We consider an optimal control problem for the two-dimensional viscous Cahn--Hilliard--Oberbeck--Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equation governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.
{"title":"Optimal Borel measure-valued controls to the viscous Cahn--Hilliard--Oberbeck--Boussinesq phase-field system on two-dimensional bounded domains","authors":"Gilbert Peralta","doi":"10.1051/cocv/2023025","DOIUrl":"https://doi.org/10.1051/cocv/2023025","url":null,"abstract":"We consider an optimal control problem for the two-dimensional viscous Cahn--Hilliard--Oberbeck--Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equation governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"52 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74861678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emerson Abreu, Leandro G. Fernandes, Joel Cruz Ramirez
We consider a class of monomial weights $x^{A}=vert x_{1}vert^{a_{1}}ldotsvert x_{N}vert^{a_{N}}$ in $mathbb{R}^{N}$, where $a_{i}$ is a nonnegative real number for each $iin{1,ldots,N}$, and we establish the $varepsilon-varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for $N=2$, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.
{"title":"The $varepsilon$ - $varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights\u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 \u0000\u0000 ","authors":"Emerson Abreu, Leandro G. Fernandes, Joel Cruz Ramirez","doi":"10.1051/cocv/2023023","DOIUrl":"https://doi.org/10.1051/cocv/2023023","url":null,"abstract":"We consider a class of monomial weights $x^{A}=vert x_{1}vert^{a_{1}}ldotsvert x_{N}vert^{a_{N}}$ in $mathbb{R}^{N}$, where $a_{i}$ is a nonnegative real number for each $iin{1,ldots,N}$, and we establish the $varepsilon-varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for $N=2$, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"6 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81844355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shengda Zeng, Yunru Bai, Vicentiu D. Rădulescu, Patrick Winkert
In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map which admits weakly compact values. Then, by employing the Aubin-Cellina convergence theorem and the theory of nonsmooth analysis, we prove that the parameter-to-solution map is bounded and continuous in the sense of Kuratowski. Finally, a generalized regularization framework for the inverse problem is developed and a new existence theorem is provided.
{"title":"An inverse problem for a double phase implicit obstacle problem with multivalued terms","authors":"Shengda Zeng, Yunru Bai, Vicentiu D. Rădulescu, Patrick Winkert","doi":"10.1051/cocv/2023022","DOIUrl":"https://doi.org/10.1051/cocv/2023022","url":null,"abstract":"In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map which admits weakly compact values. Then, by employing the Aubin-Cellina convergence theorem and the theory of nonsmooth analysis, we prove that the parameter-to-solution map is bounded and continuous in the sense of Kuratowski. Finally, a generalized regularization framework for the inverse problem is developed and a new existence theorem is provided.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"78 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86447007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to: HELLINGER-KANTOROVICH BARYCENTER BETWEEN DIRAC MEASURES","authors":"O. Minevich","doi":"10.1051/cocv/2023018","DOIUrl":"https://doi.org/10.1051/cocv/2023018","url":null,"abstract":"","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"9 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85616941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficient a = a(x) in the principal part of the parabolic equation is such that the function x → x/a(x) is in Lp(0,1) for some p > 1. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.
{"title":"Null controllability of strongly degenerate parabolic equations\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"L. Rosier, Antoine Benoit, R. Loyer","doi":"10.1051/cocv/2023016","DOIUrl":"https://doi.org/10.1051/cocv/2023016","url":null,"abstract":"We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficient a = a(x) in the principal part of the parabolic equation is such that the function x → x/a(x) is in Lp(0,1) for some p > 1. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"6 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81655526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.
{"title":"Optimal control of a parabolic equation with memory","authors":"E. Casas, J. Yong","doi":"10.1051/cocv/2023013","DOIUrl":"https://doi.org/10.1051/cocv/2023013","url":null,"abstract":"An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"11 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88726801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Leone, G. Scilla, Francesco Solombrino, A. Verde
Abstract. A regularity result for free-discontinuity energies defined on the space SBV p(·) of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Ho¨lder continuity for the variable exponent p(x). Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper [24], dealing with a constant exponent.
{"title":"Regularity of minimizers for free-discontinuity problems with p(·)-growth","authors":"C. Leone, G. Scilla, Francesco Solombrino, A. Verde","doi":"10.1051/cocv/2023062","DOIUrl":"https://doi.org/10.1051/cocv/2023062","url":null,"abstract":"Abstract. A regularity result for free-discontinuity energies defined on the space SBV p(·) of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Ho¨lder continuity for the variable exponent p(x). Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper [24], dealing with a constant exponent.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83669229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.
{"title":"Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian","authors":"Alessandra De Luca, V. Felli, Giovanni Siclari","doi":"10.1051/cocv/2023045","DOIUrl":"https://doi.org/10.1051/cocv/2023045","url":null,"abstract":"We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non-homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80531292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. Based on the uniqueness of solution to a coupled system of wave equations associated with incomplete internal observations, we establish the approximate internal synchronization by groups, the induced internal synchronization and the approximate internal synchronization in the pinning sense.
{"title":"Approximate internal controllability and synchronization of a coupled system of wave equations\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"B. Rao, Tatsien Li","doi":"10.1051/cocv/2023008","DOIUrl":"https://doi.org/10.1051/cocv/2023008","url":null,"abstract":"Abstract. Based on the uniqueness of solution to a coupled system of wave equations associated with incomplete internal observations, we establish the approximate internal synchronization by groups, the induced internal synchronization and the approximate internal synchronization in the pinning sense.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82999846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Enrique Fernandez-Cara es>, Juan Bautista Límaco, Denilson Menezes, Yuri Thamsten
This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.
{"title":"Local null controllability of a quasi-linear system and related numerical experiments","authors":"Enrique Fernandez-Cara es>, Juan Bautista Límaco, Denilson Menezes, Yuri Thamsten","doi":"10.1051/cocv/2023009","DOIUrl":"https://doi.org/10.1051/cocv/2023009","url":null,"abstract":"This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"39 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79871959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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