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Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

Author

Listed:
  • Amar Debbouche

    (Guelma University)

  • Juan J. Nieto

    (Universidad de Santiago de Compostela
    King Abdulaziz University)

  • Delfim F. M. Torres

    (University of Aveiro)

Abstract

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev-type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multicontrols, and the functional in suitable topologies.

Suggested Citation

  • Amar Debbouche & Juan J. Nieto & Delfim F. M. Torres, 2017. "Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 7-31, July.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:1:d:10.1007_s10957-015-0743-7
    DOI: 10.1007/s10957-015-0743-7
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    References listed on IDEAS

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    1. M. Benchohra & E.P. Gatsori & S.K. Ntouyas, 2003. "Controllability Results for Semilinear Evolution Inclusions with Nonlocal Conditions," Journal of Optimization Theory and Applications, Springer, vol. 118(3), pages 493-513, September.
    2. Shen, Yongjun & Yang, Shaopu & Sui, Chuanyi, 2014. "Analysis on limit cycle of fractional-order van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 94-102.
    3. Michal Fec̆kan & JinRong Wang & Yong Zhou, 2013. "Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 79-95, January.
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    Cited by:

    1. Rosa, Silvério & Torres, Delfim F.M., 2018. "Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 142-149.
    2. JinRong Wang & Michal Fečkan & Amar Debbouche, 2019. "Time Optimal Control of a System Governed by Non-instantaneous Impulsive Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 573-587, August.
    3. Chen Bin & Xiao Yu Liang & Emil Minchev & Sergey A. Timoshin, 2023. "Optimization of a Prey–Predator Model with Hysteresis and Convection," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 347-371, July.

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