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Evolution Equations and Control Theory最新文献

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Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients 非lipschitz系数中立型随机泛函微分方程的不变测度
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-11-11 DOI: 10.3934/eect.2022005
A. Stanzhytskyi, Oleksandr Stanzhytskyi, Oleksandr Misiats
{"title":"Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients","authors":"A. Stanzhytskyi, Oleksandr Stanzhytskyi, Oleksandr Misiats","doi":"10.3934/eect.2022005","DOIUrl":"https://doi.org/10.3934/eect.2022005","url":null,"abstract":"In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88272467","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}
引用次数: 3
Blow-up of solutions to semilinear wave equations with a time-dependent strong damping 具有时变强阻尼的半线性波动方程解的爆破
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-11-02 DOI: 10.3934/eect.2022006
A. Fino, M. Hamza
{"title":"Blow-up of solutions to semilinear wave equations with a time-dependent strong damping","authors":"A. Fino, M. Hamza","doi":"10.3934/eect.2022006","DOIUrl":"https://doi.org/10.3934/eect.2022006","url":null,"abstract":"<p style='text-indent:20px;'>The paper investigates a class of a semilinear wave equation with time-dependent damping term (<inline-formula><tex-math id=\"M1\">begin{document}$ -frac{1}{{(1+t)}^{beta}}Delta u_t $end{document}</tex-math></inline-formula>) and a nonlinearity <inline-formula><tex-math id=\"M2\">begin{document}$ |u|^p $end{document}</tex-math></inline-formula>. We will show the influence of the parameter <inline-formula><tex-math id=\"M3\">begin{document}$ beta $end{document}</tex-math></inline-formula> in the blow-up results under some hypothesis on the initial data and the exponent <inline-formula><tex-math id=\"M4\">begin{document}$ p $end{document}</tex-math></inline-formula> by using the test function method. We also study the local existence in time of mild solution in the energy space <inline-formula><tex-math id=\"M5\">begin{document}$ H^1(mathbb{R}^n)times L^2(mathbb{R}^n) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74927623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain 在周期域上得到色散PDE线性族精确可控性和稳定性的两个简单判据
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-10-05 DOI: 10.3934/eect.2021062
F. V. Leal, A. Pastor
{"title":"Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain","authors":"F. V. Leal, A. Pastor","doi":"10.3934/eect.2021062","DOIUrl":"https://doi.org/10.3934/eect.2021062","url":null,"abstract":"<p style='text-indent:20px;'>In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id=\"M1\">begin{document}$ H_{p}^{s}(mathbb{T}) $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M2\">begin{document}$ sin mathbb{R}. $end{document}</tex-math></inline-formula> We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id=\"M3\">begin{document}$ H_{p}^{s}(mathbb{T}) $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M4\">begin{document}$ sin mathbb{R}. $end{document}</tex-math></inline-formula></p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75271790","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}
引用次数: 4
Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces 各向异性Hölder空间中分数阶各向异性抛物方程的Cauchy问题
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-09-23 DOI: 10.3934/eect.2022029
S. Degtyarev
{"title":"Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces","authors":"S. Degtyarev","doi":"10.3934/eect.2022029","DOIUrl":"https://doi.org/10.3934/eect.2022029","url":null,"abstract":"We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another conditions we prove unique solvability of the Cauchy problem in the same spaces.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75631962","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}
引用次数: 0
Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions 一般动态边界条件下热方程的Stackelberg-Nash零可控性
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-09-06 DOI: 10.3934/eect.2021044
I. Boutaayamou, L. Maniar, O. Oukdach
{"title":"Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions","authors":"I. Boutaayamou, L. Maniar, O. Oukdach","doi":"10.3934/eect.2021044","DOIUrl":"https://doi.org/10.3934/eect.2021044","url":null,"abstract":"This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78047505","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}
引用次数: 4
Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity 具有非局部弱阻尼、抗阻尼和临界非线性的波动方程的渐近性质
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-08-17 DOI: 10.3934/eect.2022025
