Efficient Solution of State-Constrained Distributed Parabolic Optimal Control Problems

Abstract

We consider a space-time finite element method for the numerical solution of a distributed tracking-type optimal control problem subject to the heat equation with state constraints. The cost or regularization term is formulated in an anisotropic Sobolev norm for the state, and the optimal state is then characterized as the unique solution of a first kind variational inequality. We discuss an efficient realization of the anisotropic Sobolev norm in the case of a space-time tensor-product finite element mesh, and the iterative solution of the resulting discrete variational inequality by means of a semi-smooth Newton method, i.e., using an active set strategy.

Figures (1)

• Figure 1: Plot of the constrained solution $u_{\varrho h}$ along the line $x_1=x_2=x_3=0.51$.