Fuzzy Differential Equations

Abstract

Fuzzy differential equations (FDEs) appear as a natural way to model the propagation of epistemic uncertainty in a dynamical environment. There are several interpretations of a fuzzy differential equation. The first one historically was based on the Hukuhara derivative introduced in Puri-Ralescu [123] and studied in several papers (Wu-Song-Lee [150], Kaleva [83], Ding-Ma-Kandel [46], Rodriguez-Lopez [125]). This interpretation has the disadvantage that solutions of a fuzzy differential equation have always an increasing length of the support. This fact implies that the future behavior of a fuzzy dynamical system is more and more uncertain in time. This phenomenon does not allow the existence of periodic solutions or asymptotic phenomena. That is why different ideas and methods to solve fuzzy differential equations have been developed. One of them solves differential equations using Zadeh’s extension principle (Buckley-Feuring [30]), while another approach interprets fuzzy differential equations through differential inclusions. Differential inclusions and Fuzzy Differential Inclusions are two topics that are very interesting but they do not constitute the subject of the present work (see Diamond [45], Lakshmikantham-Mohapatra [98]). Recently new approaches have been developed based on generalized fuzzy derivatives discussed in the previous chapter. In the present work we will work with the interpretations based on Hukuhara differentiability, Zadeh’s extension principle and the strongly generalized differentiability concepts.