We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $end{document} in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.
We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $end{document} in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.
{"title":"A note on control of one-dimensional heterogeneous reaction-diffusion equations","authors":"M. Sônego, R. Roychowdhury","doi":"10.3934/eect.2022035","DOIUrl":"https://doi.org/10.3934/eect.2022035","url":null,"abstract":"<p style='text-indent:20px;'>We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form <inline-formula><tex-math id=\"M1\">begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $end{document}</tex-math></inline-formula> in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82544633","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":"Uniform stabilization for a string/point mass system via arbitrary local memory effects versus frictional damping","authors":"Kun‐Peng Jin, Chan Li","doi":"10.3934/eect.2022056","DOIUrl":"https://doi.org/10.3934/eect.2022056","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88220065","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 prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic begin{document}$ p $end{document}-Laplacian and a weighted anisotropic begin{document}$ q $end{document}-Laplacian (called the weighted anisotropic begin{document}$ (p,q) $end{document}-Laplacian), a multivalued reaction term depending on the gradient, two multivalued boundary conditions and an obstacle constraint. We, first, employ the theory of nonsmooth analysis and a surjectivity theorem for pseudomonotone operators to prove the existence of a nontrivial solution of the anisotropic elliptic obstacle problem, which relies on the first eigenvalue of the Steklov eigenvalue problem for the begin{document}$ p_$end{document}-Laplacian. Then, we introduce the parameter-to-solution map for the anisotropic elliptic obstacle problem, and establish a critical convergence result of the Kuratowski type to parameter-to-solution map. Finally, a general framework is proposed to examine the solvability of the nonlinear inverse problem.
The prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic begin{document}$ p $end{document}-Laplacian and a weighted anisotropic begin{document}$ q $end{document}-Laplacian (called the weighted anisotropic begin{document}$ (p,q) $end{document}-Laplacian), a multivalued reaction term depending on the gradient, two multivalued boundary conditions and an obstacle constraint. We, first, employ the theory of nonsmooth analysis and a surjectivity theorem for pseudomonotone operators to prove the existence of a nontrivial solution of the anisotropic elliptic obstacle problem, which relies on the first eigenvalue of the Steklov eigenvalue problem for the begin{document}$ p_$end{document}-Laplacian. Then, we introduce the parameter-to-solution map for the anisotropic elliptic obstacle problem, and establish a critical convergence result of the Kuratowski type to parameter-to-solution map. Finally, a general framework is proposed to examine the solvability of the nonlinear inverse problem.
{"title":"Inverse problems for anisotropic obstacle problems with multivalued convection and unbalanced growth","authors":"Shengda Zeng, Yunru Bai, Vicenţiu D. Rădulescu","doi":"10.3934/eect.2022051","DOIUrl":"https://doi.org/10.3934/eect.2022051","url":null,"abstract":"<p style='text-indent:20px;'>The prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic <inline-formula><tex-math id=\"M1\">begin{document}$ p $end{document}</tex-math></inline-formula>-Laplacian and a weighted anisotropic <inline-formula><tex-math id=\"M2\">begin{document}$ q $end{document}</tex-math></inline-formula>-Laplacian (called the weighted anisotropic <inline-formula><tex-math id=\"M3\">begin{document}$ (p,q) $end{document}</tex-math></inline-formula>-Laplacian), a multivalued reaction term depending on the gradient, two multivalued boundary conditions and an obstacle constraint. We, first, employ the theory of nonsmooth analysis and a surjectivity theorem for pseudomonotone operators to prove the existence of a nontrivial solution of the anisotropic elliptic obstacle problem, which relies on the first eigenvalue of the Steklov eigenvalue problem for the <inline-formula><tex-math id=\"M4\">begin{document}$ p_$end{document}</tex-math></inline-formula>-Laplacian. Then, we introduce the parameter-to-solution map for the anisotropic elliptic obstacle problem, and establish a critical convergence result of the Kuratowski type to parameter-to-solution map. Finally, a general framework is proposed to examine the solvability of the nonlinear inverse problem.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73836430","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 analyse the existence and uniqueness of a time-periodic solution to chemotaxis-shallow water system in a periodic domain. Under the assumption of some smallness and symmetric external force, the existence of periodic solution is established using the method of parabolic regularization and limit process. The uniqueness of the periodic solution is proved by energy estimates.
