Fuzzy Systems and its Applications

Fuzzy Systems and its Applications

Expressing the concept of confidence level on quadratic problems of fuzzy fractional optimal control

Document Type : Original Article

Author
Jahrom University - Jahrom - Iran.
10.22034/jfsa.2025.503071.1258
Abstract
The fuzzy fractional optimal control problem can be transformed into two fractional optimal control problems using the concept of α-cut. Then, by applying the Pontryagin’s minimum principle for the fractional optimal control problem, one can obtain the optimal trajectory and optimal control. Based on this, by expressing the concept of degree of confidence for this category of problems, the decision-maker can compute the optimal trajectory and optimal control of the given problem as an interval by choosing a confidence level α. It should be noted that as the value of α approaches one, the upper and lower bounds of the obtained intervals for the optimal trajectory and optimal control get closer to the optimal trajectory and control of the exact (non-fuzzy) case of the problem. At the end of this paper, we will solve a numerical example using the method presented in this paper with different α-cuts, as well as fractional derivatives of various orders β, and show that the optimal control and optimal trajectory obtained for the case of α=β=1 correspond with the optimal solutions obtained in the non-fuzzy case and with first-order derivatives.
Keywords
Subjects


[1]Abbasbandy, S., Nieto J. J., Alavi M., (2005) Tuning of reachable set in one dimensional fuzzy differential inclusions, Chaos, Solitons & Fractals, 26, 1337–1341.
[2]Abbasbandy, S. (2004) Numerical methods for fuzzy differential inclusions, Computing Mathematics Application, 48, 1633–1641.
[3]Abbod, M. F., Von Keyserlingk, D. G., Linkens, D. A., Mahfouf, M. (2001) Survey of utilisation of fuzzy technology in medicine and healthcare, Fuzzy Sets Syst, 120, 331-349.
[4]Agrawal, O. P. (2004) A general formulation and solutions scheme for fractional optimal control problem, Nonlinear Dynamics, 38, 323-337.
[5]Agrawal, O. P. (2007) A formulation and numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13, 1269-1281.
[6]Agarwal, R. P., O’Regan, D.,Lakshmikantham, V. (2005) Viability theory and fuzzy differential equations, Fuzzy Sets and Systems, 151, 563-580.
[7]Agarwal, R. P., O’Regan, D., Lakshmikantham, V. (2005) Maximal solutions and existence theory for fuzzy differential and integral equations, J. Appl. Anal, 11, 171-186.
[8]Agarwal, R. P., O’Regan, D., Lakshmikantham, V. (2003) A stacking theorem approach for fuzzy differential equations, Nonlinear Anal, 55, 299-312.
[9]Bede, B., Gal, S. G. (2010) Solutions of fuzzy differential equations based on generalized differen- tiability, Commun. Math. Anal, 9, 22-41.
[10]Bourdin, L. (2012) A class of fractional optimal control problems and fractional Pontryagin’s sys- tems. Existence of a fractional Noether’s theorem, arXiv preprint arXiv:1203.1422.

[11]Diamond, P., Kloeden, P. E. (1994) “Metric space of Fuzzy sets Theory And Applications”, World scientific publishing.
[12]Filev, D., Angelove, P., (1992) Fuzzy optimal control, Fuzzy Sets and Systems, 47, 151-156.
[13]Gal, S. G. (2000) Approximation Theory in Fuzzy Setting, In Anastassiou, G.A. (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics, ch. 13, pp. 617–666. Chapman & Hall⧸CRC, Boca Raton .
[14]Karimyar, J. M., Fakharzadeh, J. A. (2018) Fractional Fuzzy Optimal Control Problem Governed by Fuzzy System, International Journal of Pharmaceutical Reasearch and Allied sciences,151-158.
[15]Kilbas, A., Srivastava, H., Trujillo, J. (2006) Theory and Applications of Fractional Differential Equations.
[16]Kloeden, P. (1991) Remarks on peano-like theorems for fuzzy differential equations, Fuzzy Sets and Systems, 44, 161-164.
[17]Lotfi A., Dehghan M. and Yousefi S. A. (2011) A numerical tehnique for solving the fractional optimal control problems, Computers and Mathematics with Applications, 62, 1055-1067.
[18]Mostafa, A. M., Osman, M. (2021) A Numerical Technique for Solving Fuzzy Fractional Optimal Control Problems, Journal of Computational Analysis and Applications, 29, 413-430.
[19]Mortezaee, M. (2022) An application of a fuzzy system for solving time delay fractional optimal control problems with Atangana–Baleanu derivative, Optimal Control Applications and Methods, 5, 1753-1777.
[20]Najariyan, M., Farahi, M. H. (2013) Optimal control of fuzzy controlled system with fuzzy initial conditions, Iranian Journal of Fuzzy Systems, 10, 21-35.
[21]Najariyan, M., Farahi, M. H., Alavian, M. (2011) Optimal control of HIV infection by using fuzzy dynamical systems, The Journal of Mathematics and Computer Science, 2, 639-649.
[22]Naschie, M. S. (2004) The concepts of E-infinite: an elementary introduction to the Cantorianfractal theory of quantum physics, Chaos, Solitons & Fractals, 22, 495-511.
[23]Naschie, M. S. (2004) A review of E-infinite theory and the mass spectrum of high energy particle physics, Chaos, Solitons and Fractals, 19, 209-236.
[24]Naschie, M. S. (2005) On a fuzzy Kahler manifold which is consistent with slit experiment, Inter- national Journal of Nonlinear Science and Numerical Simulation, 6, 95-98.
[25]Naschie, M. S. (2005) From experimental quantum optics to quantum gravity via a fuzzy Kahler manifold, Chaos,Solitons and Fractals, 25, 969-977.
[26]Nguyen, H. T. (1978) A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64, 369-380.

[27]Nieto, J., Khastan, A., Ivaz, K. (2009) Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear Anal. Hybrid Syst, 3, 700-707.
[28]Park, J. H., Park, J. S., Kwun, Y. C. (2006) Controllability for the semi linear fuzzy integro- differential equations with nonlocal conditions, Lecture Notes in Artificial Intelligence, LNAI 4223, 221-230.
[29]Pearson, D. W. (1997) A property of linear fuzzy differential equations, Appl. Math. Letter, 10, 99-103.
[30]Seikkala, S. (1987) On the fuzzy initial value problem, Fuzzy Sets Syst, 24, 319–330.
[31]Seikkala, S., Vorobiev, D. (2002) Towards the theory of fuzzy differential equations, Fuzzy Sets and Systems, 125, 231-237.
[32]Xu, J., Liao, Z., Nieto, J. (2010) A class of differential dynamical systems with fuzzy matrices, J. Math. Anal. Appl., 368, 54-68.
[33]Zhao, Y., Zhu, Y. (2010) Fuzzy optimal control of linear quadratic models, Computers and Mathe- matics with Applications, 60, 67-73.
[34]Zimmermann, H. J. (1987) fuzzy programming and linear programming with several objective func- tion, Fuzzy set and System, 1, 45-55.
Volume 7, Issue 2 - Serial Number 15
Open Access Statement
December 2024
Pages 181-200

  • Receive Date 29 January 2025
  • Revise Date 10 March 2025
  • Accept Date 03 April 2025