{"title":"利用测量数据对抛物线优化控制进行有限元误差估计","authors":"Xun Yang, Xianbing Luo","doi":"10.1016/j.rinam.2024.100456","DOIUrl":null,"url":null,"abstract":"<div><p>A prior error estimate is considered for the finite element (FE) approximation of a parabolic optimal control (POC) with spatial measurement data. We use conforming linear finite element to discretize the space for the state, piecewise constant for the control, and Euler method to discretize the time. The convergence order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></mrow></math></span> in the <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>,</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>-norm of state variable, co-state, and control variable are obtained. To validate our theory, numerical tests are executed.</p></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"22 ","pages":"Article 100456"},"PeriodicalIF":1.4000,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2590037424000268/pdfft?md5=ad43a30f2f725635956ac9b65de5891f&pid=1-s2.0-S2590037424000268-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Finite element error estimation for parabolic optimal control with measurement data\",\"authors\":\"Xun Yang, Xianbing Luo\",\"doi\":\"10.1016/j.rinam.2024.100456\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A prior error estimate is considered for the finite element (FE) approximation of a parabolic optimal control (POC) with spatial measurement data. We use conforming linear finite element to discretize the space for the state, piecewise constant for the control, and Euler method to discretize the time. The convergence order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></mrow></math></span> in the <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>,</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>-norm of state variable, co-state, and control variable are obtained. To validate our theory, numerical tests are executed.</p></div>\",\"PeriodicalId\":36918,\"journal\":{\"name\":\"Results in Applied Mathematics\",\"volume\":\"22 \",\"pages\":\"Article 100456\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-04-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S2590037424000268/pdfft?md5=ad43a30f2f725635956ac9b65de5891f&pid=1-s2.0-S2590037424000268-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Results in Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2590037424000268\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2590037424000268","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Finite element error estimation for parabolic optimal control with measurement data
A prior error estimate is considered for the finite element (FE) approximation of a parabolic optimal control (POC) with spatial measurement data. We use conforming linear finite element to discretize the space for the state, piecewise constant for the control, and Euler method to discretize the time. The convergence order in the -norm of state variable, co-state, and control variable are obtained. To validate our theory, numerical tests are executed.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
