Finite element error estimation for parabolic optimal control problems with time delay

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2025-02-05 DOI:10.1016/j.apnum.2025.02.002

Xindan Zhang , Jianping Zhao , Yanren Hou

引用次数: 0

Abstract

In this paper, we develop a priori error estimates for the finite element approximations of parabolic optimal control problems with time delay and pointwise control constraints. At first, we derive the first-order optimality systems for the control problems and the corresponding regularity results. Then, to approximate the problem we use the piecewise linear and continuous finite elements for the space discretization of the state, while the piecewise constant discontinuous Galerkin method is used for the time discretization. For the control discretization, we consider variational discretization. We show

O (k + h^{2})

order of convergence rate for the control in the

L^{2}

norm, which is new to the best of our knowledge. Finally, some numerical examples are provided to confirm our theoretical results.

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带时滞抛物型最优控制问题的有限元误差估计

在本文中，我们开发了具有时间延迟和点控制约束的抛物型最优控制问题的有限元逼近的先验误差估计。首先，我们导出了控制问题的一阶最优系统和相应的正则性结果。然后，采用分段线性连续有限元法对状态进行空间离散，采用分段常数不连续伽辽金法对状态进行时间离散。对于控制离散化，我们考虑变分离散化。我们在L2范数中显示了O（k+h2）阶的收敛率，这是我们所知的新知识。最后，通过数值算例验证了理论结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.