A distributed parabolic control with mixed boundary conditions

J.L. MENALDI – D.A. TARZIA

ASYMPTOTIC ANALYSIS

IOS Press

Año: 2007 vol. 52 p. 227 - 241

0921-7134

We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions.  The parameter $alpha$ intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion $Gamma_1$ of the boundary of a given regular $n$-dimensional domain.  For each $alpha,$ the distributed parabolic control problem optimizes the internal energy $g.$   It is proven that the optimal control $hat{g}_{alpha}$ with optimal state $u_{hat{g}_alphaalpha}$ and optimal adjoint state $p_{hat{g}_{alpha}alpha}$ are convergent as $alpha oinfty$ (in norm of a suitable Sobolev parabolic space) to $hat{g},$ $u_{hat{g}}$ and $p_{hat{g}},$ respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on $Gamma_1.$ The main techniques used are derived from the parabolic variational inequality theory.