On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

Moreau-Yosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to in nity or zero, respectively. In particular, the strong convergence of global and local solutions is addressed. Moreover, strong regularity of the Lavrentiev-regularized optimality system is shown under certain assumptions, which in particular allows to show that locally optimal solutions of the Lavrentiev regularized problems are locally unique. This analysis is based on a second-order su cient optimality condition and a separation assumption on almost active sets.