Regularization error estimates for distributed control problems in energy spaces

For tracking type distributed optimal control problems subject to second-order elliptic partial differential equations, we analyze the regularization error of the state u _ϱ and the target (Formula presented.) with respect to the regularization parameter ϱ. The main focus is on the regularization in the energy space H ⁻¹(Ω), but we also consider the regularization in L ²(Ω) for comparison. While there is no difference in the regularization error estimates when considering suitable target functions (Formula presented.), we obtain a higher-order convergence in the relaxation parameter ϱ when considering the control in the energy space H ⁻¹(Ω), which also affects the approximation of the target (Formula presented.) by the state u _ϱ.