Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control – theory and numerical realization

Abstract

We consider control-constrained linear-quadratic optimal control problems on evolving hypersurfaces in ${\mathbb{R}}^{n+1}$. In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of vector-valued distributions. We then carry out and prove convergence of the variational discretization of a distributed optimal control problem. In the process, we investigate the convergence of a fully discrete approximation of the state equation, and obtain optimal orders of convergence under weak regularity assumptions. We conclude with a numerical example.