Differential Equations with Fuzzy Parameters via Differential Inclusions

Abstract

We give a definition of solutions of ordinary differential equations in n containing parameters which are described by changing in time fuzzy sets. They are defined as fuzzy subsets of the space of absolutely continuous functions. We introduce a hypograph metric in the space of fuzzy sets and prove a theorem on continuous dependence of fuzzy solutions on parameters and initial conditions with respect to that metric.

Fuzzy differential equations and the extension principle

“…Note that L t depends on x 0 and w, and it is the solution of deterministic Eq. (22). Since we are considering that X 0 and W are non-interactive [6], L t must be extended both in x 0 and w even if x 0 ¼ w in (22).…”

“…From Corollary 1, b L t ðX 0 ; W Þ ¼ A t ðX 0 Þ is the attainable set of problem (22). Note that L t depends on x 0 and w, and it is the solution of deterministic Eq.…”

“…In [22] problem (2) is studied via differential inclusions (Hü llermeier interpretation), with an analysis of continuous dependence of fuzzy solutions on parameters and initial conditions concerning the hypograph metric.…”

Normalized expression for solutions to linear fuzzy differential equations under combination of differences

“…The analysis of uncertain systems has been developed from different points of view which include the use of several concepts of differences and derivatives, such as the Hukuhara difference and derivative [15,20], strongly generalized derivative [2,3,4,8,21], generalized Hukuhara difference and differentiability [5,11,37,38], but also from different approaches such as differential inclusions (starting in [17] and continuing, for instance, in [10,18,35]), the use of different representations of fuzzy numbers [39], the embedding of fuzzy sets on spaces with a linear structure [34], the detailed analysis of the implications of Zadeh's extension principle [7,27,33], the use of generalized metric spaces [30] or the study of the connections between different approaches [6,8,9] and the perspectives of the problem [1].…”

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

“…They play an important role in an increasing number of system models in biology [1], engineering [2], civil engineering [3], bioinformatics and computational biology [4], quantum optics and gravity [5], and hydraulic [6,7] and modeling of mechanical systems [8]. Many investigations in this area were developed using different approaches for formulations of differential problems in a fuzzy setting (see, e.g., [1,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references therein). Historically, the earliest approach for deterministic fuzzy differential equations was based on a generalization of the Hukuhara derivative of a set-valued function.…”