Hierarchic Control for the Wave Equation

  • Fágner Dias Araruna
  • Enrique Fernández-Cara
  • Luciano Cipriano da Silva
Article

Abstract

This paper deals with the hierarchical control of the wave equation. We use Stackelberg–Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves an exact controllability problem. We consider linear and semilinear equations.

Keywords

Wave equation Exact controllability Stackelberg–Nash strategy Carleman inequalities 

Mathematics Subject Classification

35L05 90C29 93B05 

Notes

Acknowledgements

Fágner Dias Araruna has been partially supported by INCTMat, CAPES, CNPq (Brazil), and MathAmSud COSIP. Enrique Fernández-Cara has been partially supported by Grant MTM2016-76990-P (DGI-MICINN, Spain) and CAPES (Brazil). Luciano Cipriano da Silva has been partially supported by CAPES (Brazil).

References

  1. 1.
    Lions, J.-L.: Hierarchic control. Proc. Indian Acad. Sci. Math. Sci. 104(4), 295–304 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Díaz, J.I., Lions, J.-L.: On the Approximate Controllability of Stackelberg–Nash strategies. Ocean Circulation and pollution Control: A Mathematical and Numerical Investigations, Madrid (1997), pp. 17–27. Springer, Berlin (2004)CrossRefGoogle Scholar
  3. 3.
    Guillén-González, F., Marques-Lopes, F.P., Rojas-Medar, M.A.: On the approximate controllability of Stackelberg–Nash strategies for Stokes equation. Proc. Am. Math. Soc. 141(5), 1759–1773 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Araruna, F.D., Fernández-Cara, E., da Silva, L.C.: Hierarchic control for exact controllability of parabolic equations with distributed and boundary controls. (2018) (preprint)Google Scholar
  5. 5.
    Araruna, F.D., Fernández-Cara, E., Guerrero, S., Santos, M.C.: New results on the Stackelberg–Nash exact controllability of linear parabolic equations. Syst. Control Lett. 104, 78–85 (2017)CrossRefMATHGoogle Scholar
  6. 6.
    Araruna, F.D., Fernández-Cara, E., Santos, M.C.: Stackelberg–Nash exact controllability for linear and semilinear parabolic equations. ESAIM: Control Optim. Calc. Var. 21, 835–856 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fabre, C., Puel, J.-P., Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. Sect. A 125(1), 31–61 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1–41 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fu, X., Yong, J., Zhang, X.: Exact controllability for multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46(5), 1578–1614 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dpto. de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil
  2. 2.Dpto. EDAN and IMUSUniversidad de SevillaSevillaSpain

Personalised recommendations