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Existence, uniqueness and  L2 t (Hx2) ∩ L∞ t  (Hx1) ∩ Ht1(L2 x ) regularity of the gradient flow of the Ambrosio-Tortorelli functional 安布罗西奥-托托雷利函数梯度流的存在性、唯一性和 L2 t (Hx2) ∩ L∞ t (Hx1) ∩ Ht1(L2 x ) 正则性
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-08-09 DOI: 10.1051/cocv/2024060
Tommaso Cortopassi
{"title":"Existence, uniqueness and  L2 t (Hx2) ∩ L∞ t  (Hx1) ∩ Ht1(L2 x ) regularity of the gradient flow of the Ambrosio-Tortorelli functional","authors":"Tommaso Cortopassi","doi":"10.1051/cocv/2024060","DOIUrl":"https://doi.org/10.1051/cocv/2024060","url":null,"abstract":"We consider the gradient flow of the Ambrosio-Tortorelli functional, proving existence, uniqueness and L2t (Hx2) ∩ L∞t (Hx1) ∩ Ht1(L2x) regularity of the solution in dimension 2. Such functional is an approximation in Γ-convergence of the Mumford-Shah functional often used in problems of image segmentation and fracture mechanics. The strategy of the proof essentially follows the one of [9] but the crucial estimate is attained employing a different technique, and in the end it allows us to prove better estimates than the ones obtained in [9]. In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain, the initial data (u0,z0) ∈ [H1(U)]2 and 0 ≤ z0 ≤ 1, then for every T > 0 there exists a unique gradient flow (u(t),z(t)) of the Ambrosio-Tortorelli functional such that\u0000(u,z) ∈ [L2(0,T;H2(U)) ∩ L∞(0,T;H1(U)) ∩ H1(0,T;L2(U))]2, while previously such regularity was known only for short times.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"81 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dispersive problem and control sets of linear control systems on Lie groups 李群上线性控制系统的分散问题和控制集
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-07-24 DOI: 10.1051/cocv/2024054
Josiney Souza
{"title":"Dispersive problem and control sets of linear control systems on Lie groups","authors":"Josiney Souza","doi":"10.1051/cocv/2024054","DOIUrl":"https://doi.org/10.1051/cocv/2024054","url":null,"abstract":"This paper is dedicated to the study of dispersiveness and controllability of linear control systems on Lie groups. Dispersiveness means absence of recursiveness, that is contrary to the existence of control set. A linear control system on a Lie group associates with a derivation operator. For a linear system with stable derivation, it is shown that the system is dispersive if and only if the trajectories through the neutral element have no limit at infinity. As a concequence, a nonempty limit set at the neutral element is a necessary condition for the linear system to admit a control set or to be controllable. In the nondispersive case, the control sets are described by the Lie subgroup of all recurrent points of the automorphism flow of the system. If the derivation operator is asymptoticaly stable, the central limit set at the neutral element is the unique control set of the system.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"66 43","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141806485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control  problems 基于对角线化的线性二次优化控制问题并行实时求解新方法
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-07-16 DOI: 10.1051/cocv/2024051
Matthias Heinkenschloss, Nathaniel J. Kroeger
{"title":"A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control  problems","authors":"Matthias Heinkenschloss, Nathaniel J. Kroeger","doi":"10.1051/cocv/2024051","DOIUrl":"https://doi.org/10.1051/cocv/2024051","url":null,"abstract":"A new diagonalization technique for the parallel-in-time solution of  linear-quadratic optimal control problems with  time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization  efficiently transforms the discretized optimality system into K  (=number of time steps) decoupled complex valued 2n y  by 2n y  systems, where n y  is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization  technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"9 22","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141640615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local quadratic convergence of the SQP method for an optimal control problem governed by a regularized fracture propagation model 正则化断裂传播模型优化控制问题的 SQP 方法的局部二次收敛性
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-07-16 DOI: 10.1051/cocv/2024052
Ira Neitzel, Andreas Hehl
