We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrangian multiplier. Our final formulation fulfills the Babu\v{s}ka-Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a-posteriori error estimator which is efficient and reliable. Numerical experiments confirm our theoretical findings.
翻译:我们对受抛物线部分偏差方程式制约的分布式最佳控制问题提出了一种数字近似法,我们通过最近开发的对抛物线方程式进行空间-时间变异式配方来补充第一级最佳控制条件,这种配方在能源规范中具有强制性,以及拉格朗加的乘数。我们的最后配方在连续和离散水平上无限制地满足了Babu\v{s{s}ka-Brezzi的条件。因此,我们可以允许最终的预期状态,并获得高效和可靠的超时误差估计器。数字实验证实了我们的理论结论。