Lê Thanh Tùng , Trần Thiện Khải Trịnh Tùng *

* Tác giả liên hệ (trinhtung.c3mtt@soctrang.edu.vn)

Abstract

This paper is intended to investigate fuzzy variational problems of several dependent variables with nonholonomic constraints. Firstly, the necessary optimality conditions for fuzzy variational problems with nonholonomic constraints are established. Then, sufficient optimality conditions are obtained under some convexity assumptions.

Keywords: Fuzzy variational problems, nonholonomic constraints, optimality conditions, convexity

Tóm tắt

Bài báo này nhằm mục đích nghiên cứu các bài toán tối ưu hóa hàm tích phân mờ của nhiều biến số phụ thuộc với các ràng buộc hệ phương trình vi phân cấp một. Trước hết, các điều kiện cần tối ưu cho các bài toán tối ưu hóa hàm tích phân mờ với các ràng buộc hệ phương trình vi phân được thiết lập. Sau đó, các điều kiện đủ tối ưu được khảo sát sử dụng một số giả thiết lồi.

Từ khóa: Bài toán tối ưu hóa hàm tích phân mờ, ràng buộc hệ phương trình vi phân cấp một, điều kiện tối ưu, giả thiết lồi

Article Details

Tài liệu tham khảo

Ahmad, I., Jayswal, A., Al-Homidan, S., & Banerjee, J. (2019). Sufficiency and duality in interval-valued variational programming, Neural Comput Appl, 31, 4423-4433.
https://doi.org/10.1007/s00521-017-3307-y

Almeida, R., Torres, D. F. (2011). Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul, 16(3), 1490-1500.
https://doi.org/10.1016/j.cnsns.2010.07.016

Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment, Manag. Sci. 17, 141-164.
https://doi.org/10.1287/mnsc.17.4.B141

Clarke, F. (2013). Functional Analysis, Calculus of Variations and Optimal Control, Springer, New York.
https://doi.org/10.1007/9781-4471-4820-3

Dong, N. P., Long, H. V., & Khastan, A. (2020). Optimal control of a fractional order model for granular SEIR epidemic with uncertainty, Commun. Nonlinear Sci. Numer. Simul, 88, 105312.
https://doi.org/10.1016/j.cnsns.2020.105312

Farhadinia, B. (2011). Necessary optimality conditions for fuzzy variational problems, Inform. Sci, 181, 1348-1357.
https://doi.org/10.1016/j.ins.2010.11.027

Giaquinta, M., & Hildebrandt, S. (1996). Calculus of Variations I, Springer, Berlin.
https://doi.org/10.1007/978-3-662-032787

Hestenes, M. R. (1966). Calculus of Variations and Optimal Control Theory, Wiley, New York.
https://doi.org/10.1007/978-3-0348-7539-4

Hestenes, M. R. (1975). Optimization Theory: The Finite Dimensional Case, Wiley, New York.
https://doi.org/10.1137/1019126

Jacob, N., & Evans, K.P. (2018). Course In Analysis, A-Vol. IV: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations, World Scientific.
https://doi.org/10.1142/11078

Mazandarani, M., Pariz, N., & Kamyad, A. V. (2017). Granular differentiability of fuzzy-number-valued functions, IEEE Trans. Fuzzy Syst, 26, 310-323.
https://doi.org/10.1109/TFUZZ.2017.2659731

Mustafa, A. M., Gong, Z., & Osman, M. (2021). The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concept, Int. J.Comput. Math, 98, 1495-1520.
https://doi.org/10.1080/00207160.2020.1823974

Piegat, A., & Landowski, M. (2015). Horizontal membership function and examples of its applications, Int. J. Fuzzy Syst, 17, 22-30.
https://doi.org/10.1007/s40815-015-0013-8

Piegat, A., & Plucinski, M. (2021). The differences between the horizontal membership function used in multidimensional fuzzy arithmetic and the inverse membership function used in gradual arithmetic, Granul. Comput, 2021, 1-10.
https://doi.org/10.1007/s41066-021-00293-z

Rayanki, V., Ahmad, I., &Kummari, K. (2023). Interval-valued variational programming problem with Caputo-Fabrizio fractional derivative, Math Methods Appl Sci, 475 (46), 17485-17510.
https://doi.org/10.1002/mma.9512

Soolaki, J., Fard, O. S., & Borzabadi, A. H. (2016). Generalized Euler-Lagrange equations for fuzzy variational problems, SeMA J, 73, 131-148.
https://doi.org/10.1007/s40324-015-0060-y

Son, N. T. K., Long, H. V., & Dong, N. P. (2019). Fuzzy delay differential equations under granular differentiability with applications, Comp. Appl. Math, 38, 107.
https:// doi.org/10.1007/s40314-019-0881

Torres, D. F. M., & Malinowska, A. B. (2012). Introduction to the Fractional Calculus of Variations, World Scientic, Singapore.
https://doi.org/10.1142/p871

Tung, L. T., & Tam, D. H. (2022). Necessary and sufficient optimality conditions for semi-infinite programming with multiple fuzzy-valued objective functions, Stat. Optim. Inf.Comput, 10, 410-425.
https://doi.org/10.19139/soic-2310-5070-1088

Tung, L.T., & Tam, D.H. (2022). Optimality conditions and duality for continuous-time programming with multiple interval-valued objective functions, Comput Appl Math, 41, 1–28.
https://doi.org/10.1007/s40314-022-02059-y

Tung, L.T., & Tam, D. H. (2023). Necessary and sufficient optimality conditions for fuzzy variational problems of several dependent variables in terms of granular derivatives, Int J Uncertain Fuzziness Knowlege-Based Syst, 31, 825-857.
https://doi.org/10.1142/S0218488523500381

Wu, H. C. (2009). The Karush-Kuhn-Tucker optimality conditions for multi-objective pro-gramming problems with fuzzy-valued objective functions, Fuzzy Optimization and Decision Making, 8, 1-28.
https://doi.org/10.1007/s10700-009-9049-2