Model Predictive Control for the Fokker-Planck Equation

对于随机控制系统的最优控制，文献中出现了两种不同的方法。第一种更传统的方法考虑了沿未来轨迹的期望值或更高阶矩的最优化。第二种方法考虑根据某个目标对概率密度函数(PDF)进行控制，在最简单的情况下，该目标包括将PDF控制到预先指定的参考PDF。后一种方法在几个方面更加强大和多才多艺，因为它允许塑造未来轨迹的整个分布，而不是只影响某些时刻。然而，它也更难实现。由于PDF可以通过Fokker-Planck方程-抛物型偏微分方程(PDE)的解来表示，因此PDF的最优控制可以假设为具有抛物型偏微分方程约束的(确定性)最优控制问题。由于PDE的最优控制和模型预测控制的最新进展，这些方法为PDF的控制开辟了一条可行的途径，Annunziato和Borz？在最近的论文中展示了令人振奋的结果。模型预测控制(MPC)能够将一个长时间或无限时间的最优控制问题分解成一系列较短时间内的最优控制问题，从而更容易求解，因此模型预测控制(MPC)是一种可供选择的方法。分析MPC方案的两个关键问题是稳定性和性能。“稳定性”表示受控PDF收敛到期望的参考PDF的事实，而“性能”衡量的是MPC方法相对于长期或无限水平上的真正最优解的最优性损失。如果参考PDF不是先验的，并以跟踪类型泛函编码，而是隐含地从更一般的优化目标中导出--这一设置最近在经济货币政策委员会的名义下引起了相当大的关注，则后一种情况尤其重要。这项提案的理论部分的目标是推导出关于基于福克-普朗克的MPC的稳定性和性能的严格声明。关键的挑战是无限维偏微分方程组的设置和识别潜在的随机控制系统的有意义的结构性质所带来的困难，以允许如此严格的结果。在每个MPC方案的核心，需要一个快速可靠的数值算法来计算短时程子问题的解。为此，根据适当的最佳性条件发展有效的数值方法将补充作为本提议的第二个目标的理论研究。除了本身的贡献，有效的数值代码的可用性还将用于基于模拟的识别适合于我们的理论结果的假设，并用于验证在本提案中推导的MPC方案的效率。

For the optimal control of stochastic control systems two different approaches appear in the literature. The first, more traditional approach considers the optimization of expected values or higher moments along future trajectories. The second approach considers the control of the probability density function (PDF) according to some objective, which in the simplest case consists of controlling the PDF to a prespecified reference PDF. The latter approach is in several respects more powerful and versatile, as it allows to shape the entire distribution of the future trajectories opposed to influencing only some moments. However, it is also more difficult to realize. As the PDF can be expressed via the solution of the Fokker-Planck equation - a parabolic partial differential equation (PDE) - the optimal control of PDFs can be posed as a (deterministic) optimal control problems with PDE constraints. Due to the recent progress in optimal control of PDEs as well as in model predictive control, these methods open a feasible way to the control of PDFs, as illustrated by promising results in recent papers by Annunziato and Borzì. Model predictive control (MPC) is the method of choice due to its ability to split up an optimal control problem on a long or infinite time horizon into a series of optimal control problems on shorter horizons which are thus much easier to solve. Two of the key issues in the analysis of MPC schemes are stability and performance. While "stability" expresses the fact that the controlled PDF converges to a desired reference PDF, "performance" measures the loss of optimality of the MPC approach with respect to the true optimal solution on the long or infinite horizon. The latter is particularly important if the reference PDF is not given a priori and encoded in a tracking type functional but implicitly derived from a more general optimization objective - a setting which recently attracted considerable attention under the name of economic MPC. The objective of the theoretical part of this proposal is to derive rigorous statements about stability and performance of Fokker-Planck based MPC. The key challenges are the difficulties introduced by the infinite dimensional PDE setting and the identification of meaningful structural properties of the underlying stochastic control system allowing for such rigorous results.At the core of each MPC scheme a fast and reliable numerical algorithm is needed in order to compute the solutions of the short horizon subproblems. To this end, the development of efficient numerical methods based on suitable optimality conditions will complement the theoretical investigations as a second objective of this proposal. Besides constituting a contribution in its own right, the availability of efficient numerical codes will also serve for the simulation based identification of suitable assumptions for our theoretical results and for verifying the efficiency of the MPC schemes derived in this proposal.