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Fuzzy differential equations without fuzzy convexity

机译：无模糊凸的模糊微分方程

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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 Hiillermeier'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.
机译：用Hukuhara导数定义的经典模糊微分方程严格取决于能级集的凸性并导致能级集的扩展。在这里，希勒迈尔（Hiillermeier）关于通过微分包含来定义每个水平集上的模糊微分方程的建议与Aubin关于形态方程的思想相结合，这种形态方程允许非局部集演化，从而消除了模糊凸的假设，从而允许为非凸定义模糊微分方程。 -凸水平集。这种方法使用可达集作为集积分的一种更通用的形式，并且与Aumann集积分相反，它不一定会产生凸集。本文介绍的结果更加笼统，因为它们涉及模糊集，只需要封闭它们而无需额外假设凸度，紧致度或正态性。特别是，在假设右手侧满足单侧Lipschitz条件而不是更强的Lipschitz条件的情况下，建立了存在和唯一性定理。从这个新的角度也考虑了模糊延迟微分方程。

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