A New Optimal Solution Concept for Fuzzy Optimal Control Problems


1 Damghan university

2 Department of Control Engineering, Islamic Azad University, Science and Research Boroujerd, Boroujerd, Iran.


In this paper, we propose the new concept of optimal solution for fuzzy variational problems based on the possibility and necessity measures. Inspired by the well–known embedding theorem, we can transform the fuzzy variational problem into a bi–objective variational problem. Then the optimal solutions of fuzzy variational problem can be obtained by solving its corresponding biobjective variational problem.


1. O.P. Agrawal, Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative, Journal of Vibration and Control, 13, 1217–1237 (2007).

2. T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions Mathematical Programming, 122, 301-347 (2010).

3. B. Bede, L. Stefanini, Generalized differentiability of fuzzy–valued functions, Fuzzy Sets and Systems, 230, 119–41 (2013).

4. G.A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press (1963).

5. J.J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105, 241-248 (1999).

6. C.L. Dym, I.H. Shames, Solid Mechanics: A Variational Approach, New York, McGraw-Hill (1973).

7. J. Engwerda, Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games, SIAM Journal on Control and Optimization, 48, 3859-3881 (2010).

8. O.S. Fard, A.H. Borzabadi, and M. Heidari, On fuzzy Euler–Lagrange equations, Annals of Fuzzy Mathematics and Informatics, 7, 447–461 (2014).

9. O.S. Fard, M. Salehi, A survey on fuzzy fractional variational problems, Journal of Computational and Applied Mathematics, 271, 71–82 (2014).

10. O.S. Fard, M.S. Zadeh, Note on ”Necessary optimality conditions for fuzzy variational problems”, Journal of Advanced Research in Dynamical and Control Systems, 4, 1–9, (2012).

11. B. Farhadinia, Necessary optimality conditions for fuzzy variational problems, Information Sciences, 181, 1348-1357 (2011).

12. I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall (1963).

13. R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 31–43 (1986).

14. J. Gregory, C. Lin, Constrained Optimization in the Calculus of Variations and Optimal Control Theory, Van Nostrand-Reinhold (1992).

15. N.V. Hoa, Fuzzy fractional functional differential equations under Caputo gH–differentiability, Communications in Nonlinear Science and Numerical Simulation, 22, 1134-1157 (2015).

16. O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35, 389-396 (1990).

17. A. Khastan, F. Bahrami, and K. Ivaz, New results on multiple solutions for nth-order fuzzy differential equations under generalized differentiability, Boundary Value Problems, Article ID 395714, doi:10.1155/2009/395714 (2009).

18. A.B. Malinowska, D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Computers and Mathematics with Applications, 59, 3110–3116 (2010).

19. B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, Springer, Berlin (2006).

20. M.L. Puri, D.A. Ralesc,u Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422 (1986).

21. A. Sophos, E. Rotstein, and G. Stephanopoulos, Multiobjective analysis in modeling the petrochemical industry, Chemical Engineering Science, 35, 2415-2426 (1980).

22. B. Van Brunt, The Calculus of Variations, Springer–Verlag, Heidelberg (2004). 23. C. X. Wu, M. Ma, Embedding problem of fuzzy number space: part I, Fuzzy Sets and Systems, 44, 33-38 (1991).

24. J. Xu, Z. Liao, and J.J. Nieto, A class of linear differential dynamical systems with fuzzy matrices, Journal of mathematical analysis and applications, 368, 54–68 (2010).