P. P. Carvalho, J. Límaco, Denilson Menezes, Yuri Thamsten
We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.
我们研究了数学上描述一些非牛顿不可压缩粘性流动动力学的系统的零可控性。我们研究的主要模型是由O. a . Ladyzhenskaya提出的，尽管我们在这里开发的技术适用于具有剪切依赖粘度的其他流体。利用庞特里亚金最小原理，我们利用自启动论证来证明对强制线性化Stokes问题存在足够光滑的控制，只要初始数据反过来具有足够的规律性。由此，我们将结果推广到非线性问题。作为副产品，我们设计了一个准牛顿算法来计算状态和控制，我们证明了它在适当的意义上是收敛的。我们用一些数值实验来完成这项工作。
{"title":"Local null controllability of a class of non-Newtonian incompressible viscous fluids","authors":"P. P. Carvalho, J. Límaco, Denilson Menezes, Yuri Thamsten","doi":"10.3934/eect.2021043","DOIUrl":"https://doi.org/10.3934/eect.2021043","url":null,"abstract":"We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"22 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73081150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show how it is possible to define and study black holes of arbitrary shapes in the framework of Newtonian mechanics.
我们展示了如何在牛顿力学的框架下定义和研究任意形状的黑洞。
{"title":"A Newtonian approach to general black holes","authors":"A. Haraux","doi":"10.3934/eect.2023019","DOIUrl":"https://doi.org/10.3934/eect.2023019","url":null,"abstract":"We show how it is possible to define and study black holes of arbitrary shapes in the framework of Newtonian mechanics.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"58 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74963150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.
{"title":"Riccati-based solution to the optimal control of linear evolution equations with finite memory","authors":"P. Acquistapace, F. Bucci","doi":"10.3934/eect.2023035","DOIUrl":"https://doi.org/10.3934/eect.2023035","url":null,"abstract":"In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"88 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75709347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Cauchy problem for the Hartree type semilinear Schrödinger equation in the de Sitter spacetime","authors":"Mikio Ichimiya, M. Nakamura","doi":"10.3934/eect.2023028","DOIUrl":"https://doi.org/10.3934/eect.2023028","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"125 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91135163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Freitas, M. J. Santos, Dilberto S. Almeida Jr., Mauro L. Santos, A. Ramos
{"title":"Singular limits and dynamics for the abstract Timoshenko system","authors":"M. Freitas, M. J. Santos, Dilberto S. Almeida Jr., Mauro L. Santos, A. Ramos","doi":"10.3934/eect.2023037","DOIUrl":"https://doi.org/10.3934/eect.2023037","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"19 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87714510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cone-bounded feedback laws for linear second order systems","authors":"Abdellaziz Binid, A. M. Taboye, M. Laabissi","doi":"10.3934/eect.2023006","DOIUrl":"https://doi.org/10.3934/eect.2023006","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89767637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stabilization of an interconnected system of Schrödinger and wave equations with boundary coupling","authors":"Latifa Moumen, Fatima Zohra Sidiali, S. Rebiai","doi":"10.3934/eect.2023022","DOIUrl":"https://doi.org/10.3934/eect.2023022","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"39 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80221486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hyers-Ulam stability result for Hilfer fractional integrodifferential stochastic equations with fractional noises and non-instantaneous impulses","authors":"J. Priyadharsini, P. Balasubramaniam","doi":"10.3934/eect.2023042","DOIUrl":"https://doi.org/10.3934/eect.2023042","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"102 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79392079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximation of mild solutions of delay integro-differential equations on Banach spaces","authors":"Mohammed Elghandouri, K. Ezzinbi","doi":"10.3934/eect.2023029","DOIUrl":"https://doi.org/10.3934/eect.2023029","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"31 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79621803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundary exact controllability for the longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiros","authors":"M. Miranda, A. Lourêdo","doi":"10.3934/eect.2023024","DOIUrl":"https://doi.org/10.3934/eect.2023024","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"98 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85174237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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