A Posteriori Error Estimates for Parabolic Optimal Control Problems with Controls Acting on Lower Dimensional Manifolds

This article concerns a posteriori error estimates for the fully discrete finite element approximation to the optimal control problem governed by parabolic partial differential equations where the control is acting on lower dimensional manifolds. The manifold considered in this paper involves either a point, or a curve or a surface which is lying completely in the space domain. Further, the manifold is assumed to be either time independent or evolved with the time. The space discretization consists of piecewise linear and continuous finite elements for the state and co-state variables and the piecewise constant functions are employed to approximate the control variable. Moreover, the time derivative is approximated by using the backward Euler scheme. We derive a posteriori error estimates for the various dimensions of the manifold. Our numerical results exhibit the effectiveness of the derived error estimators.