Fuzzy differential equations without fuzzy convexity

Abstract

Classical fuzzy differential equations defined in terms of the Hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets. Here Hüllermeier's suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of Aubin on morphological equations, which allow nonlocal set evolution, to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non-convex level sets. This approach uses reachable sets as a more general form of set integration and, in contrast to the Aumann set integral, does not necessarily give rise to convex sets. The results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity, compactness or even normality. In particular, an existence and uniqueness theorem is established under the assumption that the right-hand sides satisfy a one-sided Lipschitz condition rather than a much stronger Lipschitz condition. Fuzzy delay differential equations are also considered from this new perspective.