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沿着沿着未来轨迹的期望值或更高的矩的优化。第二种方法考虑控制的概率密度函数（PDF），根据一些目标，在最简单的情况下，包括控制的PDF预先指定的参考PDF。后一种方法在几个方面更强大和通用，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。但是，实现起来也比较困难。由于PDF可以通过Fokker-Planck方程的解表示-抛物型偏微分方程（PDE）-PDF的最优控制可以被视为具有PDE约束的（确定性）最优控制问题。由于最优控制的偏微分方程以及在模型预测控制的最新进展，这些方法打开了一个可行的方法来控制的PDF文件，所示的Annunziato和Borzovich在最近的论文有前途的结果。模型预测控制（MPC）是一种选择方法，因为它能够将长时间或无限时间范围内的最优控制问题分解为一系列较短时间范围内的最优控制问题，因此更容易解决。MPC方案分析中的两个关键问题是稳定性和性能。虽然“稳定性”表示受控PDF收敛到期望的参考PDF的事实，但“性能”测量MPC方法相对于长或无限范围上的真正最优解的最优性的损失。后者是特别重要的，如果参考PDF不是先验的，并编码在跟踪型功能，但隐含地从一个更一般的优化目标-最近引起了相当大的关注，在经济MPC的名义下的设置。本建议的理论部分的目的是得到严格的声明的稳定性和性能的福克-普朗克为基础的MPC。关键的挑战是由无限维偏微分方程设置和有意义的结构特性的基础随机控制系统的识别，允许这样的严格results. In每个MPC计划的核心，需要一个快速和可靠的数值算法的困难，以计算的解决方案的短期子问题。为此，发展有效的数值方法的基础上适当的最优性条件将补充理论研究作为本建议的第二个目标。除了在其本身的权利构成的贡献，有效的数值代码的可用性也将用于基于模拟的识别合适的假设，我们的理论结果和验证的MPC计划在本建议中得出的效率。

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.