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 targetu ‾ 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 functionsu ‾ ∈ H 0 1 ( Ω ) , 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 targetu ‾ by the state u ϱ .