Specialized Adaptive Algorithms for Model Predictive Control of PDEs

模型预测控制是一种将无限或无限长范围内的最优控制问题的解分解为相对较短的有限时间范围内的最优控制问题的连续解的控制方法。然后，我们只使用每个最优控制的第一部分，以便在无限视界上综合得到的控制函数。在适当的条件下，可以证明该方法在无限视界上产生近似最优控制函数。此外，由于连续的再优化，结果控制是一种反馈型形式，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，因此可以预期，对于其数值计算，仅在优化区间的开始需要高精度，而在其结束时较低的精度就足够了。提出的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，该算法通过目标导向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出的最优控制相对于动力学扰动的灵敏度。我们期望在接近控制区间的开始处得到一个相对精细的离散化，在接近区间的结束处混合成越来越粗的离散化。当然，随着时间的推移，以前的网格会被重用。这将为抛物型偏微分方程模型预测控制提供一种有效的整体方法，该方法能以较低的计算量获得接近最优的无限水平性能。我们的策略是首先建立我们的ode方法，然后转向线性和非线性抛物型pde。算法的发展将结合理论研究最优控制的第一部分对动力学扰动的敏感性。特别是，我们将推导出一些条件，在这些条件下，我们可以严格证明最优控制的灵敏度随着时间的推移而降低。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.