A distributed parabolic control with mixed boundary conditions
J. L. Menaldi, Domingo Alberto Tarzia · Asymptotic Analysis · 2007
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 α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion $\varGamma _{1}$ of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control $\hat{g}_{\alpha}$ with optimal state $u_{\hat{g}_{\alpha}\alpha}$ and optimal adjoint state $p_{\hat{g}_{\alpha}\alpha}$ are convergent as α→∞ (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 $\varGamma _{1}.$ The main techniques used are derived from the parabolic variational inequality theory.