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Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. (English) Zbl 1128.35020
Global Carleman estimates for one-dimensional linear parabolic equations with a coefficient of bounded variations are derived. First, the author constructs limit weight functions by approaching the bounded variation coefficient $$c$$ by piecewise constant coefficients $$c_{\varepsilon}$$. Then he proves a Carleman estimate associated to $$\partial_t \pm \partial_x (c\partial_x)$$ by proving that the constants in the Carleman estimate $$\partial_t \pm \partial_x (c_{\varepsilon}\partial_x)$$ can be taken uniform with respect to the parameter $$\varepsilon$$ and passing to the limit in each term of the estimate. In the conclusive part of the work, the author derives a Carleman estimate for a linear parabolic system with the right-hand side in $$L^2(0,T,H^{-1}(\Omega))$$. This estimate is used for the analysis of the controllability of a semilinear system.

##### MSC:
 35B37 PDE in connection with control problems (MSC2000) 93B05 Controllability 35K20 Initial-boundary value problems for second-order parabolic equations 35B45 A priori estimates in context of PDEs
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##### References:
 [1] Aubin, J.-P., Applied functional analysis, (1979), Wiley New York [2] Barbu, V., Exact controllability of the superlinear heat equation, Appl. math. optim., 42, 73-89, (2000) · Zbl 0964.93046 [3] Benabdallah, A.; Dermenjian, Y.; Le Rousseau, J., Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, (2005), Preprint: LATP, Université d’Aix-Marseille I [4] Benabdallah, A.; Dermenjian, Y.; Le Rousseau, J., Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications, Comptes rendus Mécanique, 334, 582-586, (2006) · Zbl 1182.35057 [5] Bressan, A., Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem, (2000), Oxford Univ. Press · Zbl 0997.35002 [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, 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] Imanuvilov, O.Y.; Yamamoto, M., Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture notes in pure and appl. math., vol. 218, (2001), Dekker New York, pp. 113-137 · Zbl 0977.93041 [15] Kolmogorov, A.; Fomin, S.V., Eléments de la théorie des fonctions et de l’analyse fonctionnelle, (1974), Editions MIR [16] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod · Zbl 0189.40603 [17] Lions, J.-L.; Magenes, E., Problèmes aux limites non homogènes, vol. 1, (1968), Dunod · Zbl 0165.10801 [18] Pazy, A., Semigroups of linear operators and applications to partial differential euations, (1983), Springer-Verlag New York · Zbl 0516.47023 [19] Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. appl. math., 52, 189-221, (1973) · Zbl 0274.35041
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