A. Stanzhytskyi, Oleksandr Stanzhytskyi, Oleksandr Misiats
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.
{"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":"43 1","pages":""},"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}
The paper investigates a class of a semilinear wave equation with time-dependent damping term (begin{document}$ -frac{1}{{(1+t)}^{beta}}Delta u_t $end{document}) and a nonlinearity begin{document}$ |u|^p $end{document}. We will show the influence of the parameter begin{document}$ beta $end{document} in the blow-up results under some hypothesis on the initial data and the exponent begin{document}$ p $end{document} by using the test function method. We also study the local existence in time of mild solution in the energy space begin{document}$ H^1(mathbb{R}^n)times L^2(mathbb{R}^n) $end{document}.
The paper investigates a class of a semilinear wave equation with time-dependent damping term (begin{document}$ -frac{1}{{(1+t)}^{beta}}Delta u_t $end{document}) and a nonlinearity begin{document}$ |u|^p $end{document}. We will show the influence of the parameter begin{document}$ beta $end{document} in the blow-up results under some hypothesis on the initial data and the exponent begin{document}$ p $end{document} by using the test function method. We also study the local existence in time of mild solution in the energy space begin{document}$ H^1(mathbb{R}^n)times L^2(mathbb{R}^n) $end{document}.
{"title":"Blow-up of solutions to semilinear wave equations with a time-dependent strong damping","authors":"A. Fino, M. Hamza","doi":"10.3934/eect.2022006","DOIUrl":"https://doi.org/10.3934/eect.2022006","url":null,"abstract":"<p style='text-indent:20px;'>The paper investigates a class of a semilinear wave equation with time-dependent damping term (<inline-formula><tex-math id=\"M1\">begin{document}$ -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) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"57 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74927623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in begin{document}$ H_{p}^{s}(mathbb{T}) $end{document} with begin{document}$ sin mathbb{R}. $end{document} We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in begin{document}$ H_{p}^{s}(mathbb{T}) $end{document} with begin{document}$ sin mathbb{R}. $end{document}
In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in begin{document}$ H_{p}^{s}(mathbb{T}) $end{document} with begin{document}$ sin mathbb{R}. $end{document} We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in begin{document}$ H_{p}^{s}(mathbb{T}) $end{document} with begin{document}$ sin mathbb{R}. $end{document}
{"title":"Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain","authors":"F. V. Leal, A. Pastor","doi":"10.3934/eect.2021062","DOIUrl":"https://doi.org/10.3934/eect.2021062","url":null,"abstract":"<p style='text-indent:20px;'>In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id=\"M1\">begin{document}$ 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":"36 1","pages":""},"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}
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.
{"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":"3 1","pages":""},"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}
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.
{"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":"27 1","pages":""},"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}
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.
本文证明了具有非局部弱阻尼、非局部抗阻尼和临界非线性的波动方程的全局吸引子的存在性。
{"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":"11 1","pages":""},"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}
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.
{"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":"93 1","pages":""},"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}
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.
{"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":"24 1","pages":""},"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}
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 begin{document}$ nge 2 $end{document}, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.
In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension begin{document}$ nge 2 $end{document}, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.
{"title":"Stability estimate for a partial data inverse problem for the convection-diffusion equation","authors":"Soumen Senapati, Manmohan Vashisth","doi":"10.3934/eect.2021060","DOIUrl":"https://doi.org/10.3934/eect.2021060","url":null,"abstract":"<p style='text-indent:20px;'>In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension <inline-formula><tex-math id=\"M1\">begin{document}$ nge 2 $end{document}</tex-math></inline-formula>, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"60 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84520465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the begin{document}$ L^2 $end{document}-norm of the curl and the det-grad measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.
We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the begin{document}$ L^2 $end{document}-norm of the curl and the det-grad measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.
{"title":"A shape optimization problem constrained with the Stokes equations to address maximization of vortices","authors":"J. Simon, H. Notsu","doi":"10.3934/eect.2022003","DOIUrl":"https://doi.org/10.3934/eect.2022003","url":null,"abstract":"We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the begin{document}$ 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":"35 1","pages":""},"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}
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