Continuity of approximate reasoning using center of sums defuzzification method

In this work, a mathematical structure based on the functional analysis for studying the existence of fuzzy optimal control, which the feedback is evaluated through approximate reasoning using the center of sums defuzzification method on IF-THEN fuzzy rules. The framework consists of two propositions: To guarantee the convergence of optimal solution, a set of fuzzy membership functions (admissible fuzzy controller) which are selected out of continuous function space is compact metrizable. And assuming approximate reasoning to be a functional on the set of membership functions, its continuity is proved. Then, we show the existence of a fuzzy controller which minimize (maximize) the integral performance function of the nonlinear feedback fuzzy system.