{"title":"Spiral-Like Extremals near a Singular Surface in a Rocket Control Problem","authors":"Mariya I. Ronzhina, Larisa A. Manita","doi":"10.1134/S1560354723020028","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc.\nWe study extremals in the neighbourhood of singular points of the second order.\nOur approach is based on applying the method of a descending system of Poisson\nbrackets and the Zelikin – Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhood\nof any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 2","pages":"148 - 161"},"PeriodicalIF":0.8000,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354723020028","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc.
We study extremals in the neighbourhood of singular points of the second order.
Our approach is based on applying the method of a descending system of Poisson
brackets and the Zelikin – Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhood
of any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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