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Membership functions in L/sup 2/ space and its applications to fuzzy optimal control

The fuzzy control discussed here is nonlinear feedback control in which the feedback laws are determined by if-then type fuzzy rules through the product-sum-gravity method. Then the integral cost (benefit) function of the fuzzy control can be considered as a continuous functional on an appropriate set of membership functions. We study the continuity of the product-sum-gravity method defined on some fuzzy membership function space, and show the existence of a membership function which minimizes (maximizes) the integral cost (benefit) function.