Heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients

In this paper, we investigate the heterogeneous multiscale method (HMM) for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients. The state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, while the control variable is discretized by the piecewise constants. By applying the well-known Lions' Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates are derived for the state, co-state and the control with uniform bounded constants. Finally, numerical results are presented to illustrate our theoretical findings.