{"title":"Local quadratic convergence of the SQP method for an optimal control problem governed by a regularized fracture propagation model","authors":"Ira Neitzel, Andreas Hehl","doi":"10.1051/cocv/2024052","DOIUrl":"https://doi.org/10.1051/cocv/2024052","url":null,"abstract":"We prove local quadratic convergence of the sequential quadratic programming (SQP) method for an optimal control problem of tracking type governed by one time step of the Euler-Lagrange equation of a time discrete regularized fracture or damage energy minimization problem. This lower-level energy minimization problem % describing the fracture process\u0000    contains a penalization term for violation of the irreversibility condition in the fracture growth process and a viscous regularization term. Conditions on the latter, corresponding to a time step restriction, guarantee strict convexity and thus unique solvability of the Euler Lagrange equations. Nonetheless, these are quasilinear and the control problem is nonconvex. For the convergence proof with $L^infty$ localization of the SQP-method, we follow the approach from cite{Troeltzsch:1999}, utilizing strong regularity of generalized equations and arguments from cite{HoppeNeitzel:2020} for $L^2$-localization.\u0000\u00002020 Mathematics Subject Classification\u0000\u000090C55, 49M41, 49M15.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"10 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141644073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Interior point methods in optimal control 最优控制中的内点法
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-06-17 DOI: 10.1051/cocv/2024049
P. Malisani
{"title":"Interior point methods in optimal control","authors":"P. Malisani","doi":"10.1051/cocv/2024049","DOIUrl":"https://doi.org/10.1051/cocv/2024049","url":null,"abstract":"This paper deals with Interior Point Methods (IPMs) for Optimal Control Problems (OCPs) with pure state and mixed constraints. This paper establishes a complete proof of convergence of IPMs for a general class of OCPs. Convergence results are proved for primal variables, namely state and control variables, and for dual variables, namely, the adjoint state, and the constraints multipliers. In addition, the presented convergence result does not rely on a strong convexity assumption. Finally, this paper compares the performances of a primal and a primal-dual implementation of IPMs in optimal control in three examples.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"5 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141335334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Minimal time sampled-data controls for the ordinary differential system 常微分系统的最小时间采样数据控制
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-05-16 DOI: 10.1051/cocv/2024044
Li Wang, Lijuan Wang
{"title":"Minimal time sampled-data controls for the ordinary differential system","authors":"Li Wang, Lijuan Wang","doi":"10.1051/cocv/2024044","DOIUrl":"https://doi.org/10.1051/cocv/2024044","url":null,"abstract":"In this paper, we design a minimal time optimal control problem for an ordinary differential system with sampled-data controls, and use it to approximate a time optimal control problem for the ordinary differential system with distributed controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the minimal time  sampled-data control problem and the minimal time distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic behavior of null controllability cost for parabolic equations with vanishing diffusivity under Robin and Neumann boundary conditions 在罗宾和诺伊曼边界条件下，具有消失扩散性的抛物方程的无效可控性成本的渐近行为
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-05-15 DOI: 10.1051/cocv/2024042
Fouad Et-tahri, Jon Asier Bárcena-Petisco, I. Boutaayamou, Lahcen Maniar
{"title":"Asymptotic behavior of null controllability cost for parabolic equations with vanishing diffusivity under Robin and Neumann boundary conditions","authors":"Fouad Et-tahri, Jon Asier Bárcena-Petisco, I. Boutaayamou, Lahcen Maniar","doi":"10.1051/cocv/2024042","DOIUrl":"https://doi.org/10.1051/cocv/2024042","url":null,"abstract":"In this paper we study the null-controllability cost of a transport-diffusion system under Robin conditions with distributed control and in which the transport coefficient is a gradient field. First, we provide some conditions on transport coefficient and boundary potential to show that the control cost decays exponentially when the viscosity vanishes and the control time is sufficiently large. On the other hand, if the range of the control region by the transport flow does not cover that of Ω, we prove that the control cost explodes exponentially for the Neumann conditions case with vanishing viscosity and all control time.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"69 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140973890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
