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In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. Furthermore, we also propose a space-time mixed finite element discretization scheme to approximate the space-time elliptic equations, and derive a priori error estimates for the state and the adjoint state. Numerical examples are presented to illustrate our theoretical findings and the performance of our approach. [ABSTRACT FROM AUTHOR]

A semi-linear elliptic control problems with distributed control and pointwise inequality constraints on the control and the state is considered. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. This class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data. [ABSTRACT FROM AUTHOR]

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and ad joint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations. [ABSTRACT FROM AUTHOR]