Dirichlet control of elliptic state constrained problems

We study a state constrained Dirichlet optimal control problem and derive a priori error estimates for its finite element discretization. Additional control constraints may or may not be included in the formulation. The pointwise state constraints are prescribed in the interior of a convex polygonal domain. We obtain a priori error estimates for the $$L^2(\varGamma )$$ L 2 ( Γ ) -norm of order $$h^{1-1/p}$$ h 1 - 1 / p for pure state constraints and $$h^{3/4-1/(2p)}$$ h 3 / 4 - 1 / ( 2 p ) when additional control constraints are present. Here, p is a real number that depends on the largest interior angle of the domain. Unlike in e.g. distributed or Neumann control problems, the state functions associated with $$L^2$$ L 2 -Dirichlet control have very low regularity, i.e. they are elements of $$H^{1/2}(\varOmega )$$ H 1 / 2 ( Ω ) . By considering the state constraints in the interior we make use of higher interior regularity and separate the regularity limiting influences of the boundary on the one-hand, and the measure in the right-hand-side of the adjoint equation associated with the state constraints on the other hand. We note in passing that in case of control constraints, these may be interpreted as state constraints on the boundary. Copyright Springer Science+Business Media New York 2016