Regularization error estimates for distributed control problems in energy spaces

• Published in: Mathematical methods in the applied sciences Vol. 44; no. 5; pp. 4176 - 4191
• Main Authors: Neumüller, Martin, Steinbach, Olaf
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.03.2021
• Subjects: distributed control problem; Elliptic functions; Error analysis; Optimal control; Parameters; Partial differential equations; Regularization; regularization error estimates; Tracking control
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7021

For tracking type distributed optimal control problems subject to second‐order elliptic partial differential equations, we analyze the regularization error of the state uϱ and the target u‾ 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 L2(Ω) for comparison. While there is no difference in the regularization error estimates when considering suitable target functions u‾∈H01(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 target u‾ by the state uϱ.