Mathematical Control and Related Fields (MCRF)

Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability

Pages: 203 - 259, Volume 4, Issue 2, June 2014      doi:10.3934/mcrf.2014.4.203

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Thuy N. T. Nguyen - Université d'Orléans, Bâtiment de Mathématiques (MAPMO), B.P. 6759, 45067 Orléans cedex 2, France (email)

Abstract: In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x )$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.

Keywords:  Parabolic operator, semi-discrete Carleman estimates, nonsmooth coefficients, observability inequality, null-controllability.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: January 2013;      Revised: July 2013;      Available Online: February 2014.