Finite Element Approximations of Parabolic Optimal Control Problems with Controls Acting on a Lower Dimensional Manifold

This paper is devoted to the study of finite element approximations to parabolic optimal control problems with controls acting on a lower dimensional manifold. The manifold can be a point, a curve or a surface which may be independent of time or evolve in the time horizon, and is assumed to be strictly contained in the space domain. At first, we obtain the first order optimality conditions for the control problems and the corresponding regularity results. Then, for the control problems we consider the fully discrete finite element approximations based on the dG(0) scheme for time discretization and piecewise linear finite elements for space discretization, and variational discretization to the control variable. A priori error estimates are finally obtained for the fully discretised control problems and supported by numerical examples.