Modelling Fuzzy Control Problems with Toll Sets

We shall introduce in this paper toll sets and apply it to dynamical and control systems. We shall reformulate viability theorems for 'fuzzy controlled problem". 'Fuzzification" of dynamical and control systems led us to use indicators instead of characteristic functions for imbedding subsets into functions. Therefore, instead of obtaining the genuine fuzzy sets by a convexification procedure, we obtain the 'toll sets", the membership function being a cost function taking values between 0 and . Hence we can define 'fuzzy" or 'toll" – differential inclusions and control systems and solve the viability problem or the toll viability problem by building fuzzy controllers which regulate viable solutions (satisfying at each instant viability and toll viability constraints).