最优控制理论中变分偏微分方程的数值解

IF 2.5 3区数学 Q1 MATHEMATICS, APPLIED

Computational & Applied Mathematics Pub Date : 2012-01-01 DOI:10.1590/S1807-03022012000100003

V. Costanza, M. I. Troparevsky, P. Rivadeneira

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引用次数: 0

摘要

提出了一种基于Picard初始值问题的迭代方法，用于求解一阶矩阵值未知的拟线性偏微分方程，特别是最近发现的最优控制Hamilton方程中缺失边值的变分偏微分方程。作为说明，将迭代数值解与有限维非线性系统和正则拉格朗日最优控制问题的解析解以及通过标准数学软件得到的数值解进行了比较。由于LQR在具有广义代价的非线性系统的两自由度控制策略中所起的关键作用，本文讨论了与n维有限视界时变线性二次问题相关的(n + 1)维变分偏微分方程的应用。数学学科分类:初级:35F30;二级:93 10大。

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Numerical solution of the variational PDEs arising in optimal control theory

An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.

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来源期刊

Computational & Applied Mathematics Mathematics-Computational Mathematics

CiteScore

4.50

自引率

11.50%

发文量

352

审稿时长

>12 weeks

期刊介绍： Computational & Applied Mathematics began to be published in 1981. This journal was conceived as the main scientific publication of SBMAC (Brazilian Society of Computational and Applied Mathematics). The objective of the journal is the publication of original research in Applied and Computational Mathematics, with interfaces in Physics, Engineering, Chemistry, Biology, Operations Research, Statistics, Social Sciences and Economy. The journal has the usual quality standards of scientific international journals and we aim high level of contributions in terms of originality, depth and relevance.