A Systematic Review on the Advancement in the Study of Fuzzy Variational Problems

Abstract

The study of fuzzy variational problems has received significant attention over the past decade due to its successful applications in numerous fields, such as image segmentation and optimal control theory. The fuzzy Euler-Lagrange equations provide the necessary optimality conditions for solving fuzzy variational problems explicitly and have been studied under several differentiability conditions. In this paper, we provide a systematic review to recap the history of variational principle in the calculus of variations and compare it with the existing techniques in the fuzzy setting. We begin with the preliminary concepts and definitions of fuzzy theory and scrutinize the Euler-Lagrange’s strategy via systematically searched studies concerning fuzzy variational problems to highlight the importance of improving the existing methods. Finally, we set up the main open problems regarding the limitations of the current approaches, shedding light on future directions.

Sufficient Conditions for the Existence and Uniqueness of Minimizers for Variational Problems under Uncertainty

“…Although Heidari et al [31] attempted to derive some sufficiency results, the study considered the convexity notion, which is inadequate in the fuzzy setting as discussed below. Moreover, it was based on differential inclusion and not on the fuzzy setting [16].…”

“…Similarly, Fard and Salehi [14] and Zhang et al [15] addressed the fuzzy fractional variational problems based on the Caputo-type fuzzy fractional derivative. However, the current literature is based on the intuition that every extremal problem will have a solution and this assumption is not always valid, since there is no guarantee of the existence of a solution, as observed by Verma et al [16]. The existing studies employed various differentiability concepts to derive the necessary optimality conditions, namely the Euler-Lagrange equations, which lead to local extrema.…”

“…However, as discussed above, there is no sufficient result to guarantee the existence of minimizers of such problems. Hence, it is advantageous to establish the sufficient conditions for the existence and uniqueness of minimizers in the fuzzy setting [16,18].…”

Some Characterizations of Certain Complex Fuzzy Subgroups

“…Gulzar [44] et al presented the novel concept of complex fuzzy subfields. Verma [45] et al discussed a systematic review on the development in the inquiry of fuzzy variational problems. Masmali [46] et al introduced the concept of µ-fuzzy subgroups and discussed many algebraic properties of fact.…”

Algebraic Properties of ω , θ -Complex Fuzzy Subgroups

“…Gulzar [35] et al presented the novel concept of complex fuzzy subfields. Verma [36] et al discussed a systematic review on the advancement in study of fuzzy variational problems. Alolaiyan et al [38] developed the structure of (α, β)-complex fuzzy subgroups.…”