{"title":"On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance","authors":"K. A. Shchelchkov","doi":"10.1134/s0012266124010105","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the problem of stabilization to zero under disturbance in terms of\na differential pursuit game. The dynamics is described by a nonlinear autonomous system of\ndifferential equations. The set of control values of the pursuer is finite, and that of the evader\n(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring the\ntrajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.\nTo construct the control, the pursuer knows only the state coordinates at some discrete times, and\nthe choice of the disturbance’s control is unknown. In the paper, we obtain conditions for the\nexistence of a neighborhood of zero from each point of which a capture occurs in the indicated\nsense. A winning control is constructed constructively and has an additional property specified in\na theorem. In addition, an estimate of the capture time sharp in some sense is produced.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"115 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124010105","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the problem of stabilization to zero under disturbance in terms of
a differential pursuit game. The dynamics is described by a nonlinear autonomous system of
differential equations. The set of control values of the pursuer is finite, and that of the evader
(disturbance) is a compact set. The control objective, i.e., the pursuer’s goal, is to bring the
trajectory to any predetermined neighborhood of zero in finite time regardless of the disturbance.
To construct the control, the pursuer knows only the state coordinates at some discrete times, and
the choice of the disturbance’s control is unknown. In the paper, we obtain conditions for the
existence of a neighborhood of zero from each point of which a capture occurs in the indicated
sense. A winning control is constructed constructively and has an additional property specified in
a theorem. In addition, an estimate of the capture time sharp in some sense is produced.
期刊介绍:
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.