Abstract. We prove the uniqueness of the solution for a non-strictly convex problem in the Calculus of Variations of the form [[EQUATION]] φ (∇ v ) − λv . Here, φ is a convex function not differentiable at the origin and λ is a Lipschitz function. To prove this result we show that under fairly general assumptions, the minimizers are globally Lipschitz continuous.
{"title":"A uniqueness result for a non-strictly convex problem in the calculus of variations","authors":"Benjamin Lledos","doi":"10.1051/cocv/2023079","DOIUrl":"https://doi.org/10.1051/cocv/2023079","url":null,"abstract":"Abstract. We prove the uniqueness of the solution for a non-strictly convex problem in the Calculus of Variations of the form [[EQUATION]] φ (∇ v ) − λv . Here, φ is a convex function not differentiable at the origin and λ is a Lipschitz function. To prove this result we show that under fairly general assumptions, the minimizers are globally Lipschitz continuous.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476472","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 the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.
{"title":"Global exact controllability of the viscous and resistive MHD system in a rectangle thanks to the lateral sides and to distributed phantom forces","authors":"Jiajiang Liao","doi":"10.1051/cocv/2023078","DOIUrl":"https://doi.org/10.1051/cocv/2023078","url":null,"abstract":"We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"78 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135221270","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 the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's $S(theta)$ sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.
{"title":"Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition","authors":"Mégane Bournissou","doi":"10.1051/cocv/2023077","DOIUrl":"https://doi.org/10.1051/cocv/2023077","url":null,"abstract":"We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's $S(theta)$ sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"111 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869346","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}
Patrizia Di Gironimo, Sara Monsurrò, Gabriella Zecca
In this paper we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction.
{"title":"Dirichlet problem for noncoercive nonlinear elliptic equations with singular drift term in unbounded domains","authors":"Patrizia Di Gironimo, Sara Monsurrò, Gabriella Zecca","doi":"10.1051/cocv/2023076","DOIUrl":"https://doi.org/10.1051/cocv/2023076","url":null,"abstract":"In this paper we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"39 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268398","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}
Ivan Gudoshnikov, Oleg Makarenkov, Dmitrii Rachinskii
We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron X of a fixed shape. We provide examples of elasto-plastic spring models leading to structurally stable situations where X is a one or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.
{"title":"Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input","authors":"Ivan Gudoshnikov, Oleg Makarenkov, Dmitrii Rachinskii","doi":"10.1051/cocv/2023074","DOIUrl":"https://doi.org/10.1051/cocv/2023074","url":null,"abstract":"We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron X of a fixed shape. We provide examples of elasto-plastic spring models leading to structurally stable situations where X is a one or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077762","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}
Laetitia Giraldi, Pierre Lissy, Clément Moreau, Jean-Baptiste Pomet
We consider affine control systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions of local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in $mathrm{L}^{infty}$ or small in $mathrm{W}^{1,infty}$. These results are illustrated with several examples.
{"title":"Necessary conditions for local controllability of a particular class of systems with two scalar controls","authors":"Laetitia Giraldi, Pierre Lissy, Clément Moreau, Jean-Baptiste Pomet","doi":"10.1051/cocv/2023073","DOIUrl":"https://doi.org/10.1051/cocv/2023073","url":null,"abstract":"We consider affine control systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions of local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in $mathrm{L}^{infty}$ or small in $mathrm{W}^{1,infty}$. These results are illustrated with several examples.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077695","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 discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see [14]. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of globally optimal solutions for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.
{"title":"A discretize-then-optimize approach to PDE-constrained shape optimization","authors":"Roland Herzog, Estefanía Loayza-Romero","doi":"10.1051/cocv/2023071","DOIUrl":"https://doi.org/10.1051/cocv/2023071","url":null,"abstract":"We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see [14]. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of globally optimal solutions for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547516","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}
Nicolas Burq, Belhassen Dehman, Jérôme Le Rousseau
For the linear wave and Schr"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency with a dyadic scale.
{"title":"Semi-classical observation suffices for observability: wave and Schr\"odinger equations","authors":"Nicolas Burq, Belhassen Dehman, Jérôme Le Rousseau","doi":"10.1051/cocv/2023069","DOIUrl":"https://doi.org/10.1051/cocv/2023069","url":null,"abstract":"For the linear wave and Schr\"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency with a dyadic scale.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904370","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, Everaldo Medeiros, Marcos Montenegro
This paper deals with properties of non-negative solutions of the boundary value problem in the presence of diffusion a and source f in a bounded domain Ω ⊂ Rn, n ≥ 1, where a and f are non-decreasing continuous functions on [0,L0) and f is positive. Part of the results are new even if we restrict ourselves to the Gelfand type case L0 = ∞, a(t) = t and f is a convex function. We study the behavior of related extremal parameters and solutions with respect to L0 and also to a and f in the C0 topology. The work is carried out in a unified framework for 0 < L0 ≤ ∞ under some interactive conditions between a and f.
{"title":"The effect of diffusions and sources on semilinear elliptic problems ","authors":"Emerson Abreu, Everaldo Medeiros, Marcos Montenegro","doi":"10.1051/cocv/2023068","DOIUrl":"https://doi.org/10.1051/cocv/2023068","url":null,"abstract":"This paper deals with properties of non-negative solutions of the boundary value problem in the presence of diffusion a and source f in a bounded domain Ω ⊂ Rn, n ≥ 1, where a and f are non-decreasing continuous functions on [0,L0) and f is positive. Part of the results are new even if we restrict ourselves to the Gelfand type case L0 = ∞, a(t) = t and f is a convex function. We study the behavior of related extremal parameters and solutions with respect to L0 and also to a and f in the C0 topology. The work is carried out in a unified framework for 0 < L0 ≤ ∞ under some interactive conditions between a and f.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815343","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 construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in~cite{fQuantJump} is optimal.
{"title":"Whispering gallery modes for a transmission problem","authors":"Spyridon Filippas","doi":"10.1051/cocv/2023067","DOIUrl":"https://doi.org/10.1051/cocv/2023067","url":null,"abstract":"We construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in~cite{fQuantJump} is optimal.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011252","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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