Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

Abstract

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

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

“…The necessary optimality conditions given by the Euler-Lagrange equations were first derived by Farhadinia [13] using the Buckley-Fuering derivative approach [33] under twice differentiability assumptions. For similar results under different differentiability notions, one may refer to [12,14,42].…”

“…Theorem 11 has also been studied under the gHdifferentiability by Fard et al [12]. Subsequently, it was further extended to derive the fuzzy fractional Euler-Lagrange equations for fuzzy fractional variational problems using the Caputo-type fuzzy fractional and Atangana-Baleanu Journal of Function Spaces fractional derivatives by Fard and Salehi [42] and Zhang et al [14], respectively. All such studies employed different differentiability concepts to obtain the necessary optimal conditions for fuzzy variational or fractional variational problems, namely the derivation of the Euler-Lagrange in the fuzzy setting.…”

“…Since most practical optimization problems can be modelled as fuzzy energy functionals, it is therefore advantageous to study fuzzy variational problems for decision-making. However, the existing studies in fuzzy variational problems are mostly aimed at obtaining the explicit solution by deriving the Euler-Lagrange equations in the fuzzy setting [12][13][14], which are essentially based on the intuition that every extremal problem will have a solution. This is in contrast to the standard variational principle in the calculus of variations that focuses on the existence of minimizers or more generally the existence of solutions.…”

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

“…As far as the uncertainties in variational problems are concerned, their necessary optimality conditions were first introduced by Farhadinia [12] and extended by Fard et al [13], where they focused on the necessary optimality conditions of the fuzzy variational problem for finding its extrema by deriving its Euler-Lagrange equations. 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].…”

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

“…In [37], it has been proposed the fractional generalization by means of H-differentiable. Recently, the authors [38,39] defined a novel operator of fractional derivatives based on Atangana-Baleanu-Caputo (ABC) in view of fuzzy valued function with form of parametric interval, called ABC gH-differentiability. In this paper, we intend to study the effect of ABC gH-differentiability on the solution of different types of fuzzy fractional integrodifferential equations (FFIDEs).…”