基于动态规划粘性解及特征正交分解降维方法的偏微分方程最优控制

项目名称： 基于动态规划粘性解及特征正交分解降维方法的偏微分方程最优控制

项目编号： No.11471036

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 孙兵

作者单位： 北京理工大学

项目金额： 75万元

中文摘要： 控制系统设计的每一个步骤都离不开优化与反馈，而构造最优反馈控制是控制理论研究中最激动人心的课题之一，是控制理论梦寐以求的目标。然而，无论应用Pontryagin极大值原理还是Bellman古典动态规划方法，对于一般非线性系统，特别是无穷维系统，最优反馈律的解析求解是不可能的，数值求解是唯一可能且具有实践意义的途径。鉴于目前主要采用的基于极大值原理的打靶法要猜测初值，求出的控制是开环的。利用古典动态规划原理，数值求解高维HJB方程会引发维数灾等问题。本项目将基于DPVS-POD方法，在粘性解的框架下，数值求解经POD降维得到的低维HJB方程获得最优反馈控制。粘性解的引入给古典动态规划方法建立了严格的数学基础，数值求解HJB方程帮我们构造出最优反馈控制，POD方法的使用从本质上克服了维数灾难。研究中，我们将提出一般的算法并证明其收敛性，并将其应用到尽可能多的由偏微分方程描述的最优控制问题中去。

中文关键词： 动态规划粘性解；最优反馈控制；特征正交分解降维；数值解；收敛性

英文摘要： The control system design can not do without the optimization and feedback. Moreover, to synthesize the optimal feedback control is always one of the most exciting topics in control theory. And it is the Holy Grail of control theory. However, whether by the Pontryagin maximum principle or the Bellman classical dynamic programming approach, it is definitely not possible to find the analytical solutions of the optimal feedback law for the general nonlinear system, especially the infinite dimensional systems. The numerical solution is the only and practically effective approach. The currently corresponding numerical method mainly adopts the multiple shooting method, while it has happened the difficulty in guess of the initial data and the obtained control is of open-loop. On the other hand, the classical dynamic programming principle will evoke the curse of dimensionality in solving the Hamilton-Jacobi-Bellman (HJB) equation in high dimensions. In view of these facts above, this proposal will, on the basis of the DPVS-POD approach and under the frame of the viscosity solution, numerically solve the HJB equation in reduced dimensions by the POD method to obtain the optimal feedback control. The definition of the viscosity solution builds the rigorous mathematical foundations for the classical dynamic programming approach. And the numerical solution to the HJB equation helps us construct the optimal feedback control. Moreover, the POD method overcomes the curse of dimensionality in essence. In the investigation, we will give the effective algorithms and the corresponding proof of convergence. Then the algorithms will be used to investigate more optimal control problems governed by the partial differential equations.

英文关键词： Dynamic Programming Viscosity Solutions (DPVS);Optimal Feedback Control;Proper Orthogonal Decomposition (POD) Model Reduction;Numerical Solutions;Convergence