a differential relation \(\dot x(t)=x(t){\mathbb{O}}R,\) where R is a ''fuzzy relation'' and \({\mathbb{O}}\) is a law of composition. Some basic properties (which are quite different from the usual probabilistic process) are obtained in this paper, and more are promised in a forthcoming paper.
scientific article; zbMATH DE number 3852870
scientific article