Continuity of product-sum-gravity method on L2 space using fuzzy number for premise variable

The authors present a mathematical framework for studying a fuzzy feedback control system, which is constructed by if-then type fuzzy rules through a kind of product-sum-gravity method. Moreover, fuzzy numbers are used instead of definite values (crisp numbers) as premise variables in if-then fuzzy rule. To guarantee the convergence of optimal solution the compactness of the set of membership functions in L² space is proved. And assuming approximate reasoning to be a functional on the set of membership functions, its continuity is proved. Then, we show that the system has an optimal feedback control by essential use of compactness of sets of fuzzy membership functions.