SOLVING A TYPE OF FUZZY LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

Abstract

In this study, a type of fuzzy linear systems of differential equations (FLSDE) is analyzed. The dynamics of the system is definite (crisp) but initial value is a fuzzy set the alpha-cuts of which are ellipsoids. Many practice problems are modeled with FLSDE naturally. For example, the motion equation of a satellite, a free-falling body or a charged particle is definite, but its initial position and velocity are usually uncertain. In the researches made up to now, FLSDE the initial value of which is given with a vector of fuzzy numbers is analyzed. It means that each component of the initial position vector has independent uncertainty. Such an assumption simplifies soft calculations, but it causes either serious data loss or excessive data handling in fuzzy model. For example, in two-dimensional case, approximation of a circle with the most appropriate square (or, in the more general case, approximation of an ellipse with the most appropriate rectangle) will leave out the regions that are necessary to examine or vice versa. The initial value given as a fuzzy ellipsoid makes solution of FLSDE significantly difficult and impossible to apply the methods proposed in the previous researches. In this study, a method of solution for such type of FLSDE is suggested. The method is based on the fact that a linear transformation maps an ellipsoid to another ellipsoid. At any time the system's solution constitutes a fuzzy set, alpha-cuts of which are nested ellipsoids. The suggested approach and method of solution were explained on different examples.