A stabilized mixed finite element approximation of bilinear state optimal control problems

We propose a stabilized mixed finite element approximation for optimal control problems governed by bilinear state equations. It is proved that the resulting mixed bilinear formulation is coercive and also continuous, which avoids the difficulty in choosing the mixed finite element spaces, i.e., the Ladyzhenkaya-Babuska-Brezzi matching condition for the mixed finite element spaces is unnecessary. Under pointwise bilateral constraint on the control variable, we deduce the optimality conditions at both continuous and discrete levels for the optimal control problems under consideration. Then an a priori error analysis in a weighted norm is discussed, with relatively low regularity requirements for the solutions to the optimal control problems. Finally, numerical experiments are given to confirm the efficiency and reliability of the proposed stabilized mixed finite element method.