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Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. (English) Zbl 1189.35349
Summary: We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise \({\mathcal C}^1\)). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.

MSC:
35R05 PDEs with low regular coefficients and/or low regular data
93B07 Observability
93B05 Controllability
35R30 Inverse problems for PDEs
35B45 A priori estimates in context of PDEs
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[1] Ammar Khodja, F.; Benabdallah, A.; Dupaix, C., Null controllability of some reaction – diffusion systems with one control force, J. math. anal. appl., 320, 928-943, (2006) · Zbl 1157.93004
[2] Ammar Khodja, F.; Benabdallah, A.; Dupaix, C.; Kostin, I., Null-controllability of some systems of parabolic type by one control force, ESAIM control optim. calc. var., 11, 426-448, (2005) · Zbl 1125.93005
[3] Aubin, J.-P., Applied functional analysis, (1979), John Wiley & Sons New York
[4] Barbu, V., Exact controllability of the superlinear heat equation, Appl. math. optim., 42, 73-89, (2000) · Zbl 0964.93046
[5] A. Benabdallah, P. Gaitan, J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., in press · Zbl 1155.35485
[6] Brezis, H., Analyse fonctionnelle, (1983), Masson Paris · Zbl 0511.46001
[7] Doubova, A.; Fernandez-Cara, E.; Gonzales-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, 798-819, (2002) · Zbl 1038.93041
[8] Doubova, A.; Osses, A.; Puel, J.-P., Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM control optim. calc. var., 8, 621-661, (2002) · Zbl 1092.93006
[9] Fabre, C.; Puel, J.-P.; Zuazua, E., Approximate controllability of the semilinear heat equation, Proc. roy. soc. Edinburgh sect. A, 125, 31-61, (1995) · Zbl 0818.93032
[10] Fernández-Cara, E.; Guerrero, S., Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. control optim., 45, 4, 1395-1446, (2006) · Zbl 1121.35017
[11] Fernández-Cara, E.; Zuazua, E., Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. inst. H. Poincaré anal. non linéaire, 17, 583-616, (2000) · Zbl 0970.93023
[12] Fernández-Cara, E.; Zuazua, E., On the null controllability of the one-dimensional heat equation with BV coefficients, Comput. appl. math., 21, 167-190, (2002) · Zbl 1119.93311
[13] Fursikov, A.; Imanuvilov, O.Yu., Controllability of evolution equations, Lecture notes, vol. 34, (1996), Seoul National University Korea · Zbl 0862.49004
[14] M. Gonzales-Burgos, private communication, 2005
[15] Lions, J.-L., Quelques méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod · Zbl 0189.40603
[16] Lions, J.-L., Contrôlabilité exacte perturbations et stabilisation de systèmes distribués, vol. 1, (1988), Masson Paris · Zbl 0653.93002
[17] Lions, J.-L.; Magenes, E., Problèmes aux limites non homogènes, vol. 1, (1968), Dunod · Zbl 0165.10801
[18] Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies appl. math., 52, 189-221, (1973) · Zbl 0274.35041
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