Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. By applying some error estimates techniques of standard mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present some numerical examples which confirm our theoretical results.