variational discretization and mixed methods for semilinear parabolic optimal control problems with integral constraint

the aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. the state and co-state are approximated by the lowest order raviart-thomas mixed finite element spaces and the control is not discreted. optimal error estimates in l2 are established for the state and the control variable. as a result, it can be proved that the discrete solutions possess the convergence property of order. finally, a numerical example is presented which confirms the theoretical results.