{"title":"On Damping a Control System of Arbitrary Order with Global Aftereffect on a Tree","authors":"S. A. Buterin","doi":"10.1134/s0001434624050249","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study a problem of damping a control system described by functional-differential equations of natural order <span>\\(n\\)</span> and neutral type with nonsmooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order <span>\\(2n\\)</span> with nonlocal quasi-derivatives and multidirectional shifts of the argument as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050249","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study a problem of damping a control system described by functional-differential equations of natural order \(n\) and neutral type with nonsmooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order \(2n\) with nonlocal quasi-derivatives and multidirectional shifts of the argument as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved.
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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