{"title":"Time periodic solution to chemotaxis-shallow water system in a periodic domain","authors":"Qingfang Shi, Xinli Zhang","doi":"10.3934/eect.2022044","DOIUrl":"https://doi.org/10.3934/eect.2022044","url":null,"abstract":"In this paper, we analyse the existence and uniqueness of a time-periodic solution to chemotaxis-shallow water system in a periodic domain. Under the assumption of some smallness and symmetric external force, the existence of periodic solution is established using the method of parabolic regularization and limit process. The uniqueness of the periodic solution is proved by energy estimates.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89213477","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 studies the dynamical behavior of a radio-dependent predator-prey model with age structure and two delays. The model is first formulated as an abstract non-densely defined Cauchy problem and the conditions for existence of the positive equilibrium point are derived. Then, through determining the distribution of eigenvalues, the globally asymptotic stability of the boundary equilibrium and the locally asymptotic stability for the positive equilibrium are obtained, respectively. In addition, it is also shown that a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears under some conditions. Finally, some numerical examples are provided to illustrate the obtained results.
{"title":"Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling Ⅲ functional response and delays","authors":"Dongxue Yan, Yuan Yuan, Xianlong Fu","doi":"10.3934/eect.2022034","DOIUrl":"https://doi.org/10.3934/eect.2022034","url":null,"abstract":"This paper studies the dynamical behavior of a radio-dependent predator-prey model with age structure and two delays. The model is first formulated as an abstract non-densely defined Cauchy problem and the conditions for existence of the positive equilibrium point are derived. Then, through determining the distribution of eigenvalues, the globally asymptotic stability of the boundary equilibrium and the locally asymptotic stability for the positive equilibrium are obtained, respectively. In addition, it is also shown that a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears under some conditions. Finally, some numerical examples are provided to illustrate the obtained results.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76511483","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 consider a mechanochemical model in biological patterns in begin{document}$ mathbb{R}^N $end{document}, begin{document}$ Ngeq 5 $end{document}. We first prove the existence of time periodic solution in begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}.
In this paper, we consider a mechanochemical model in biological patterns in begin{document}$ mathbb{R}^N $end{document}, begin{document}$ Ngeq 5 $end{document}. We first prove the existence of time periodic solution in begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}.
{"title":"Time periodic solution to a mechanochemical model in biological patterns","authors":"Chengxin Du, Changchun Liu","doi":"10.3934/eect.2022039","DOIUrl":"https://doi.org/10.3934/eect.2022039","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we consider a mechanochemical model in biological patterns in <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}^N $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ Ngeq 5 $end{document}</tex-math></inline-formula>. We first prove the existence of time periodic solution in <inline-formula><tex-math id=\"M3\">begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}</tex-math></inline-formula>. Then we obtain the existence, uniqueness and regularity of the mild solution of the problem. Finally, we prove that the mild solution can become strong solution in <inline-formula><tex-math id=\"M4\">begin{document}$ BC(mathbb{R}; L^{N,infty}(Omega)) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74902982","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 consider continuous dependence and optimal control of a dynamic elastic-viscoplastic contact model with Clarke subdifferential boundary conditions. Since the constitutive law of elastic-viscoplastic materials has an implicit expression of the stress field, the weak form of the model is an evolutionary hemivariational inequality coupled with an integral equation. By providing some equivalent weak formulations, we prove the continuous dependence of the solution on external forces and initial conditions in the weak topologies. Finally, the existence of optimal solutions to a boundary optimal control problem is established.
