Numerical analysis of a stabilized optimal control problem governed by a parabolic convection--diffusion equation

We consider an optimal control problem on a bounded domain $\Omega\subset{\mathbb{R}}^2,$ governed by a parabolic convection--diffusion equation with pointwise control constraints. We follow the optimize-then-discretize-approach, where for the state and the co-state variable, we consider the piecewise finite element method alongside with the algebraic flux correction method for its stabilization and the for temporal discretization, we use the backward Euler for the state variable and explicit Euler for the co-state variable. The discrete control variable is obtained by the projection of the discretized adjoint state on the set of admissible controls. The resulting stabilized fully--discrete scheme is nonlinear and a fixed point argument is used in order to prove its existence and uniqueness under a mild condition between the time step $k$ and the mesh step $h,$ e.g., $k = \mathcal{O}(h^{1+\epsilon}),\,0<\epsilon<1.$ Further, for sufficiently regular solution, we derive error estimates in $L^2$ and $H^1$ norm with respect on space and $\ell^\infty$ norm in time for the state and the co-state variable. For the control variable we also derive an $L^2$ estimate for its error with respect to spatial variable and $\ell^\infty$ in time. Finally, we present numerical experiments that validate the the order of convergence of the stabilized fully--discrete scheme via algebraic flux correction method.