​Mean reflected BSDE driven by a marked point process and application in insurance risk management 平均值反映了 BSDE，由保险风险管理中的标记点流程和应用驱动
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-05-01 DOI: 10.1051/cocv/2024040
Zihao Gu, Yiqing Lin, Xu Kun
{"title":"​Mean reflected BSDE driven by a marked point process and application in insurance risk management","authors":"Zihao Gu, Yiqing Lin, Xu Kun","doi":"10.1051/cocv/2024040","DOIUrl":"https://doi.org/10.1051/cocv/2024040","url":null,"abstract":"This paper aims to solve a super-hedging problem along with  insurance re-payment under running risk management constraints. The initial endowment for the super-heding problem is characterized by a class of mean reflected backward stochastic differential equation driven by a marked point process (MPP) and a Brownian motion. By Lipschitz assumptions on the generators and proper integrability on the terminal value, we give the well-posedness of this kind of BSDEs by combining a representation theorem with the fixed point argument.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":"228 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141039739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Existence of bounded solutions to multiplicative Poisson equations under mixing property 混合特性下乘法泊松方程有界解的存在性
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-04-18 DOI: 10.1051/cocv/2024038
Marcin Pitera, Łukasz Stettner
{"title":"Existence of bounded solutions to multiplicative Poisson equations under mixing property","authors":"Marcin Pitera, Łukasz Stettner","doi":"10.1051/cocv/2024038","DOIUrl":"https://doi.org/10.1051/cocv/2024038","url":null,"abstract":"We study the problem of Multiplicative Poisson Equation (MPE) bounded solution existence in a generic probabilistic discrete-time setup with preimposed mixing. In particular, we consolidate results based on the span-contraction framework and derive an explicit sharp bound that must be imposed on the cost function to guarantee the existence of a bounded MPE solution under mixing. Also, we study properties that the probability kernel must satisfy to ensure the existence of an MPE solution for any generic risk-reward function and characterise process behaviour in the complement of the invariant measure support. Finally, we present numerous examples and stochastic dominance based arguments that help to better understand the problems that arise when the mean is replaced by the entropy in the ergodic setup.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":" 48","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140690129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Linear-quadratic two-person differential game: Nash game versus Stackelberg game, local information versus global information 线性二次双人微分博弈：纳什博弈与斯塔克尔伯格博弈，局部信息与全局信息
ESAIM: Control, Optimisation and Calculus of Variations Pub Date : 2024-04-17 DOI: 10.1051/cocv/2024037
Xinwei Feng, Ying Hu, Jianhui Huang
{"title":"Linear-quadratic two-person differential game: Nash game versus Stackelberg game, local information versus global information","authors":"Xinwei Feng, Ying Hu, Jianhui Huang","doi":"10.1051/cocv/2024037","DOIUrl":"https://doi.org/10.1051/cocv/2024037","url":null,"abstract":"In this paper, we present a unified framework to study a variety of two-person dynamic decision problems, including stochastic (zero-sum, non zero-sum) Nash game, Stackelberg game with global information. For these games, the solvability of these problems is discussed via progressive formulations respectively: the abstract quadratic functional, Hamiltonian system for open-loop, and Riccati equation for closed-loop (feedback) representation. Based on the unified framework, time consistency/inconsistency property of related equilibrium is studied. Then we introduce a new type of game, Stackelberg game with local information. For this, the classical best-response machinery adopted for global information is no longer workable. As resolution, a repeated game approach is employed to construct the equilibrium strategies via a backward- and forward-procedure. Moreover, connection of local information pattern to time-inconsistency is also revealed. Finally, relations among zerosum Nash game, zero-sum Stackelberg game with global information and local information are also identified.","PeriodicalId":512605,"journal":{"name":"ESAIM: Control, Optimisation and Calculus of Variations","volume":" 38","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140692773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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