Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients

Anna Doubova; A. Osses; J.-P. Puel

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 621-661
  • ISSN: 1292-8119

Abstract

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The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y|log3/2(1+|y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.

How to cite

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Doubova, Anna, Osses, A., and Puel, J.-P.. "Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 621-661. <http://eudml.org/doc/90663>.

@article{Doubova2010,
abstract = { The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y|log3/2(1+|y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry. },
author = {Doubova, Anna, Osses, A., Puel, J.-P.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carleman inequalities; controllability; transmission problems.; transmission problems},
language = {eng},
month = {3},
pages = {621-661},
publisher = {EDP Sciences},
title = {Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients},
url = {http://eudml.org/doc/90663},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Doubova, Anna
AU - Osses, A.
AU - Puel, J.-P.
TI - Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 621
EP - 661
AB - The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f(y) grows slower than |y|log3/2(1+|y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.
LA - eng
KW - Carleman inequalities; controllability; transmission problems.; transmission problems
UR - http://eudml.org/doc/90663
ER -

References

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  1. S. Anita and V. Barbu, Null controllability of nonlinear convective heat equation. ESAIM: COCV5 (2000) 157-173.  Zbl0938.93008
  2. D.G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal.25 (1967) 81-122.  Zbl0154.12001
  3. D.G. Aronson and J. Serrin, A maximum principle for nonlinear parabolic equations. Ann. Scuola Norm. Sup. Pisa3 (1967) 291-305 Zbl0148.34803
  4. J.P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson (1984).  Zbl0551.90001
  5. V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim.42 (2000) 73-89.  Zbl0964.93046
  6. T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires. Ellipses, Paris, Mathématiques & Applications (1990).  
  7. S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J.44 (1995) 545-573.  Zbl0847.35078
  8. A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM: SICON (to appear).  Zbl1038.93041
  9. C. Fabre, J.-P. Puel and E. Zuazua, (a) Approximate controllability for the semilinear heat equation. C. R. Acad. Sci. Paris Sér. I Math.315 (1992) 807-812; (b) Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A125 (1995) 31-61.  Zbl0818.93032
  10. C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability for the linear heat equation with controls of minimal L∞ norm. C. R. Acad. Sci. Paris Sér. I Math.316 (1993) 679-684.  Zbl0799.35094
  11. E. Fernández-Cara, Null controllability of the semilinear heat equation. ESAIM: COCV2 (1997) 87-107.  Zbl0897.93011
  12. E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations5 (2000) 465-514.  Zbl1007.93034
  13. E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire17 (2000) 583-616.  Zbl0970.93023
  14. E. Fernández-Cara and E. Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients (to appear).  Zbl1119.93311
  15. A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Seoul National University, Korea, Lecture Notes34 (1996).  
  16. O.Yu. Imanuvilov, Controllability of parabolic equations. Mat. Sb.186 (1995) 102-132.  
  17. O.Yu. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. Lecture Notes in Pure Appl. Math.218 (2001) 113-137.  Zbl0977.93041
  18. O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uraltzeva, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskow (1967).  
  19. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983).  Zbl0516.47023
  20. D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math.52 (1973) 189-211.  Zbl0274.35041
  21. F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in Lp. Indiana Univ. Math. J.29 (1980) 79-102.  Zbl0443.35034
  22. F.B. Weissler, Semilinear evolution equations in Banach spaces. J. Funct. Anal.32 (1979) 277-296.  Zbl0419.47031
  23. E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and their Applications, Vol. X, edited by H. Brezis and J.-L. Lions. Pitman (1991) 357-391.  
  24. E. Zuazua, Finite dimensional controllability for the semilinear heat equations. J. Math. Pures76 (1997) 570-594.  Zbl0877.35053
  25. E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control and Cybernetics28 (1999) 665-683.  Zbl0959.93025

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