A distributed parabolic control with mixed boundary conditions

Article type: Research Article

Authors: Menaldi, Jose-Luis | Tarzia, Domingo Alberto

Affiliations: Wayne State University, Department of Mathematics, Detroit, MI 48202, USA E-mail: [email protected] | Universidad Austral, Departamento de Matemática—Conicet, FCE, Paraguay 1950, S2000FZF Rosario, Argentina E-mail: [email protected]

Abstract: 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.

Keywords: parabolic variational inequalities, distributed evolution optimal control, mixed boundary conditions, adjoint state, optimality condition, asymptotic

Journal: Asymptotic Analysis, vol. 52, no. 3-4, pp. 227-241, 2007