A distributed parabolic control with mixed boundary conditions

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 Γ1 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 ĝα with optimal state uĝαα and optimal adjoint state pĝαα are convergent as α → ∞ (in norm of a suitable Sobolev parabolic space) to ĝ, uĝ and pĝ, respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on Γ1. The main techniques used are derived from the parabolic variational inequality theory.