Compactness of a set of fuzzy membership functions and existence of optimal solution in fuzzy-output feedback
Yu‐Guang Yang, Yasunari Shidama, K. Ohkubo, Hiroo Yamaura · 1997
This paper presents a framework for studying the existence of optimal control based on fuzzy rules. The framework consists of two propositions: 1) a set of fuzzy membership functions with bounded gradients is a uniformly bounded, equi-continuous closed set, and so the set of fuzzy membership functions is sequentially compact by the Ascoli-Arzela theorem; and 2) the defuzzification function is continuous on the set of fuzzy membership functions with bounded gradients. The existence of fuzzy optimal control is proved.