This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carath´eodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.
{"title":"Variational problems concerning sub-Finsler metrics in Carnot groups","authors":"Fares Essebei, Enrico Pasqualetto","doi":"10.1051/cocv/2023006","DOIUrl":"https://doi.org/10.1051/cocv/2023006","url":null,"abstract":"This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carath´eodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"59 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91325520","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}
In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number of Fourier modes. In order to obtain the irreducibility, we first investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a finite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator, we establish the irreducibility of Kuramoto-Sivashinsky equation.
{"title":"Irreducibility of Kuramoto-Sivashinsky equation driven by degenerate noise","authors":"Peng Gao","doi":"10.1051/cocv/2022014","DOIUrl":"https://doi.org/10.1051/cocv/2022014","url":null,"abstract":"In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number of Fourier modes. In order to obtain the irreducibility,\u0000\u0000we first investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a finite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach.\u0000\u0000Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator,\u0000\u0000we establish the irreducibility of Kuramoto-Sivashinsky equation.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"228 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85567280","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 \"well-poseness of evolutionary Navier-Stokes equations with forces of low regularity on two-dimensional domains\"","authors":"K. Kunisch, E. Casas","doi":"10.1051/cocv/2022012","DOIUrl":"https://doi.org/10.1051/cocv/2022012","url":null,"abstract":"No Abstract","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"62 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78639597","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}
B. M. Calsavara, E. Fernández-Cara, Luz de Teresa, José Antonio Villa
This paper deals with the application of multiple strategies to control some parabolic PDEs. We assume that we can act on the system through a hierarchy of distributed controls: with a first control (a follower), we drive the state exactly to zero; then, with an additional control (the leader), we minimize a prescribed cost functional. That means that we invert the roles played by leaders and followers in the recent literature. We study linear and semilinear problems. More precisely, we prove the existence (and uniqueness in the linear case) of a leader-follower couple. Then, we deduce an appropriate optimality system that must be satisfied by the controls and the corresponding state and adjoint states. We also indicate some generalizations to other controls, PDEs and systems. In particular, we establish similar existence and optimality results for hierarchical-biobjective (Pareto-Stackelberg) control problems, where there are two cost functionals and two independent leader controls whose main task is to find an associated Pareto equilibrium and one common follower in charge of null controllability.
{"title":"New results concerning the hierarchical control of linear and semilinear parabolic equations\u0000 ","authors":"B. M. Calsavara, E. Fernández-Cara, Luz de Teresa, José Antonio Villa","doi":"10.1051/cocv/2022011","DOIUrl":"https://doi.org/10.1051/cocv/2022011","url":null,"abstract":"This paper deals with the application of multiple strategies to control some parabolic PDEs.\u0000 We assume that we can act on the system through a hierarchy of distributed controls: with a first control (a follower), we drive the state exactly to zero; then, with an additional control (the leader), we minimize a prescribed cost functional.\u0000 That means that we invert the roles played by leaders and followers in the recent literature.\u0000 We study linear and semilinear problems.\u0000 More precisely, we prove the existence (and uniqueness in the linear case) of a leader-follower couple.\u0000 Then, we deduce an appropriate optimality system that must be satisfied by the controls and the corresponding state and adjoint states. We also indicate some generalizations to other controls, PDEs and systems.\u0000 In particular, we establish similar existence and optimality results for hierarchical-biobjective (Pareto-Stackelberg) control problems, where there are two cost functionals and two independent leader controls whose main task is to find an associated Pareto equilibrium and one common follower in charge of null controllability.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"51 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86859611","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 study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin's maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.
{"title":"Extended mean-field control problem with partial observation","authors":"Tianyang Nie, Keyu Yan","doi":"10.1051/cocv/2022010","DOIUrl":"https://doi.org/10.1051/cocv/2022010","url":null,"abstract":"We study an extended mean-field control problem with partial observation, where the dynamic of the state is given by a forward-backward stochastic differential equation of McKean-Vlasov type. The cost functional, the state and the observation all depend on the joint distribution of the state and the control process. Our problem is motivated by the recent popular subject of mean-field games and related control problems of McKean-Vlasov type. We first establish a necessary condition in the form of Pontryagin's maximum principle for optimality. Then a verification theorem is obtained for optimal control under some convex conditions of the Hamiltonian function. The results are also applied to studying linear-quadratic mean-filed control problem in the type of scalar interaction.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"64 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84059577","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}
In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.
