Constrained Dirichlet Boundary Control in $L^2$ for a Class of Evolution Equations

Optimal Dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed. This approach allows us to consider the boundary controls in $L^2$, which has advantages over approaches which consider control in Sobolev spaces involving (fractional) derivatives. Pointwise constraints on the boundary are incorporated by the primal-dual active set strategy. Its global and local superlinear convergences are shown. A discretization based on space-time finite elements is proposed and numerical examples are included.