IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i19p4218-d1256290.html
   My bibliography  Save this article

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

Author

Listed:
  • Faïçal Ndaïrou

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

  • Delfim F. M. Torres

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

Abstract

We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

Suggested Citation

  • Faïçal Ndaïrou & Delfim F. M. Torres, 2023. "Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems," Mathematics, MDPI, vol. 11(19), pages 1-12, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4218-:d:1256290
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/19/4218/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/19/4218/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    2. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    3. Ndaïrou, Faïçal & Khalighi, Moein & Lahti, Leo, 2023. "Ebola epidemic model with dynamic population and memory," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rahaman, Mostafijur & Mondal, Sankar Prasad & Alam, Shariful & Metwally, Ahmed Sayed M. & Salahshour, Soheil & Salimi, Mehdi & Ahmadian, Ali, 2022. "Manifestation of interval uncertainties for fractional differential equations under conformable derivative," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    2. Marina Plekhanova & Guzel Baybulatova, 2020. "Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems," Mathematics, MDPI, vol. 8(4), pages 1-9, April.
    3. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    4. Dmytro Sytnyk & Barbara Wohlmuth, 2023. "Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type," Mathematics, MDPI, vol. 11(10), pages 1-35, May.
    5. Hansin Bilgili & Chwen Sheu, 2022. "A Bibliometric Review of the Mathematics Journal," Mathematics, MDPI, vol. 10(15), pages 1-17, July.
    6. Aneesh S. Deogan & Roeland Dilz & Diego Caratelli, 2024. "On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media," Mathematics, MDPI, vol. 12(7), pages 1-17, March.
    7. Kumar, Ajay & Kumar, Sunil & Momani, Shaher & Hadid, Samir, 2024. "A chaos study of fractal–fractional predator–prey model of mathematical ecology," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 857-888.
    8. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.
    9. Fiaz, Muhammad & Aqeel, Muhammad & Marwan, Muhammad & Sabir, Muhammad, 2022. "Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    10. Raza, Ali & Ghaffari, Abuzar & Khan, Sami Ullah & Haq, Absar Ul & Khan, M. Ijaz & Khan, M. Riaz, 2022. "Non-singular fractional computations for the radiative heat and mass transfer phenomenon subject to mixed convection and slip boundary effects," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    11. El-Nabulsi, Rami Ahmad & Khalili Golmankhaneh, Alireza & Agarwal, Praveen, 2022. "On a new generalized local fractal derivative operator," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    12. Kai Diethelm, 2022. "A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme," Mathematics, MDPI, vol. 10(8), pages 1-12, April.
    13. Özköse, Fatma & Yavuz, Mehmet & Şenel, M. Tamer & Habbireeh, Rafla, 2022. "Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    14. Artion Kashuri & Muhammad Samraiz & Gauhar Rahman & Zareen A. Khan, 2022. "Some New Beesack–Wirtinger-Type Inequalities Pertaining to Different Kinds of Convex Functions," Mathematics, MDPI, vol. 10(5), pages 1-20, February.
    15. Mangal, Shiv & Misra, O.P. & Dhar, Joydip, 2023. "Fractional-order deterministic epidemic model for the spread and control of HIV/AIDS with special reference to Mexico and India," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 82-102.
    16. Cao, Dewei & Chen, Hu, 2023. "Error analysis of a finite difference method for the distributed order sub-diffusion equation using discrete comparison principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 109-117.
    17. J. Alberto Conejero & Jonathan Franceschi & Enric Picó-Marco, 2022. "Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?," Mathematics, MDPI, vol. 10(15), pages 1-18, August.
    18. Li, Hang & Shen, Yongjun & Han, Yanjun & Dong, Jinlu & Li, Jian, 2023. "Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    19. María Pilar Velasco & David Usero & Salvador Jiménez & Luis Vázquez & José Luis Vázquez-Poletti & Mina Mortazavi, 2020. "About Some Possible Implementations of the Fractional Calculus," Mathematics, MDPI, vol. 8(6), pages 1-22, June.
    20. Isah, Sunday Simon & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "On bivariate fractional calculus with general univariate analytic kernels," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4218-:d:1256290. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.