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 $f\left(y\right)$ grows slower than $\left|y\right|{log}^{3/2}(1+|y\left|\right)$ 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.

Classification: 35B37

Keywords: Carleman inequalities, controllability, transmission problems

@article{COCV_2002__8__621_0, author = {Doubova, Anna and Osses, A. and Puel, J.-P.}, title = {Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, pages = {621-661}, doi = {10.1051/cocv:2002047}, zbl = {1092.93006}, mrnumber = {1932966}, language = {en}, url = {http://www.numdam.org/item/COCV_2002__8__621_0} }

Doubova, Anna; Osses, A.; Puel, J.-P. Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 621-661. doi : 10.1051/cocv:2002047. http://www.numdam.org/item/COCV_2002__8__621_0/

[1] Null controllability of nonlinear convective heat equation. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR 1744610 | Zbl 0938.93008

and ,[2] Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal. 25 (1967) 81-122. | MR 244638 | Zbl 0154.12001

and ,[3] A maximum principle for nonlinear parabolic equations. Ann. Scuola Norm. Sup. Pisa 3 (1967) 291-305 | Numdam | MR 219901 | Zbl 0148.34803

and ,[4] L'analyse non linéaire et ses motivations économiques. Masson (1984). | Zbl 0551.90001

,[5] Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73-89. | MR 1751309 | Zbl 0964.93046

,[6] Introduction aux problèmes d'évolution semi-linéaires. Ellipses, Paris, Mathématiques & Applications (1990). | Zbl 0786.35070

and ,[7] The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545-573. | MR 1355412 | Zbl 0847.35078

and ,[8] On the controllability of parabolic system with a nonlinear term involving the state and the gradient. SIAM: SICON (to appear). | Zbl 1038.93041

, , and ,[9] | MR 1318622 | Zbl 0818.93032

, and , (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. A 125 (1995) 31-61.[10] Approximate controllability for the linear heat equation with controls of minimal ${L}^{\infty}$ norm. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 679-684. | MR 1214415 | Zbl 0799.35094

, and ,[11] Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-107. | Numdam | Zbl 0897.93011

,[12] The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5 (2000) 465-514. | Zbl 1007.93034

and ,[13] Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583-616. | Numdam | Zbl 0970.93023

and ,[14] On the null controllability of the one-dimensional heat equation with BV coefficients (to appear). | Zbl 1119.93311

and ,[15] Controllability of Evolution Equations. Seoul National University, Korea, Lecture Notes 34 (1996). | MR 1406566 | Zbl 0862.49004

and ,[16] Controllability of parabolic equations. Mat. Sb. 186 (1995) 102-132. | MR 1349016 | Zbl 0845.35040

,[17] 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. | MR 1817179 | Zbl 0977.93041

and ,[18] Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskow (1967). | Zbl 0174.15403

, and ,[19] Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

,[20] A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211. | MR 341256 | Zbl 0274.35041

,[21] Local existence and nonexistence for semilinear parabolic equations in ${L}^{p}$. Indiana Univ. Math. J. 29 (1980) 79-102. | MR 554819 | Zbl 0443.35034

,[22] Semilinear evolution equations in Banach spaces. J. Funct. Anal. 32 (1979) 277-296. | MR 538855 | Zbl 0419.47031

,[23] 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. | MR 1131832 | Zbl 0731.93011

,[24] Finite dimensional controllability for the semilinear heat equations. J. Math. Pures 76 (1997) 570-594. | MR 1441986 | Zbl 0872.93014

,[25] Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control and Cybernetics 28 (1999) 665-683. | MR 1782020 | Zbl 0959.93025

,