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

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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.

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Acknowledgments

This research was initiated during a visit of Debbouche and Torres to the Department of Mathematical Analysis of the University of Santiago of Compostela, Spain, followed by a visit of Debbouche to the Department of Mathematics of University of Aveiro, Portugal. The hospitality and the financial support provided by the host institutions in Spain and Portugal are here gratefully acknowledged. Nieto has been partially supported by the Ministerio de Economía y Competitividad of Spain under Grants MTM2010–15314 and MTM2013–43014–P, Xunta de Galicia under Grant R2014/002, and co-financed by the European Community fund FEDER. Torres was supported by funds through The Portuguese Foundation for Science and Technology (FCT), within CIDMA Project UID/MAT/04106/2013 and OCHERA Project PTDC/EEI-AUT/1450/2012, co-financed by FEDER under POFC-QREN with COMPETE reference FCOMP-01-0124-FEDER-028894.

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Authors and Affiliations

  1. Department of Mathematics, Guelma University, 24000, Guelma, Algeria

    Amar Debbouche

  2. Departamento de Análisis Matemático, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Juan J. Nieto

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Juan J. Nieto

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

    Delfim F. M. Torres

Authors
  1. Amar Debbouche
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  2. Juan J. Nieto
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  3. Delfim F. M. Torres
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Correspondence to Delfim F. M. Torres.

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Debbouche, A., Nieto, J.J. & Torres, D.F.M. Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations. J Optim Theory Appl 174, 7–31 (2017). https://doi.org/10.1007/s10957-015-0743-7

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