The authors present a discrete formulation of parabolic optimal control problems based on the reformulation of their optimality conditions as second order in time and fourth order in space elliptic boundary value problems. The authors prove existence and uniqueness of solutions. Since time and space have different physical meanings, temporal and spatial discretization are separated. A priori and a posteriori error analysis for the temporal discretization is proposed. Here, in the a posteriori error analysis part the authors construct residual based error estimators for the time discretization and keep the space variable continuous. The key idea here consists in applying residual-based a posteriori error estimation techniques for two-point boundary value problems for the state and the adjoint state. A priori error estimates for temporal semi-discretization with piecewise linear, continuous finite elements are proved. Furthermore, the authors propose a space-time mixed finite element method to approximate the state and the adjoint state for which a priori error estimates are proved.
Main result: The authors present the first order optimality conditions for parabolic optimal control problems. The space-time domain elliptic systems for the state and the adjoint state are derived. The proof of the existence and uniqueness of solutions to the space-time elliptic boundary value problems is given. A priori and a posteriori error estimates for the time discretization scheme are derived. The numerical analysis of the space-time mixed finite element approximations of the state and adjoint state is investigated. Finally, numerical examples to illustrate the authors analytical findings are presented.