Source Representation Strategy for Optimal Boundary Control Problems with State Constraints

A state-constrained optimal boundary control problem gov- erned by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater as- sumption has to be made that restricts the theory to the two-dimensional case. This diculty is overcome by a source representation of the control and combined with a Lavrentiev type regularization. Optimality condi- tions for the regularized problem are derived, where the corresponding Lagrange multipliers have L2-regularity. By the spectral theorem for compact and normal operators, the convergence result of (22) is extended to a higher dimensional case. Moreover, the convergence for vanishing regularization parameter of the adjoint state associated with the regu- larized problem is shown. Finally, the uniform boundedness of the regu- larized Lagrange multipliers in L1() is verified by a maximum principle argument.