{"title":"布辛斯克系统的堆栈伯格精确可控性","authors":"Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang","doi":"10.1007/s00030-024-00971-2","DOIUrl":null,"url":null,"abstract":"<p>We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the “leader” control is the null-controllability property whereas the objective of the “follower” control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stackelberg exact controllability for the Boussinesq system\",\"authors\":\"Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang\",\"doi\":\"10.1007/s00030-024-00971-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the “leader” control is the null-controllability property whereas the objective of the “follower” control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.</p>\",\"PeriodicalId\":501665,\"journal\":{\"name\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"volume\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00030-024-00971-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-024-00971-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Stackelberg exact controllability for the Boussinesq system
We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the “leader” control is the null-controllability property whereas the objective of the “follower” control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.