Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations

This work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations. The minimization of the tracking functional subject to implicit parabolic equations is examined. Existence of an optimal solution is proved and an optimality system of equations is derived. Semi-discrete (in space) error estimates for the finite element approximations of the optimality system are presented. These estimates are symmetric and applicable for higher-order discretizations. Finally, fully-discrete error estimates of arbitrarily high-order are presented based on a discontinuous Galerkin (in time) and conforming (in space) scheme. Two examples related to the Lagrangian moving mesh Galerkin formulation for the convection-diffusion equation are described.