Fuzzy differential inclusions

Abstract

Differential equations are considered with some unknown parameters such that there is, however, some information available about certain preferred values of the parameters. Using a formal approach to such information, in accordance with the theory of fuzzy sets (FS), we introduce the notation of a differential inclusion (DI) with a fuzzy right-hand side. The solution of a differential inclusion is defined as the FS of motions. It is established that the level set of this FS is identical with the bundle of solutions of an ordinary differential equation whose right-hand side is given by the corresponding level set of the fuzzy right-hand side. Conditions for the right-hand side of the original differential equation and for the membership function of the FS of parameters are stated which ensure that there exists a solution of the DI with a fuzzy right-hand side. A special case of a controlled linear system is considered with a matrix of coefficients defined by means of the direct product of one-dimensional FS's. In this way a new formalism for differential systems with fuzzy unknown parameters is proposed. A connection between the theory of DI's and FS's is established.