In this paper, we consider a class of Monge -Ampere equations in a free boundary domain of $mathbb{R}^2$ where the value of the unknown function is prescribed on the free boundary. From a variational point of view, these equations describe an optimal transport problem from an a priori undetermined source domain to a prescribed target domain. We prove the existence and uniqueness of a variational solution to these Monge -Ampere equations under a singularity condition on the density function on the source domain. Furthermore, we provide regularity results under some conditions on the prescribed domain.
{"title":"Free boundary Monge-Ampere equations","authors":"M. Sedjro","doi":"10.1051/cocv/2022048","DOIUrl":"https://doi.org/10.1051/cocv/2022048","url":null,"abstract":"In this paper, we consider a class of Monge -Ampere equations in a free boundary domain of $mathbb{R}^2$ where the value of the unknown function is prescribed on the free boundary. From a variational point of view, these equations describe an optimal transport problem from an a priori undetermined source domain to a prescribed target domain. We prove the existence and uniqueness of a variational solution to these Monge -Ampere equations under a singularity condition on the density function on the source domain. Furthermore, we provide regularity results under some conditions on the prescribed domain.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"34 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80701108","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 the asymptotics of singularly perturbed phase-transition functionals of the form ℱk(u) = 1/εk∫Afk(𝑥,u,εk∇u)d𝑥, where u ∈ [0, 1] is a phase-field variable, εk > 0 a singular-perturbation parameter i.e., εk → 0, as k → +∞, and the integrands fk are such that, for every x and every k, fk(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionals Fk Γ-converge (up to subsequences) to a surface functional of the form ℱ∞(u) = ∫Su∩Af∞(𝑥,𝜈u)dHn-1, where u ∈ BV(A; {0, 1}) and f∞ is characterised by the double limit of suitably scaled minimisation problems. Afterwards we extend our analysis to the setting of stochastic homogenisation and prove a Γ-convergence result for stationary random integrands.
{"title":"Γ-convergence and stochastic homogenisation of phase-transition functionals","authors":"R. Marziani","doi":"10.1051/cocv/2023030","DOIUrl":"https://doi.org/10.1051/cocv/2023030","url":null,"abstract":"In this paper, we study the asymptotics of singularly perturbed phase-transition functionals of the form\u0000ℱk(u) = 1/εk∫Afk(𝑥,u,εk∇u)d𝑥,\u0000where u ∈ [0, 1] is a phase-field variable, εk > 0 a singular-perturbation parameter i.e., εk → 0, as k → +∞, and the integrands fk are such that, for every x and every k, fk(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionals Fk Γ-converge (up to subsequences) to a surface functional of the form\u0000ℱ∞(u) = ∫Su∩Af∞(𝑥,𝜈u)dHn-1,\u0000where u ∈ BV(A; {0, 1}) and f∞ is characterised by the double limit of suitably scaled minimisation problems. Afterwards we extend our analysis to the setting of stochastic homogenisation and prove a Γ-convergence result for stationary random integrands.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"52 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84479177","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 prove controllability of the Schr"odinger equation in $mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schr"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.
{"title":"Controllability of the Schrödinger equation on unbounded domains without geometric control condition","authors":"Matthias Taufer","doi":"10.1051/cocv/2023037","DOIUrl":"https://doi.org/10.1051/cocv/2023037","url":null,"abstract":"We prove controllability of the Schr\"odinger equation in $mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets.\u0000 This demonstrates in particular that controllability of the Schr\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation.\u0000 Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90244358","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 paper considers dynamic optimization problems for a class of control average mean-field stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation (MF-SDE) with individual noise and common noise, and the weight coefficients in the cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation (MF-FBSDE). Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence (NCE) equation systems are also discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.
{"title":"Dynamic optimization problems for mean-field stochastic large-population systems","authors":"Min Li, Na Li, Zhanghua Wu","doi":"10.1051/cocv/2022044","DOIUrl":"https://doi.org/10.1051/cocv/2022044","url":null,"abstract":"This paper considers dynamic optimization problems for a class of control average mean-field stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation (MF-SDE) with individual noise and common noise, and the weight coefficients in the cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation (MF-FBSDE). Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence (NCE) equation systems are also discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88919952","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 the small-time global null controllability of the generalized Burgers' equations $y_t + gamma |y|^{gamma-1}y_x-y_{xx}=u(t)$ on the segment $[0,1]$. The scalar control $u(t)$ is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition $y(t,1)=0$, and allow a left boundary control $y(t,0)=v(t)$. Under the assumption $gamma>3/2$ we prove that the system is small-time global null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.
