Properties of several metric spaces of fuzzy sets

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We first give some relations among the endograph metric, the sendograph metric and the $\Gamma$-convergence, and then investigate the level characterizations of the endograph metric and the $\Gamma$-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of fuzzy sets whose $\alpha$-cuts are compact when $\alpha>0$ equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Statistical depth for fuzzy sets

Statistical depth functions provide a way to order the elements of a space by their centrality in a probability distribution. That has been very successful for generalizing non-parametric order-based statistical procedures from univariate to multivariate and (more recently) to functional spaces. We introduce two general definitions of statistical depth which are adapted to fuzzy data. For that purpose, two concepts of symmetric fuzzy random variables are introduced and studied. Furthermore, a generalization of Tukey's halfspace depth to the fuzzy setting is presented and proved to satisfy the above notions, through a detailed study of its properties.A. Nieto-Reyes and L. Gonzalez are supported by the Spanish Ministerio de Economía, Industria y Competitividad grant MTM2017-86061-C2-2-P. P. Terán is supported by the Ministerio de Economía y Competitividad grant MTM2015-63971-P, the Ministerio de Ciencia, Innovación y Universidades grant PID2019-104486GB-I00 and the Consejería de Empleo, Industria y Turismo del Principado de Asturias grant GRUPIN-IDI2018-000132

Approximation of fuzzy numbers by convolution method

In this paper we consider how to use the convolution method to construct approximations, which consist of fuzzy numbers sequences with good properties, for a general fuzzy number. It shows that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers which have finite non-differentiable points. In the previous work, this convolution method only can be used to construct differentiable approximations for continuous fuzzy numbers whose possible non-differentiable points are the two endpoints of 1-cut. The constructing of smoothers is a key step in the construction process of approximations. It further points out that, if appropriately choose the smoothers, then one can use the convolution method to provide approximations which are differentiable, Lipschitz and preserve the core at the same time.Comment: Submitted to Fuzzy Sets and System at Sep 18 201