Advertisement

Differential Equations

, Volume 54, Issue 9, pp 1125–1136 | Cite as

Theoretical Analysis of the Magnetic Cloaking Problem Based on an Optimization Method

  • G. V. Alekseev
  • Yu. E. Spivak
Partial Differential Equations
  • 4 Downloads

Abstract

Control problems are considered for a model of magnetic scattering on a permeable anisotropic obstacle shaped as a spherical layer. Such problems arise in developing technologies for designing magnetic cloaking devices when the corresponding inverse problems are solved by an optimization method. The solvability of the direct and extremal problems for the model in question is proved and the optimality system is derived. Its analysis permits obtaining sufficient conditions on the initial data which ensure the local uniqueness and stability of the optimal solutions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dolin, L.S., On the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 1961, vol. 4, pp. 964–967.Google Scholar
  2. 2.
    Pendry, J.B., Shurig, D., and Smith, D.R., Controlling electromagnetic fields, Science, 2006, vol. 312, pp. 1780–1782.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Leonhardt, U., Optical conformal mapping, Science, 2006, vol. 312, pp. 1777–1780.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cummer, S.A. and Shurig, D., One path to acoustic cloaking, New J. Phys., 2007, vol. 9, p. 45.CrossRefGoogle Scholar
  5. 5.
    Wood, B. and Pendry, J.B., Metamaterials at zero frequency, J. Phys. Condens. Matter, 2007, vol. 19, p. 076208.CrossRefGoogle Scholar
  6. 6.
    Sanchez, A., Navau, C., Prat-Camps, J., and Chen, D.-X., Antimagnets: controlling magnetic fields with superconductor-metamaterial hybrids, New J. Phys., 2011, vol. 13, p. 093034.CrossRefGoogle Scholar
  7. 7.
    Gömöry, F., Solovyov, M., Souc, J. et al., Experimental realization of a magnetic cloak, Science, 2012, vol. 335, pp. 1466–1468.CrossRefGoogle Scholar
  8. 8.
    Guenneau, S., Amra, C., and Veynante, D., Transformation thermodynamics: Cloaking and concentrating heat flux, Opt. Express, 2012, vol. 20, pp. 8207–8218.CrossRefGoogle Scholar
  9. 9.
    Tikhonov, A.N. and Arsenin, V.Ya., Metody resheniya nekorrektnykh zadach (Methods for Solving Ill-Posed Problems), Moscow: Nauka, 1986.zbMATHGoogle Scholar
  10. 10.
    Xu, S., Wang, Y., Zhang, B., and Chen, H., Invisibility cloaks from forward design to inverse design, Sci. China Inf. Sci., 2013, vol. 56, p. 120408.Google Scholar
  11. 11.
    Alekseev, G.V., Problema nevidimosti v akustike, optike i teploperenose (Invisibility Problem in Acoustics, Optics, and Heat Transfer), Vladivostok: Dalnauka, 2016.Google Scholar
  12. 12.
    Popa, B.-I. and Cummer, S.A., Cloaking with optimized homogeneous anisotropic layers, Phys. Rev. A, 2009, vol. 79, p. 023806.CrossRefGoogle Scholar
  13. 13.
    Xi, S., Route to low-scattering cylindrical cloaks with finite permittivity and permeability, Phys. Rev. B, 2009, vol. 79, p. 155122.CrossRefGoogle Scholar
  14. 14.
    Alekseev, G.V., Cloaking via impedance boundary condition for 2-D Helmholtz equation, Appl. Anal., 2014, vol. 93, no. 2, pp. 254–268.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Alekseev, G.V., Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method, Comput. Math. Math. Phys., 2013, vol. 53, no. 12, pp. 1853–1869.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Alekseev, G.V., Stability estimates in the problem of cloaking material bodies for Maxwell’s equations, Comput. Math. Math. Phys., 2014, vol. 54, no. 12, pp. 1788–1803.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Alekseev, G.V., Analysis and optimization in problems of cloaking material bodies for the Maxwell equations, Differ. Equations, 2016, vol. 52, no. 3, pp. 361–372.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Anikonov, D.S., Nazarov, V.G., and Prokhorov, I.V., Visible and invisible media in tomography, Dokl. Math., 1997, vol. 56, no. 3, pp. 955–958.zbMATHGoogle Scholar
  19. 19.
    Melenk, J.M., Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comput., 2010, vol. 79, pp. 1871–1914.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fursikov, A.V., Optimal’noe upravlenie raspredelennymi sistemami. Teoriya i priblizheniya (Optimal Control of Distributed Systems. Theory and Applications), Novosibirsk: Nauchnaya Kniga, 1999.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics of the Far Eastern Branch of the Russian Academy of SciencesVladivostokRussia
  2. 2.Far Eastern Federal UniversityVladivostokRussia

Personalised recommendations