Advertisement

Local Null Controllability of a 1D Stefan Problem

  • E. Fernández-Cara
  • F. HernándezEmail author
  • J. Límaco
Article
  • 70 Downloads

Abstract

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

Keywords

Null controllability Free-boundary problems 1D heat equation Carleman estimates 

Mathematics Subject Classification

93B05 93C20 35K20 35R35 

Notes

Acknowledgements

E. F-C.  was partially supported by MINECO (Spain), Grant MTM2013-41286-P. We would like to express our thanks to the anonymous referee for their helpful comments.

References

  1. Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Consultants Bureau, New York (1987)CrossRefzbMATHGoogle Scholar
  2. Avalos, G., Lasiecka, I.: Boundary controllability of thermoelastic plates via the free boundary conditions. SIAM J. Control Optim. 38(2), 337–383 (2000). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Clark, H.R., Fernández-Cara, E., Limaco, J., Medeiros, L.A.: Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities. Appl. Math. Comput. 223, 483–505 (2013)MathSciNetzbMATHGoogle Scholar
  4. Doubova, A., Fernández-Cara, E.: Some control results for sim-plified one-dimensional models of fluid–solid interaction. Math. Models Methods Appl. Sci. 15(5), 783–824 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Doubova, A., Fernández-Cara, E., González-Burgos, M., Zuazua, E.: On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41(3), 798–819 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fattorini, H.O., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43(4), 272–292 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fernández-Cara, E., Límaco, J., Menezes, S.B.: On the controllability of a free-boundary problem for the 1D heat equation. Syst. Control Lett. 87, 29–35 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5(4–6), 465–514 (2000)MathSciNetzbMATHGoogle Scholar
  10. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)zbMATHGoogle Scholar
  11. Friedman, A.: Variational Principles and Free-boundary Problems. Wiley, New York (1982)zbMATHGoogle Scholar
  12. Friedman, A. (ed.): Tutorials in Mathematical Biosciences, III. Cell Cycle, Proliferation, and Cancer. Lecture Notes in Mathematics, vol. 1872, Mathematical Biosciences Subseries. Springer, Berlin (2006)Google Scholar
  13. Friedman, A.: PDE problems arising in mathematical biology. Netw. Heterog. Media 7(4), 691–703 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fursikov, A.V., Imanuvilov, O.Yu.: Controllability of Evolution Equations. Lectures Notes Series, vol. 34. National University, RIM, Seoul (1996)Google Scholar
  15. Hansen, S.W., Imanuvilov, O.Yu.: Exact controllability of a multilayer Rao–Nakra plate with free boundary conditions. Math. Control Relat. Fields 1(2), 189–230 (2011)Google Scholar
  16. Hermans, A.J.: Water Waves and Ship Hydrodynamics. An Introduction, 2nd edn. Springer, Dordrecht (2011)CrossRefzbMATHGoogle Scholar
  17. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs. American Mathematical Society, Providence (1968)CrossRefGoogle Scholar
  18. Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20(1–2), 335–356 (1995)CrossRefzbMATHGoogle Scholar
  19. Liu, Y., Takahashi, T., Tucsnak, M.: Single input controllability of a simplified fluid–structure interaction model. ESAIM Control Optim. Calc. Var. 19(1), 20–42 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Mercan, M., Nakoulima, O.: Control of Stackelberg for a two-stroke problem. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 22(6), 441–463 (2015)MathSciNetzbMATHGoogle Scholar
  21. Nakoulima, O.: Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Math. Acad. Sci. Paris 339(6), 405–410 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Puel, J.-P., Yamamoto, M.: On a global estimate in a linear inverse hyperbolic problem. Inverse Probl. 12(6), 995–1002 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Stoker, J.J.: Water Waves. Interscience, New York (1957)zbMATHGoogle Scholar
  24. Vázquez, J.L., Zuazua, E.: Large time behavior for a simplified 1D model of fluid–solid interaction. Commun. Partial Differ. Equ. 28(9–10), 1705–1738 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Wrobel, L.C., Brebbia, C.A. (eds.): Computational Modelling of Free and Moving Boundary Problems, vol. 1, Fluid flow. In: Proceedings of the First International Conference, 2–4 July, 1991. Computational Mechanics Publications, Southampton (1991)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Dpto. E.D.A.N.Universidad de SevillaSevilleSpain
  2. 2.Inst. MatemáticaUniversidade Federal FluminenseNiteróiBrazil

Personalised recommendations