The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concept

Abstract

In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler–Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM).

KEYWORDS:

• Horizontal membership function (HMF)
• granular differentiability
• fuzzy variational problem (FVP)
• fuzzy optimal control problem (FOCP)
• variational iteration method (VIM)

2010 Mathematics Subject Classifications:

• 49Kxx
• 49Mxx

Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their helpful and constructive suggestions, which led to improving the quality of the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 61763044].