{"title":"Small-time global null controllability of generalized Burgers' equations","authors":"Rémi Robin","doi":"10.1051/cocv/2023021","DOIUrl":"https://doi.org/10.1051/cocv/2023021","url":null,"abstract":"In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + gamma |y|^{gamma-1}y_x-y_{xx}=u(t)$ on the segment $[0,1]$. The scalar control $u(t)$ is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition $y(t,1)=0$, and allow a left boundary control $y(t,0)=v(t)$. Under the assumption $gamma>3/2$ we prove that the system is small-time global null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"12 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75303253","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}
Randers distances are an asymmetric generalization of Riemannian distances, and arise in optimal control problems subject to a drift term, among other applications. We show that Randers eikonal equation can be approximated by a logarithmic transformation of an anisotropic second order linear equation, generalizing Varadhan's formula for Riemannian manifolds. Based on this observation, we establish the convergence of a numerical method for computing Randers distances, from point sources or from a domain's boundary, on Cartesian grids of dimension two and three, which is consistent at order two thirds, and uses tools from low-dimensional algorithmic geometry for best efficiency. We also propose a numerical method for optimal transport problems whose cost is a Randers distance, exploiting the linear structure of our discretization and generalizing previous works in the Riemannian case. Numerical experiments illustrate our results.
{"title":"A linear finite-difference scheme for approximating Randers distances on Cartesian grids","authors":"F. Bonnans, G. Bonnet, J. Mirebeau","doi":"10.1051/cocv/2022043","DOIUrl":"https://doi.org/10.1051/cocv/2022043","url":null,"abstract":"Randers distances are an asymmetric generalization of Riemannian distances, and arise in optimal control problems subject to a drift term, among other applications. We show that Randers eikonal equation can be approximated by a logarithmic transformation of an anisotropic second order linear equation, generalizing Varadhan's formula for Riemannian manifolds. Based on this observation, we establish the convergence of a numerical method for computing Randers distances, from point sources or from a domain's boundary, on Cartesian grids of dimension two and three, which is consistent at order two thirds, and uses tools from low-dimensional algorithmic geometry for best efficiency. We also propose a numerical method for optimal transport problems whose cost is a Randers distance, exploiting the linear structure of our discretization and generalizing previous works in the Riemannian case. Numerical experiments illustrate our results.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"13 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82090751","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}
Optimal transport has recently proved to be a useful tool in various machine learning applications, needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wassserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.
{"title":"Regularization for Wasserstein distributionally robust optimization\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":"Waïss Azizian, F. Iutzeler, J. Malick","doi":"10.1051/cocv/2023019","DOIUrl":"https://doi.org/10.1051/cocv/2023019","url":null,"abstract":"Optimal transport has recently proved to be a useful tool in various machine learning applications, needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wassserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"49 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76377902","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 note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power $theta$ in $[0,1]$. The damping matrix is degenerate, which makes the the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers technique, we prove the following regularity for the underlying semigroup: begin{itemize} item The semigroup is of Gevrey class $delta$ for every $delta>1/2theta$, for each $theta$ in $(0,1/2)$. item The semigroup is analytic for $theta=1/2$. item The semigroup is of Gevrey class $delta$ for every $delta>1/2(1-theta)$, for each $theta$ in $(1/2,1)$.end{itemize} Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove that the semigroup is not differentiable for $theta=0$ or $theta=1$. Those results strongly improve upon some recent results presented in cite{ast}.
{"title":"Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case","authors":"Zhaobin Kuang, Zhuangyi Liu, L. Tébou","doi":"10.1051/cocv/2022042","DOIUrl":"https://doi.org/10.1051/cocv/2022042","url":null,"abstract":"In this note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power $theta$ in $[0,1]$. The damping matrix is degenerate, which makes the the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers technique, we prove the following regularity for the underlying semigroup:\u0000\u0000begin{itemize}\u0000\u0000item The semigroup is of Gevrey class $delta$ for every $delta>1/2theta$, for each $theta$ in $(0,1/2)$.\u0000\u0000item The semigroup is analytic for $theta=1/2$.\u0000\u0000item The semigroup is of Gevrey class $delta$ for every $delta>1/2(1-theta)$, for each $theta$ in $(1/2,1)$.end{itemize}\u0000\u0000 Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove that the semigroup is not differentiable for $theta=0$ or $theta=1$. Those results strongly improve upon some recent results presented in cite{ast}.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"60 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77888364","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 investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs are proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate.
{"title":"Two equivalent families of linear fully coupled forward backward stochastic differential equations","authors":"Ruyi Liu, Zhanghua Wu, Detao Zhang","doi":"10.1051/cocv/2022073","DOIUrl":"https://doi.org/10.1051/cocv/2022073","url":null,"abstract":"In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs are proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80692745","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 the paper, the problems of controllability and approximate controllability are studied for the heat equation $w_t=frac{1}{rho}left(kw_xright)_x+gamma w$ , $x>0$ , $tin(0,T)$ , controlled by the Dirichlet boundary condition. Control is considered in $L^infty(0,T)$ . It is proved that each initial state of this system is approximately controllable to any its end state in a given time $T>0$ .
{"title":"Controllability problems for the heat equation with variable coefficients on a half-axis","authors":"L. Fardigola, K. Khalina","doi":"10.1051/cocv/2022041","DOIUrl":"https://doi.org/10.1051/cocv/2022041","url":null,"abstract":"In the paper, the problems of controllability and approximate controllability are studied for the heat equation $w_t=frac{1}{rho}left(kw_xright)_x+gamma w$ , $x>0$ , $tin(0,T)$ , controlled by the Dirichlet boundary condition. Control is considered in $L^infty(0,T)$ . It is proved that each initial state of this system is approximately controllable to any its end state in a given time $T>0$ .","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"25 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81286983","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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