AIMS Mathematics, 2020, 5(2): 1105-1126. doi: 10.3934/math.2020077.

Research article Special Issues

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Generalized conformable variational calculus and optimal control problems with variable terminal conditions

Department of Mathematics, Balıkesir University, Balıkesir, 10145, Turkey

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

This paper provides generalized transversality conditions for the problems of variational calculus and optimal control, constructed by the conformable derivative. The generalized term is used to emphasize the problems with performance indexes containing the conformable derivative and defined by the classical integral and to distinguish them from the problems with performance indexes defined by the conformable integral. Special cases of the generalized transversality conditions both for variational calculus and optimal control are exhibited and supported by illustrative examples.
  Article Metrics

Keywords conformable derivative; fractional order; optimal control; variational calculus; generalized Euler-Lagrange equation; generalized transversality condition

Citation: Beyza Billur İskender Eroǧlu, Dilara Yapışkan. Generalized conformable variational calculus and optimal control problems with variable terminal conditions. AIMS Mathematics, 2020, 5(2): 1105-1126. doi: 10.3934/math.2020077


  • 1. R. Khalil, M. A. Horani, A. Yousef, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.    
  • 2. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, New York, 2006.
  • 3. A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.    
  • 4. V. E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci., 18 (2013), 2945-2948.    
  • 5. V. E. Tarasov, No nonlocality. No fractional derivative, Commun. Nonlinear Sci., 62 (2018), 157-163.    
  • 6. M. D. Ortigueira, J. T. Machado, What is a fractional derivative?, J. Comput. Phys., 293 (2015), 4-13.    
  • 7. G. S. Teodoro, J. T. Machado, E. C. De Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208.    
  • 8. J. Weberszpil, J. A. Helayël-Neto, Variational approach and deformed derivatives, Physica A, 450 (2016), 217-227.    
  • 9. W. Rosa, J. Weberszpil, Dual conformable derivative: Definition, simple properties and perspectives for applications, Chaos Soliton. Fract., 117 (2018), 137-141.    
  • 10. D. R. Anderson, E. Camrud, D. J. Ulness, On the nature of the conformable derivative and its applications to physics, Journal of Fractional Calculus and Applications, J. Fract. Calc. Appl., 10 (2019), 92-135.
  • 11. H. W. Zhou, S. Yang, S. Q. Zhang, Conformable derivative approach to anomalous diffusion, Physica A, 491 (2018), 1001-1013.    
  • 12. D. Avcı, B. B. İskender Eroǧlu, N. Özdemir, The Dirichlet problem of a conformable advection-diffusion equation, Therm. Sci., 21 (2017), 9-18.    
  • 13. T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.    
  • 14. F. Silva, D. Moreira, M. Moret, Conformable Laplace transform of fractional differential equations, Axioms, 7 (2018), 55.
  • 15. A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
  • 16. R. Khalil, A. Yousef, M. Al Horani, et al. Fractional analytic functions, Far East J. Math. Sci., 103 (2018), 113-123.
  • 17. S. Uçar, N. Yılmaz Özgür, B. B. İskender Eroǧlu, Complex conformable derivative, Arab. J. Geosci., 12 (2019), 201.
  • 18. M. A. Hammad and R. Khalil, Abel's formula and Wronskian for conformable fractional differential equations, Int. J. Differ. Equ. Appl., 13 (2014), 177-183.
  • 19. D. R. Anderson, D. J. Ulness, Results for conformable differential equations, preprint, 2016.
  • 20. E. Ünal, A. Gökdoǧan, E. Çelik, Solutions around a regular α singular point of a sequential conformable fractional differential equation, Kuwait J. Sci., 44 (2017), 9-16.
  • 21. N. Sene, Solutions for some conformable differential equations, Progr. Fract. Differ. Appl. 4 (2018), 493-501.
  • 22. Z. Hammouch, T. Mekkaoui, P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative, Eur. Phys. J. Plus, 133 (2018), 248-253.    
  • 23. H. Bulut, T. A. Sulaiman, H. M. Başkonuş, Dark, bright optical and other solitons with conformable space-time fractional second-order spatiotemporal dispersion, Optik, 163 (2018), 1-7.    
  • 24. M. Yavuz, Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8 (2017), 1-7.
  • 25. E. Ünal, A. Gökdoǧan, Solution of conformable fractional ordinary differential equations via differential transform method, Optik, 128 (2017), 264-273.    
  • 26. M. Yavuz, N. Özdemir, A different approach to the European option pricing model with new fractional operator, Math. Model. Nat. Pheno., 13 (2018), 12.
  • 27. F. Evirgen, Conformable fractional gradient based dynamic system for constrained optimization problem, Acta Phys. Pol. A, 132 (2017), 1066-1069.    
  • 28. W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., 290 (2015), 150-158.    
  • 29. B. B. İskender Eroǧlu, D. Avcı, N. Özdemir, Optimal control problem for a conformable fractional heat conduction equation, ACTA Phys. Pol. A, 132 (2017), 658-662.    
  • 30. J. M. Lazo, D. F. M. Torres, Variational calculus with conformable fractional derivatives, IEEE/CAA J. Autom. Sin., 4 (2017), 340-352.    
  • 31. B. B. İskender Eroǧlu, D. Yapişkan, Local generalization of transversality conditions for optimal control problem, Math. Model. Nat. Pheno., 14 (2019), 310.
  • 32. T. Chiranjeevi, R. K. Biswas, Closed-form solution of optimal control problem of a fractional order system, Journal of King Saud University-Science, 2019.
  • 33. R. K. Biswas, S. Sen, Free final time fractional optimal control problems, J. Franklin. I., 351 (2014), 941-951.    


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved