Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold (Q2801785)

The aim of this paper is to analyze 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. The motivation to consider such optimal control problems comes from the fact that the support of the controls needs to be very small compared to the total size of the domain \(\Omega\) if one is restricted by the cost of controls. The authors analyze at first the well-posedness of the state equation in order to obtain first order optimality conditions for the control problems and the corresponding regularity results. Then, they consider the fully discrete finite element approximations for the state equation based on the piecewise constant discontinuous Galerkin scheme for time discretization and piecewise linear finite elements for space discretization. A-priori error estimates for the state approximations are also derived. For the fully discretized control problems, a variational discretization in the control variable is used, and a-priori error estimates supported by numerical examples are obtained.