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Necessary conditions for \(\Psi \)-Hilfer fractional optimal control problems and \(\Psi \)-Hilfer two-step Lagrange interpolation polynomial

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Abstract

The manuscript primarily focuses on the necessary conditions for fractional optimal control problems using \(\Psi \)-Hilfer derivative. The outcomes are derived from the integration of the product of functions with respect to another function \(\Psi \). We explore the existence and uniqueness of the solutions of \(\Psi \)-Hilfer fractional optimal control problems with time-invariant system using Banach fixed point technique. We establish a new numerical scheme on \(\Psi \)-Hilfer two-step Lagrange interpolation polynomial. We employed some specific numerical computations to examine the effectiveness of the results.

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References

  1. Agrawal OP (1989) General formulation for the numerical solution of optimal control problems. Int J Control 50(2):627–638

    Article  MathSciNet  Google Scholar 

  2. Agrawal OP (2004) A General formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn 38:323–337

    Article  MathSciNet  Google Scholar 

  3. Agrawal OP (2008) A quadratic numerical scheme for fractional optimal control problems. ASME J Dyn Syst Meas Control 130(1):011010

    Article  Google Scholar 

  4. Agrawal OP (2010) Generalized variational problems and Euler-Lagrange equations. Comput Math Appl 59(5):1852–1864

    Article  MathSciNet  Google Scholar 

  5. Agrawal OP (2002) Formulation of Euler-Lagrange equations for fractional variational problems. Math Anal Appl 272:368–379

    Article  MathSciNet  Google Scholar 

  6. Rezazadeh A, Avazzadeh Z (2022) Solving a category of two-dimensional fractional optimal control problems using discrete Legendre polynomials. Asian J Control 25:551–562

    Article  MathSciNet  Google Scholar 

  7. Mali AD, Kucche KD (2020) On the Boundary Value Problems of \(\Psi \)-Hilfer Fractional Differential Equations. Cornell University, Cornell

    Google Scholar 

  8. Hamoud AA, Sharif AA, Ghadle KP (2021) Existence, uniqueness and stability results of fractional Volterra-Fredholm Integro differential equations of \(\Psi \)-Hilfer type. Discontin Nonlinearity, Complex L and T scitific Publish 10(3):535–545

    Google Scholar 

  9. Jajarmi A, Baleanu D (2019) On the fractional optimal control problems with a general derivative operator. Asian J Control 23(2):1062–1071

    Article  MathSciNet  Google Scholar 

  10. Bahaa GM (2019) Optimal control problem for variable-order fractional differential systems with time delay involving Atangana-Baleanu derivatives. Chaos, Solitons Fractals 122:129–142

    Article  ADS  MathSciNet  Google Scholar 

  11. Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D (2019) A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Choas, An Interdiscipl J Nonlinear Sci, AIP publish 29(8):083127

    Article  MathSciNet  CAS  Google Scholar 

  12. Naidu DS. “Optimal control systems”, CRC Press, New york, Wasington

  13. Kirk DE (1998) Optimal control theory: an introduction. Dover Publications Inc, New York

    Google Scholar 

  14. Elsgolts L (1970) Differential equations and calculus of variations. Mir Publishers, Moscow, Russia

    Google Scholar 

  15. Norouzi F, NGuérékata GM (2021) A study of \(\Psi \) -Hilfer fractional differential system with application in financial crisis. Chaos Solitons and Fractals 6:100056

    Article  Google Scholar 

  16. Ghanbari G, Razzaghi M (2022) Numerical solutions for distributed-order fractional optimal control problems by using generalized fractional-order Chebyshev wavelets. Nonlinear Dyn 108:265–277

    Article  Google Scholar 

  17. Jafari H, Ganji RM, Sayevand K, Baleanu D (2021) A numerical approach for solving fractional optimal control problems with mittag-leffler kernel. J Vib Control 28(19–20):2596–606

    MathSciNet  Google Scholar 

  18. Bouacida I, Kerboua M, Segni S (2023) Controllability results for Sobolev type Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Am Inst Math Sci 12:213–229

