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Optimal Potentials for Problems with Changing Sign Data

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Abstract

In this paper, we consider variational optimal control problems. The state equation is an elliptic partial differential equation of a Schrödinger type, governed by the Laplace operator with a potential, with a right-hand side that may change sign. The control variable is the potential itself that may vary in a suitable admissible class of nonnegative potentials. The cost is an integral functional, linear (but non-monotone) with respect to the state function. We prove the existence of optimal potentials, and we provide some necessary conditions for optimality. Several numerical simulations are shown.

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References

  1. Buttazzo, G., Gerolin, A., Ruffini, B., Velichkov, B.: Optimal potentials for Schördinger operators. J. Éc. Polytech. Math. 1, 71–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buttazzo, G., Velichkov, B.: A shape optimal control problem with changing sign data. SIAM J. Math. Anal. 50(3), 2608–2627 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allaire, G., Dapogny, C.: A linearized approach to worst-case design in parametric and geometric shape optimization. Math. Models Methods Appl. Sci. 24(11), 2199–2257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellido, J.C., Buttazzo, G., Velichkov, B.: Worst-case shape optimization for the Dirichlet energy. Nonlinear Anal. 153, 117–129 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser Verlag, Basel (2005)

    MATH  Google Scholar 

  6. Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions. Appl. Math. Opt. 23, 17–49 (1991)

    Article  MATH  Google Scholar 

  7. Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dal Maso, G., Mosco, U.: Wiener’s criterion and \(\Gamma \)-convergence. Appl. Math. Optim. 15, 15–63 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bucur, D.: Minimization of the \(k\)-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bucur, D., Buttazzo, G., Velichkov, B.: Spectral optimization problems for potentials and measures. SIAM J. Math. Anal. 46(4), 2956–2986 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carlen, E.A., Frank, R.L., Lieb, E.H.: Stability estimates for the lowest eigenvalue of a Schrödinger operator. Geom. Funct. Anal. 24(1), 63–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bouchitté, G., Buttazzo, G.: Optimal design problems for Schrödinger operators with noncompact resolvents. ESAIM Control Optim. Calc. Var. 23, 627–635 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buttazzo, G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Research Notes in Mathematics, vol. 207. Longman, Harlow (1989)

    MATH  Google Scholar 

  15. Allaire, G.: Numerical Analysis and Optimization. An Introduction to Mathematical Modeling and Numerical Simulation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  16. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)

    Google Scholar 

  17. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Svanberg, K.: The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work started during a visit of the second author at the Department of Mathematics of University of Pisa and continued during a stay of the authors at the Centro de Ciencias de Benasque “Pedro Pascual.” The authors gratefully acknowledge both institutions for the excellent working atmosphere provided. The work of the first author is part of the Project 2015PA5MP7 “Calcolo delle Variazioni” funded by the Italian Ministry of Research and University. The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad Project MTM2014-53309-P. The third author has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) Project GeoSpec and the project ANR CoMeDiC.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Giuseppe Buttazzo

  2. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n. Aptdo 1160, 41080, Seville, Spain

    Faustino Maestre

  3. Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38401, Saint-Martin-d’Hères, France

    Bozhidar Velichkov

Authors
  1. Giuseppe Buttazzo
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  2. Faustino Maestre
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  3. Bozhidar Velichkov
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Correspondence to Giuseppe Buttazzo.

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Buttazzo, G., Maestre, F. & Velichkov, B. Optimal Potentials for Problems with Changing Sign Data. J Optim Theory Appl 178, 743–762 (2018). https://doi.org/10.1007/s10957-018-1347-9

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  • DOI: https://doi.org/10.1007/s10957-018-1347-9

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