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On the Existence of Weak Optimal Controls in the Coefficients for a Degenerate Anisotropic p-Laplacian

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Continuous and Distributed Systems II

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 30))

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Abstract

We consider an optimal control problem for nonlinear degenerate elliptic problems involving an anisotropic p-Laplacian and Dirichlet boundary conditions. We take the matrix-valued coefficients A(x) of such system as a control in \(L^{p/2}(\varOmega ;\mathbb {R}^{\frac{N(N+1)}{2}})\). One of the important features of the admissible controls is the fact that eigenvalues of the coefficient matrices may vanish in \(\varOmega \). Equations of this type may exhibit the Lavrentiev phenomenon and nonuniqueness of weak solutions. Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of this optimal control problem in the class of weak solutions.

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Acknowledgments

Research funded by the DFG-cluster CE315: Engineering of Advanced Materials

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Authors and Affiliations

  1. Dnipropetrovsk Mining University, Karl Marks Av., 19, Dnipropetrovsk, 49005, Ukraine

    Olha P. Kupenko

  2. Institute for Applied System Analysis, National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Av., 37, Building 35, Kiev, 03056, Ukraine

    Olha P. Kupenko

  3. Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg Lehrstuhl AMII, Cauerstr. 11, 91058, Erlangen, Germany

    Günter Leugering

Authors
  1. Olha P. Kupenko
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  2. Günter Leugering
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Correspondence to Olha P. Kupenko .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Viktor A. Sadovnichiy

  2. Kiev Polytechnic Institute, National Technical University Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

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Kupenko, O.P., Leugering, G. (2015). On the Existence of Weak Optimal Controls in the Coefficients for a Degenerate Anisotropic p-Laplacian. In: Sadovnichiy, V., Zgurovsky, M. (eds) Continuous and Distributed Systems II. Studies in Systems, Decision and Control, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-19075-4_19

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  • DOI: https://doi.org/10.1007/978-3-319-19075-4_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19074-7

  • Online ISBN: 978-3-319-19075-4

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