Chunyan Zhao, C. Zhong, Zhi-Yi Tang
{"title":"Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity","authors":"Chunyan Zhao, C. Zhong, Zhi-Yi Tang","doi":"10.3934/eect.2022025","DOIUrl":"https://doi.org/10.3934/eect.2022025","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80133947","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}
引用次数: 2
The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(mathbb R^{n}) $ H^{s}(mathbb R^{n}) $中临界非齐次非线性Schrödinger方程的Cauchy问题
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-07-02 DOI: 10.3934/eect.2022059
J. An, Jinmyong Kim
{"title":"The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(mathbb R^{n}) $","authors":"J. An, Jinmyong Kim","doi":"10.3934/eect.2022059","DOIUrl":"https://doi.org/10.3934/eect.2022059","url":null,"abstract":"In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation iut +∆u = |x| f(u), u(0) = u0 ∈ H (R), where n ≥ 3, 1 ≤ s < n2 , 0 < b < 2 and f(u) is a nonlinear function that behaves like λ |u| σ u with λ ∈ C and σ = 4−2b n−2s . We establish the local well-posedness as well as the small data global well-posedness and scattering in H(R) with 1 ≤ s < n2 for the critical INLS equation under some assumption on b. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74096933","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}
引用次数: 0
On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping 基于惯性和Hessian驱动阻尼的一阶优化方法中微扰的影响
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-06-30 DOI: 10.3934/eect.2022022
H. Attouch, J. Fadili, V. Kungurtsev
{"title":"On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping","authors":"H. Attouch, J. Fadili, V. Kungurtsev","doi":"10.3934/eect.2022022","DOIUrl":"https://doi.org/10.3934/eect.2022022","url":null,"abstract":"Second-order continuous-time dissipative dynamical systems with viscous and Hessian driven damping have inspired effective first-order algorithms for solving convex optimization problems. While preserving the fast convergence properties of the Nesterov-type acceleration, the Hessian driven damping makes it possible to significantly attenuate the oscillations. To study the stability of these algorithms with respect to perturbations, we analyze the behaviour of the corresponding continuous systems when the gradient computation is subject to exogenous additive errors. We provide a quantitative analysis of the asymptotic behaviour of two types of systems, those with implicit and explicit Hessian driven damping. We consider convex, strongly convex, and non-smooth objective functions defined on a real Hilbert space and show that, depending on the formulation, different integrability conditions on the perturbations are sufficient to maintain the convergence rates of the systems. We highlight the differences between the implicit and explicit Hessian damping, and in particular point out that the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80111813","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}
引用次数: 2
Stability estimate for a partial data inverse problem for the convection-diffusion equation 对流扩散方程部分数据反问题的稳定性估计
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-04-25 DOI: 10.3934/eect.2021060
Soumen Senapati, Manmohan Vashisth
{"title":"Stability estimate for a partial data inverse problem for the convection-diffusion equation","authors":"Soumen Senapati, Manmohan Vashisth","doi":"10.3934/eect.2021060","DOIUrl":"https://doi.org/10.3934/eect.2021060","url":null,"abstract":"<p style='text-indent:20px;'>In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension <inline-formula><tex-math id=\"M1\">begin{document}$ nge 2 $end{document}</tex-math></inline-formula>, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84520465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A shape optimization problem constrained with the Stokes equations to address maximization of vortices 用Stokes方程约束的形状优化问题求解涡旋的最大化
IF 1.5 4区 数学
Evolution Equations and Control Theory Pub Date : 2021-04-20 DOI: 10.3934/eect.2022003
J. Simon, H. Notsu
{"title":"A shape optimization problem constrained with the Stokes equations to address maximization of vortices","authors":"J. Simon, H. Notsu","doi":"10.3934/eect.2022003","DOIUrl":"https://doi.org/10.3934/eect.2022003","url":null,"abstract":"We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the begin{document}$ L^2 $end{document}-norm of the curl and the det-grad measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87374033","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}
引用次数: 2
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