We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.
{"title":"Boundary controllability for a 1D degenerate parabolic equation with drift and a singular potential","authors":"Leandro Galo-Mendoza, Marcos L'opez-Garc'ia","doi":"10.3934/mcrf.2023027","DOIUrl":"https://doi.org/10.3934/mcrf.2023027","url":null,"abstract":"We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75009987","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 a wave-wave transmission problem with generalized acoustic boundary conditions","authors":"M. Dimassi, A. Wehbe, H. Yazbek, Ibtissame Zaiter","doi":"10.3934/mcrf.2023031","DOIUrl":"https://doi.org/10.3934/mcrf.2023031","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76784763","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 consider a new class of mixed linear-quadratic Nash games and social optimization for two types of interactive agents. One is called a major agent and the others are minor agents. By 'mixed', we mean that all minor agents team up with each other to compete against this major agent for their contradictory cost functions. Different from the standard setup, this major's state is governed by some linear stochastic differential equation where the diffusion term and drift term both contain a control process, while the states of these minors are all weakly-coupled and driven by some linear backward stochastic differential equations because their terminal conditions are specified. To construct decentralized strategies for these two types of agents respectively, the backward person-by-person optimization method, combining some variational method and mean-field approximation are applied. Under some suitable conditions, we also verify the asymptotic optimality of these decentralized strategies.
{"title":"Mixed Nash games and social optima for linear-quadratic forward-backward mean-field systems","authors":"Xinwei Feng, Yiwei Lin","doi":"10.3934/mcrf.2023038","DOIUrl":"https://doi.org/10.3934/mcrf.2023038","url":null,"abstract":"We consider a new class of mixed linear-quadratic Nash games and social optimization for two types of interactive agents. One is called a major agent and the others are minor agents. By 'mixed', we mean that all minor agents team up with each other to compete against this major agent for their contradictory cost functions. Different from the standard setup, this major's state is governed by some linear stochastic differential equation where the diffusion term and drift term both contain a control process, while the states of these minors are all weakly-coupled and driven by some linear backward stochastic differential equations because their terminal conditions are specified. To construct decentralized strategies for these two types of agents respectively, the backward person-by-person optimization method, combining some variational method and mean-field approximation are applied. Under some suitable conditions, we also verify the asymptotic optimality of these decentralized strategies.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135318149","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":"Stackelberg equilibrium with social optima in linear-quadratic-Gaussian mean-field system","authors":"Xinwei Feng, Lu Wang","doi":"10.3934/mcrf.2023024","DOIUrl":"https://doi.org/10.3934/mcrf.2023024","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82798802","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":"Time-inconsistent stochastic linear-quadratic optimal control problem under non-Markovian regime-switching jump-diffusion model","authors":"I. Alia, M. Alia","doi":"10.3934/mcrf.2023030","DOIUrl":"https://doi.org/10.3934/mcrf.2023030","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83482321","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":"Recovering the velocity in a 1-d non-local transport equation","authors":"S. Ervedoza, Zhiqiang Wang, Jiacheng Zhang","doi":"10.3934/mcrf.2023011","DOIUrl":"https://doi.org/10.3934/mcrf.2023011","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86933614","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":"Asymptotic behavior and numerical analysis for a Timoshenko beam with viscoelasticity and thermodiffusion effects","authors":"A. Ramos, M. Aouadi, Imed Mahfoudhi, M. Freitas","doi":"10.3934/mcrf.2023012","DOIUrl":"https://doi.org/10.3934/mcrf.2023012","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85620615","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}
Here we deal with the problem of boundary asymptotic exponential stabilization of flows through porous media. More exactly we study the porous media equation with general monotone porosity in a bounded domain of dimension $ d = 1,2,3 $. We construct an explicit, linear, of finite-dimensional structure feedback controller with Dirichlet part-boundary actuation, which stabilizes any trajectory of the system, for any given initial data. The form of the controller is based on the spectrum of the Dirichlet-Laplace operator and ensures exponential decay to zero of the fluctuation variable for any a priori prescribed decay rate. Also, we extend these results to the case of porous media equation perturbed by Itô Lipschitz noise.
