A numerical approach for solving a class of two-dimensional variable-order fractional optimal control problems using Gegenbauer operational matrix

IF 1.6 4区计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS

IMA Journal of Mathematical Control and Information Pub Date : 2023-01-01 DOI:10.1093/imamci/dnac031

Farzaneh Soufivand;Fahimeh Soltanian;Kamal Mamehrashi

{"title":"A numerical approach for solving a class of two-dimensional variable-order fractional optimal control problems using Gegenbauer operational matrix","authors":"Farzaneh Soufivand;Fahimeh Soltanian;Kamal Mamehrashi","doi":"10.1093/imamci/dnac031","DOIUrl":null,"url":null,"abstract":"This study presents a spectral method for solving the two-dimensional variable-order fractional optimal control problems (2D-VOFOCPs). In this work, a dynamic system with variable-order fractional derivatives appears. The Caputo derivative, which is one of the most widely used and essential types of fractional derivatives, has been used to construct operational matrices. The shifted Gegenbauer polynomials are used as orthogonal bases. For this purpose, at first, the control and state functions are approximated by the shifted Gegenbauer polynomials with unknown coefficients. Then, by substituting the approximated functions into initial and boundary conditions, the dynamical system and the objective function, an algebraic equation system is achieved. The solution of the obtained system of the algebraic equation is equivalent to the solution of 2D-VOFOCP. Furthermore, the convergence of the method is studied. Eventually, two numerical examples are presented to illustrate the applicability and accuracy of the proposed method.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"40 1","pages":"1-19"},"PeriodicalIF":1.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"IMA Journal of Mathematical Control and Information","FirstCategoryId":"94","ListUrlMain":"https://ieeexplore.ieee.org/document/10078814/","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}

引用次数: 1

Abstract

This study presents a spectral method for solving the two-dimensional variable-order fractional optimal control problems (2D-VOFOCPs). In this work, a dynamic system with variable-order fractional derivatives appears. The Caputo derivative, which is one of the most widely used and essential types of fractional derivatives, has been used to construct operational matrices. The shifted Gegenbauer polynomials are used as orthogonal bases. For this purpose, at first, the control and state functions are approximated by the shifted Gegenbauer polynomials with unknown coefficients. Then, by substituting the approximated functions into initial and boundary conditions, the dynamical system and the objective function, an algebraic equation system is achieved. The solution of the obtained system of the algebraic equation is equivalent to the solution of 2D-VOFOCP. Furthermore, the convergence of the method is studied. Eventually, two numerical examples are presented to illustrate the applicability and accuracy of the proposed method.

查看原文本刊更多论文

用Gegenbauer运算矩阵求解一类二维变阶分数最优控制问题的数值方法

本文提出了一种求解二维变阶分数最优控制问题的谱方法。在这项工作中，出现了一个具有变阶分数导数的动态系统。Caputo导数是应用最广泛和最基本的分数导数类型之一，已被用于构造运算矩阵。移位后的Gegenbauer多项式被用作正交基。为此，首先，通过具有未知系数的移位Gegenbauer多项式来近似控制函数和状态函数。然后，通过将近似函数代入初始条件和边界条件、动力系统和目标函数，得到代数方程组。所得到的代数方程组的解等价于2D-VOFOCP的解。此外，还研究了该方法的收敛性。最后，通过两个算例说明了该方法的适用性和准确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

IMA Journal of Mathematical Control and Information 工程技术-应用数学

CiteScore

3.30

自引率

6.70%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal is to provide an outlet for papers which are original and of high quality in mathematical control theory, systems theory, and applied information sciences. Short papers and mathematical correspondence or technical notes will be welcome, although the primary function of the journal is to publish papers of substantial length and coverage. The emphasis will be upon relevance, originality and clarify of presentation, although timeliness may well be an important feature in acceptable papers. Speculative papers that suggest new avenues for research or potential solutions to unsolved problems of control and information theory will be particularly welcome. Specific application papers will not normally be within the remit of the journal. Applications that illustrate techniques or theories will be acceptable. A prime function of the journal is to encourage the interplay between control and information theory and other mathematical sciences. All submitted papers will be judged on their merits by at least two referees and a full paper report will be available to the intending authors. Submitted articles will in general be published in an issue within six months of submission. Papers should not have previously published, nor should they be undes consideration for publication in another journal. This Journal takes publication ethics very seriously. If misconduct is found or suspected after the manuscript is published, the journal will investigate the matter and this may result in the article subsequently being retracted.