    MathSciNet  Google Scholar 

  19. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    Google Scholar 

  20. Mamehrashi K (2023) Ritz approximate method for solving delay fractional optimal control problems. J Comput Appl Math 417:114606

    Article  MathSciNet  Google Scholar 

  21. Leibniz GW (1849) Letter from Hanover, Germany to G.F.A. L’Hospital, September 30, 1695, in Mathematische Schriften, reprinted 1962, Hildesheim. Germany (Olns Verlag) 2:301–302

  22. Love ER, Young LC (1938) On fractional integration by parts. London Math Soc 2(1):1–35

    Article  MathSciNet  Google Scholar 

  23. Lee EB, Markus L (1965). In: Robert E (ed) Foundations of optimal control theory. Krieger Publishing Company Inc, Krieger

  24. Miller KS (1993) An introduction to fractional calculus and fractional differential equations. Journals on Wiley and Sons, New York

    Google Scholar 

  25. Ishfaq M, Idris A, Ali A, Fahd J, Subhash A (2021) On \(\Psi \)-Hilfer generalized proportional fractional operators. AIMS Math 7(1):82–102

    MathSciNet  Google Scholar 

  26. Almalahi MA, Abdo MS, Panchal SK (2020) \(\Psi \)-Hilfer fractional functional differential equation by Picard operator method. J Appl Nonlinear Dyn 9(4):685–702

    Article  MathSciNet  Google Scholar 

  27. Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, San Diego

    Google Scholar 

  28. Podlubny I (1999) Fractional differential equations. Academic Press, USA

    Google Scholar 

  29. Naik PA, Jian Z, Owolabi KM (2020) Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos, Solitons and Fractals 138:109826

    Article  MathSciNet  PubMed  Google Scholar 

  30. Riewe F (1996) Non conservative Lagrangian and Hamiltonian mechanics. Phys Rev 53(2):1890

    ADS  MathSciNet  CAS  Google Scholar 

  31. Samko SG, Kilbas AA, Marichev OI (1993) “Fractional integrals and derivatives: theory and applications”, Gordon and Breach, Yverdon, [orig. ed. in Russian; Nauka i Tekhnika, Minsk, 1987]

  32. Rakhshan SA, Effati S (2020) Fractional optimal control problems with time-varying delay: a new delay fractional Euler-Lagrange equations. J Franklin Inst 357(10):5954–5988

    Article  MathSciNet  Google Scholar 

  33. Sabermahani S, Yadollah O (2023) Solving distributed-order fractional optimal control problems via the Fibonacci wavelet method. J Vib Control 4:10775463221147716

    Google Scholar 

  34. Abdeljawad Thabet, Atangana Abdon, Gómez-Aguilar JF, Jarad Fahd (2019) On a more general fractional integration by parts formulae and applications. A Statis Mech Appl 536:122494

    Article  MathSciNet  Google Scholar 

  35. Da Vanterler J, Sousa C, De Capelas Oliveira E (2017) On the \(\Psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simul 60:72–91

  36. Zaky MA, Tenreiro Machado JA (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Num Simul 52:177–189

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is supported by RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt.09.10.2018.

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Authors and Affiliations

  1. Department of Mathematics, Alagappa University, Karaikudi, 630 004, India

    K. Ramalakshmi

  2. Department of Mathematics, Central University of Tamil Nadu, Thiruvarur, 610 005, India

    B. Sundaravadivoo

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  1. K. Ramalakshmi
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  2. B. Sundaravadivoo
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Both authors have contributed equally.

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Correspondence to B. Sundaravadivoo.

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Ramalakshmi, K., Sundaravadivoo, B. Necessary conditions for \(\Psi \)-Hilfer fractional optimal control problems and \(\Psi \)-Hilfer two-step Lagrange interpolation polynomial. Int. J. Dynam. Control 12, 42–55 (2024). https://doi.org/10.1007/s40435-023-01342-y

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  • DOI: https://doi.org/10.1007/s40435-023-01342-y

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