Space-Time Finite Element Approximation of Parabolic Optimal Control Problems

Michael Hinze*, Wei Gong and Zhaojie Zhou

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.

Mathematics Subject Classification: 65N30 65N50

Keywords: Parabolic optimal control, a priori error estimates, a posteriori error control

Minisymposion: Stability, Sensitivity and Error Analysis for Optimal Control Problems