Lavrentiev-prox-regularization for optimal control of PDEs with state constraints (Q843270)

An optimal control problem, governed by an elliptic partial differential equation (PDE), with pointwise state constraints is considered. In the non-prox Lavrentiev regularization a single real-valued parameter \(\lambda>0\) is introduced. For each \(\lambda\), an auxiliary problem with a mixed state-control constraint is defined. To obtain the convergence result \(\lambda\) must converge to \(0_+\). But as \(\lambda\) decreases the corresponding problems become more and more difficult to solve. The author introduces a Lavrentiev prox-regularization method. Another regularization parameter \(\varepsilon \geq 0\) is introduced in the cost functional. For a sequence of regularization parameters \((\lambda_k,\varepsilon_k)\) convergent to 0, the strong convergence, with respect to the \(L^2\)-norm, of the generated control sequence to the optimal control is demonstrated. Numerical examples are given to show that the Lavrentiev prox-regularization method gives a faster convergence than the non-prox Lavrentiev regularization one.