{"title":"Approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary conditions","authors":"Tatsien Li, B. Rao","doi":"10.1051/cocv/2021006","DOIUrl":"https://doi.org/10.1051/cocv/2021006","url":null,"abstract":"In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"125 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72814893","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}
A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form $dot x_1=u$, $dot x_j=x_1^{j-1}$, $j=2,ldots,n$, is obtained for arbitrary~$n$. In the paper we present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which gives analytic tools for finding the optimal time and optimal controls. Being homogeneous, the mentioned system approximates a certain class of affine systems in the sense of time optimality. Therefore, the obtained results can be used for solving the time-optimal control problem for systems from this class.
{"title":"Hausdorff moment problem and nonlinear time optimality","authors":"G. Sklyar, S. Ignatovich","doi":"10.1051/cocv/2022007","DOIUrl":"https://doi.org/10.1051/cocv/2022007","url":null,"abstract":"A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form $dot x_1=u$, $dot x_j=x_1^{j-1}$, $j=2,ldots,n$, is obtained for arbitrary~$n$. In the paper we present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which gives analytic tools for finding the optimal time and optimal controls. Being homogeneous, the mentioned system approximates a certain class of affine systems in the sense of time optimality. Therefore, the obtained results can be used for solving the time-optimal control problem for systems from this class.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"42 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80952320","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}
In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.
{"title":"Necessary and sufficient conditions for the nonincrease of functions of solutions to constrained differential inclusions","authors":"Mohamed Adlene Maghenem","doi":"10.1051/cocv/2022008","DOIUrl":"https://doi.org/10.1051/cocv/2022008","url":null,"abstract":"In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"56 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85931888","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 study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form begin {gather*} min biggl{ int_{Omega} F(x,w,Dw) d x : w in mathcal{K}_{psi}(Omega) biggr}, end {gather*} with $F$ double phase functional of the form begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), end {equation*} where $Omega$ is a bounded open subset of $mathbb{R}^n$ , $psi in W^{1,p}(Omega)$ is a fixed function called textit { obstacle } and $mathcal{K}_{psi}(Omega)= { w in W^{1,p}(Omega) : w geq psi text{a.e. in} Omega }$ is the class of admissible functions . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .
我们研究了形式为begin {gather*} min biggl{int_{Omega} F(x,w,Dw) d x 的变分障碍问题解梯度的高分数可微性: w in mathcal{K}_{psi}(Omega) biggr}， end {gather*}与$F$双相泛函的形式为begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q)， end {equation*}其中$Omega$是$mathbb{R}^n$的有界开放子集，$psi in w ^{1,p}(Omega)$是一个固定函数，称为 texttit {obstacle}和$mathcal{K}_{psi}(Omega)= {w in w ^{1,p}(Omega): w geq psi text{a.e。in} }$是可容许函数的类。假设障碍物的梯度属于一个合适的Besov空间，我们能够证明解的梯度保持一定的分数可微性。
{"title":"Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems","authors":"Antonio Giuseppe Grimaldi, Erica Ipocoana","doi":"10.1051/cocv/2022050","DOIUrl":"https://doi.org/10.1051/cocv/2022050","url":null,"abstract":"We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form begin {gather*} min biggl{ int_{Omega} F(x,w,Dw) d x : w in mathcal{K}_{psi}(Omega) biggr}, end {gather*} with $F$ double phase functional of the form begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), end {equation*} where $Omega$ is a bounded open subset of $mathbb{R}^n$ , $psi in W^{1,p}(Omega)$ is a fixed function called textit { obstacle } and $mathcal{K}_{psi}(Omega)= { w in W^{1,p}(Omega) : w geq psi text{a.e. in} Omega }$ is the class of admissible functions . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"89 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81472456","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}
This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D measuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]]. We analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.
{"title":"A consistent approximation of the total perimeter functional for topology optimization algorithms","authors":"S. Amstutz, C. Dapogny, À. Ferrer","doi":"10.1051/cocv/2022005","DOIUrl":"https://doi.org/10.1051/cocv/2022005","url":null,"abstract":"This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D\u0000measuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]].\u0000We analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"50 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83987858","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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