{"title":"Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions","authors":"Xilu Wang, Xiaoliang Cheng","doi":"10.3934/eect.2021064","DOIUrl":"https://doi.org/10.3934/eect.2021064","url":null,"abstract":"In this paper, we consider continuous dependence and optimal control of a dynamic elastic-viscoplastic contact model with Clarke subdifferential boundary conditions. Since the constitutive law of elastic-viscoplastic materials has an implicit expression of the stress field, the weak form of the model is an evolutionary hemivariational inequality coupled with an integral equation. By providing some equivalent weak formulations, we prove the continuous dependence of the solution on external forces and initial conditions in the weak topologies. Finally, the existence of optimal solutions to a boundary optimal control problem is established.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87575137","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 introduce a second-order equation based on a diffusion tensor combined with a bilateral total variation (BTV) term that takes the benefit from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edge and the BTV term in reducing blur. Compared to the other partial differential equations (PDE) used in the SR context, the proposed super-resolution (SR) PDE can efficiently preserve image components such as corners and flat regions with less apparition of blur in homogeneous regions. Moreover, since the SR approaches are always ill-posed, a mathematical study of the existence and uniqueness of the solution is also checked in a Sobolev space. From the numerical experiments we can observe that the proposed PDE can efficiently improve the quality of the high-resolution (HR) image. Surprisingly, the apparition of blur in the restored image is less compared to the other methods. In addition to visual evaluation, two-based metrics are computed to confirm the improvements offered by the proposed PDE.
{"title":"On the well-posedness of a tensor-based second order PDE with bilateral term for image super-resolution","authors":"El Mourabit Idriss, A. Hakim, A. Laghrib","doi":"10.3934/eect.2022047","DOIUrl":"https://doi.org/10.3934/eect.2022047","url":null,"abstract":"In this paper, we introduce a second-order equation based on a diffusion tensor combined with a bilateral total variation (BTV) term that takes the benefit from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edge and the BTV term in reducing blur. Compared to the other partial differential equations (PDE) used in the SR context, the proposed super-resolution (SR) PDE can efficiently preserve image components such as corners and flat regions with less apparition of blur in homogeneous regions. Moreover, since the SR approaches are always ill-posed, a mathematical study of the existence and uniqueness of the solution is also checked in a Sobolev space. From the numerical experiments we can observe that the proposed PDE can efficiently improve the quality of the high-resolution (HR) image. Surprisingly, the apparition of blur in the restored image is less compared to the other methods. In addition to visual evaluation, two-based metrics are computed to confirm the improvements offered by the proposed PDE.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82733620","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 manuscript the controllability for a class of time-dependent neutral stochastic integro-differential systems driven by fractional Brownian motion in a separable Hilbert space with delay is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.
{"title":"Controllability of retarded time-dependent neutral stochastic integro-differential systems driven by fractional Brownian motion","authors":"Youssef Benkabdi, E. Lakhel","doi":"10.3934/eect.2022031","DOIUrl":"https://doi.org/10.3934/eect.2022031","url":null,"abstract":"In this manuscript the controllability for a class of time-dependent neutral stochastic integro-differential systems driven by fractional Brownian motion in a separable Hilbert space with delay is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82657240","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 manuscript the controllability of Hilfer fractional neutral differential inclusions with non-instantaneous impulse in Banach space is investigated by using semi-group theory, fractional calculus, upper semi-continuous (u.s.c), multi-functions and Mönch fixed point theorem. Sufficient conditions are derived by using Hausdorff measure of non-compactness (MNC). Further, the obtained result is illustrated by an example.
{"title":"Controllability of Hilfer type fractional evolution neutral integro-differential inclusions with non-instantaneous impulses","authors":"K. Sanjay, P. Balasubramaniam","doi":"10.3934/eect.2022043","DOIUrl":"https://doi.org/10.3934/eect.2022043","url":null,"abstract":"In this manuscript the controllability of Hilfer fractional neutral differential inclusions with non-instantaneous impulse in Banach space is investigated by using semi-group theory, fractional calculus, upper semi-continuous (u.s.c), multi-functions and Mönch fixed point theorem. Sufficient conditions are derived by using Hausdorff measure of non-compactness (MNC). Further, the obtained result is illustrated by an example.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86391180","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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