Specialized Adaptive Algorithms for Model Predictive Control of PDEs

模型预测控制是将无限或无限长时间域上的最优控制问题的求解分解为相对较短的有限时间域上的最优控制问题的逐次求解的控制方法。然后，我们只使用每个最优控制的第一部分，以便在无限的水平上综合所产生的控制函数。在适当的条件下，该方法可以在无穷大范围内产生近似最优的控制函数。此外，由于连续的重新优化，所得到的控制是反馈类型的形式，该形式提供了对模型误差和扰动的鲁棒性。由于只使用每个最优控制函数的第一部分，可以预期，对于其数值计算，仅在优化区间的开始时需要高精度，而在接近其结束时较低的精度就足够了。该项目的主要目标是构建抛物型偏微分方程模型预测控制的数值算法，通过面向目标的误差估计和自适应来利用这一事实。由我们的算法构建的网格将反映在控制区间开始时计算的最优控制对动态扰动的敏感性。我们预计，我们将在控制区间开始附近获得相对精细的离散化，并在该区间结束时混合成更粗和更粗的离散化。当然，随着时间范围的推移，以前的网格会被重复使用。这将产生一种有效的抛物型偏微分方程组的全局模型预测控制方法，以较低的计算量获得接近最优的无限时间性能。我们的策略是首先建立常微分方程组的预测控制方法，然后转移到线性和非线性抛物型偏微分方程组。算法的发展将与关于最优控制的第一部分相对于动力学扰动的灵敏度的理论研究相结合。特别地，我们将推导出严格证明最优控制的灵敏度随时间递减的条件。这将加深对新创建的算法的理解，并确定可以应用这些算法的问题类别。

Model Predictive Control is a control method in which the solution of optimal control problems on infinite or indefinitely long horizons is split up into the successive solution of optimal control problems on relatively short finite time horizons. Only the first piece of each optimal control us then used in order to synthesize the resulting control function on the infinite horizon. Under suitable conditions, this method can be shown to produce approximately optimal control functions on the infinite horizon. Moreover, due to the successive re-optimization the resulting control is of a feedback type form which provides robustness against model errors and perturbations. Since only the first piece of each optimal control function is used, one can expect that for its numerical computation a high accuracy is only needed at the beginning of the optimization interval while a lower accuracy is sufficient towards its end. The main objective of the proposed project is the construction of numerical algorithms for model predictive control of parabolic partial differential equations which exploit this fact via goal oriented error estimation and adaptivity.The grids, constructed by our algorithm, will reflect the sensitivity of the computed optimal control at the beginning of the control interval with respect to perturbations of the dynamics. We expect that we will obtain a relatively fine discretization near the beginning of the control interval that blends into coarser and coarser discretizations towards the end of this interval. Of course, previous grids are reused as the time-horizon moves on. This will result in an efficient overall method for model predictive control of parabolic PDEs that obtains a near optimal infinite horizon performance with low computational effort.Our strategy is to first establish our method for ODEs and then move on to linear and non-linear parabolic PDEs. The algorithmic development will be combined with theoretical investigations on the sensitivity of the first piece of the optimal control with respect to perturbations of the dynamics. Particularly, we will derive conditions under which we can rigorously prove that the sensitivity of the optimal control decreases over time. This will create a deeper understanding of the newly created algorithms and identify classes of problems for which these algorithms can be applied.