本文研究多孔介质流动的边界渐近指数稳定问题。更确切地说，我们研究了一维$ d = 1,2,3 $的有界区域内具有一般单调孔隙率的多孔介质方程。我们构造了一个具有Dirichlet部分边界驱动的显式线性有限维结构反馈控制器，对于任何给定的初始数据，该控制器可以稳定系统的任何轨迹。控制器的形式是基于狄利克雷-拉普拉斯算子的频谱，并确保波动变量的指数衰减到零对于任何先验规定的衰减率。同时，我们将这些结果推广到受Itô Lipschitz噪声扰动的多孔介质方程。
{"title":"Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation","authors":"Ionuţ Munteanu","doi":"10.3934/mcrf.2023037","DOIUrl":"https://doi.org/10.3934/mcrf.2023037","url":null,"abstract":"Here we deal with the problem of boundary asymptotic exponential stabilization of flows through porous media. More exactly we study the porous media equation with general monotone porosity in a bounded domain of dimension $ d = 1,2,3 $. We construct an explicit, linear, of finite-dimensional structure feedback controller with Dirichlet part-boundary actuation, which stabilizes any trajectory of the system, for any given initial data. The form of the controller is based on the spectrum of the Dirichlet-Laplace operator and ensures exponential decay to zero of the fluctuation variable for any a priori prescribed decay rate. Also, we extend these results to the case of porous media equation perturbed by Itô Lipschitz noise.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136257650","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":"Risk-based optimal portfolio of an insurance firm with regime switching and noisy memory","authors":"Calisto Guambe, Rodwell Kufakunesu, Lesedi Mabitsela","doi":"10.3934/mcrf.2023023","DOIUrl":"https://doi.org/10.3934/mcrf.2023023","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73380554","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}
The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential$ begin{equation*} frac{partial boldsymbol{y}}{partial t}-mu Deltaboldsymbol{y}+(boldsymbol{y}cdotnabla)boldsymbol{y}+alphaboldsymbol{y}+beta|boldsymbol{y}|^{r-1}boldsymbol{y}+nabla p+Psi(boldsymbol{y})niboldsymbol{g}, nablacdotboldsymbol{y} = 0, end{equation*} $in a $ d $-dimensional torus is considered in this work, where $ din{2,3} $, $ mu,alpha,beta>0 $ and $ rin[1,infty) $. For $ d = 2 $ with $ rin[1,infty) $ and $ d = 3 $ with $ rin[3,infty) $ ($ 2betamugeq 1 $ for $ d = r = 3 $), we establish the existence of a unique global strong solution for the above multi-valued problem with the help of the abstract theory of $ m $-accretive operators. Moreover, we demonstrate that the same results hold local in time for the case $ d = 3 $ with $ rin[1,3) $ and $ d = r = 3 $ with $ 2betamu<1 $. We explored the $ m $-accretivity of the nonlinear as well as multi-valued operators, Yosida approximations and their properties, and several higher order energy estimates in the proofs. For $ rin[1,3] $, we quantize (modify) the Navier-Stokes nonlinearity $ (boldsymbol{y}cdotnabla)boldsymbol{y} $ to establish the existence and uniqueness results, while for $ rin[3,infty) $ ($ 2betamugeq1 $ for $ r = 3 $), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $ beta|boldsymbol{y}|^{r-1}boldsymbol{y} $. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.
下面的对流Brinkman-Forchheimer (CBF)方程(或阻尼Navier-Stokes方程)在$ d $维环面中考虑位势$ begin{equation*} frac{partial boldsymbol{y}}{partial t}-mu Deltaboldsymbol{y}+(boldsymbol{y}cdotnabla)boldsymbol{y}+alphaboldsymbol{y}+beta|boldsymbol{y}|^{r-1}boldsymbol{y}+nabla p+Psi(boldsymbol{y})niboldsymbol{g}, nablacdotboldsymbol{y} = 0, end{equation*} $，其中$ din{2,3} $, $ mu,alpha,beta>0 $和$ rin[1,infty) $。对于$ d = 2 $与$ rin[1,infty) $和$ d = 3 $与$ rin[3,infty) $ ($ 2betamugeq 1 $与$ d = r = 3 $)，我们利用$ m $ -增生算子的抽象理论，建立了上述多值问题唯一全局强解的存在性。此外，我们证明了同样的结果在$ d = 3 $与$ rin[1,3) $和$ d = r = 3 $与$ 2betamu<1 $的情况下保持局部时间。我们在证明中探讨了非线性算子和多值算子的$ m $ -活跃性，Yosida近似及其性质，以及几个高阶能量估计。对于$ rin[1,3] $，我们量化(修改)Navier-Stokes非线性$ (boldsymbol{y}cdotnabla)boldsymbol{y} $以建立存在唯一性结果，而对于$ rin[3,infty) $ ($ 2betamugeq1 $ For $ r = 3 $)，我们通过非线性阻尼项$ beta|boldsymbol{y}|^{r-1}boldsymbol{y} $来处理Navier-Stokes非线性。最后，讨论了上述理论在流不变性、时间最优控制和镇定等反馈控制问题中的应用。
{"title":"2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems","authors":"Sagar Gautam, Kush Kinra, Manil T. Mohan","doi":"10.3934/mcrf.2023034","DOIUrl":"https://doi.org/10.3934/mcrf.2023034","url":null,"abstract":"The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential$ begin{equation*} frac{partial boldsymbol{y}}{partial t}-mu Deltaboldsymbol{y}+(boldsymbol{y}cdotnabla)boldsymbol{y}+alphaboldsymbol{y}+beta|boldsymbol{y}|^{r-1}boldsymbol{y}+nabla p+Psi(boldsymbol{y})niboldsymbol{g}, nablacdotboldsymbol{y} = 0, end{equation*} $in a $ d $-dimensional torus is considered in this work, where $ din{2,3} $, $ mu,alpha,beta>0 $ and $ rin[1,infty) $. For $ d = 2 $ with $ rin[1,infty) $ and $ d = 3 $ with $ rin[3,infty) $ ($ 2betamugeq 1 $ for $ d = r = 3 $), we establish the existence of a unique global strong solution for the above multi-valued problem with the help of the abstract theory of $ m $-accretive operators. Moreover, we demonstrate that the same results hold local in time for the case $ d = 3 $ with $ rin[1,3) $ and $ d = r = 3 $ with $ 2betamu<1 $. We explored the $ m $-accretivity of the nonlinear as well as multi-valued operators, Yosida approximations and their properties, and several higher order energy estimates in the proofs. For $ rin[1,3] $, we quantize (modify) the Navier-Stokes nonlinearity $ (boldsymbol{y}cdotnabla)boldsymbol{y} $ to establish the existence and uniqueness results, while for $ rin[3,infty) $ ($ 2betamugeq1 $ for $ r = 3 $), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $ beta|boldsymbol{y}|^{r-1}boldsymbol{y} $. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135754362","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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