{"title":"Gradient Descent Method With Momentum Acceleration for \u0000 \u0000 \u0000 A\u0000 W\u0000 B\u0000 +\u0000 C\u0000 \u0000 W\u0000 T\u0000 \u0000 D\u0000 =\u0000 E\u0000 \u0000 $ AWB+CW^TD=E$\u0000 , Its Minimum Frobenius Norm Solution and Application in Time-Varying Linear Systems","authors":"Akbar Shirilord, Mehdi Dehghan","doi":"10.1049/cth2.70047","DOIUrl":"https://doi.org/10.1049/cth2.70047","url":null,"abstract":"<p>This study presents a gradient descent approach for addressing the matrix equation <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>W</mi>\u0000 <mi>B</mi>\u0000 <mo>+</mo>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>W</mi>\u0000 <mi>T</mi>\u0000 </msup>\u0000 <mi>D</mi>\u0000 <mo>=</mo>\u0000 <mi>E</mi>\u0000 </mrow>\u0000 <annotation>$AWB + CW^T D = E$</annotation>\u0000 </semantics></math>. Additionally, this method is utilized to solve the optimization problem <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>min</mi>\u0000 <mi>W</mi>\u0000 </msub>\u0000 <msup>\u0000 <mrow>\u0000 <mo>∥</mo>\u0000 <mi>A</mi>\u0000 <mi>W</mi>\u0000 <mi>B</mi>\u0000 <mo>+</mo>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>W</mi>\u0000 <mi>T</mi>\u0000 </msup>\u0000 <mi>D</mi>\u0000 <mo>−</mo>\u0000 <mi>E</mi>\u0000 <mo>∥</mo>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$ min _{W} Vert AWB + CW^T D - EVert ^2$</annotation>\u0000 </semantics></math> with the Frobenius norm. We provide a comprehensive analysis of the convergence and characteristics of these techniques. To improve the convergence rate, we incorporate a specific variant of the momentum method. To validate the effectiveness of our proposed iterative methods, we offer various numerical examples and compare the outcomes with those of existing algorithms. Lastly, we investigate an application within the context of time-varying linear systems.</p>","PeriodicalId":50382,"journal":{"name":"IET Control Theory and Applications","volume":"19 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1049/cth2.70047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144598411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}