Page 43 - programme book

AN-003
The Existence of Minimizers for the Fuzzy Variational Problems
and its Application to Optimal Control Problem

Mansi Verma 1, a) , Chuei Yee Chen 1, a) , Adem Kılıçman 1, 2, a) , Fong Peng Lim 1, a) and Gafurjan
Ibragimov 1, a)

1 Department of Mathematics & Statistics, Faculty of Science, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia.
2 Institute for Mathematical Research, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia.

a) Corresponding author: cychen@upm.edu.my

Abstract. Classical variational problems work well under the assumption that all the variables defining
the particular energy functionals are crisp or would be accessible accurately. However, uncertainty and
inadequate knowledge are inherent characteristics of modeling mathematical systems that led to the
study of fuzzy variational problems that has received significant attention over the previous decade
with its successful applications in numerous fields, such as image segmentation and optimal control
theory. The current study on fuzzy variational problems explicitly focuses on the necessary optimality
conditions for solving its extremums and has been studied under several differentiability conditions.
However, the existence results for the minimizers are not established yet. This paper studies the
existence of minimizers for fuzzy variational problems under a weaker notion of convexity, namely
pre invexity, and Buckley-Feuring differentiability. We further discuss its application in a real-world
cost minimization problem that is modeled as a fuzzy variational problem.

Keywords: Fuzzy variational problem; fuzzy Euler-Lagrange equations; existence of minimizer; invex
sets; preinvex functions