BDF2 schemes for optimal parameter control problems governed by bilinear parabolic equations

In this paper, BDF2 schemes combined with the finite element method are proposed to solve control constrained optimal parameter control problems governed by bilinear parabolic equations, where the control variable acts as a reaction coefficient. The discretize-then-optimize (DO) approach is adopted. First, the state equation is discretized and two different discrete formulas are utilized for the objective functional. Then, by using the Lagrange multiplier method, two fully discrete first-order necessary optimality systems are deduced and named as DO1 and DO2 systems, respectively. It can be found that the optimize-then-discretize (OD) and discretize-then-optimize (DO) approaches coincide in the case of the DO2 system. Furthermore, a priori error estimates are derived for the state, co-state and control variables of the DO2 system, which show that the DO2 system has second-order accuracy in time. Finally, three numerical examples are presented to illustrate the theoretical findings.