In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C∞-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls.
{"title":"Optimal control for the Paneitz obstacle problem","authors":"C. B. Ndiaye","doi":"10.1051/cocv/2023036","DOIUrl":"https://doi.org/10.1051/cocv/2023036","url":null,"abstract":"In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C∞-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"9 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87570796","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 provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: “Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?” The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.
{"title":"Robustness of polynomial stability with respect to sampling","authors":"M. Wakaiki","doi":"10.1051/cocv/2023035","DOIUrl":"https://doi.org/10.1051/cocv/2023035","url":null,"abstract":"We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: “Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?” The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87199416","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}
P. Acampora, Emanuele Cristoforoni, C. Nitsch, C. Trombetti
We study the thermal insulation of a bounded body Ω ⊂R n , under a prescribed heat source f > 0, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.
{"title":"A free boundary problem in thermal insulation with a prescribed heat source","authors":"P. Acampora, Emanuele Cristoforoni, C. Nitsch, C. Trombetti","doi":"10.1051/cocv/2022081","DOIUrl":"https://doi.org/10.1051/cocv/2022081","url":null,"abstract":"We study the thermal insulation of a bounded body Ω ⊂R n , under a prescribed heat source f > 0, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"193 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76554855","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 article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson random martingale measure, the cost function containing cross terms is quadratic, in addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. Firstly, we prove the existence and uniqueness of the optimal control, and by using Pontryagin’s maximum principle we get the dual characterization of optimal control; Secondly, by introducing the adjoint processes of the state equation, establishing a stochastic Hamiltonian system and using decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information; Thirdly, the existence and uniqueness of the solutions of the two associated integro-differential Riccati equations are proved; Finally, we discuss a special case, and by means of filtering technique, establish the corresponding the filtering state feedback representation of the optimal control.
{"title":"A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information","authors":"Yiyun Yang, M. Tang, Qingxin Meng","doi":"10.1051/cocv/2022039","DOIUrl":"https://doi.org/10.1051/cocv/2022039","url":null,"abstract":"In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson random martingale measure, the cost function containing cross terms is quadratic, in addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. Firstly, we prove the existence and uniqueness of the optimal control, and by using Pontryagin’s maximum principle we get the dual characterization of optimal control; Secondly, by introducing the adjoint processes of the state equation, establishing a stochastic Hamiltonian system and using decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information; Thirdly, the existence and uniqueness of the solutions of the two associated integro-differential Riccati equations are proved; Finally, we discuss a special case, and by means of filtering technique, establish the corresponding the filtering state feedback representation of the optimal control.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"14 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89343195","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 explicitly computes the unique control set D with non-empty interior of a linear control system on R2, when the associated matrix has complex eigenvalues. It turns out that the closure of D coincides with the region delimited by a computable periodic orbit O of the system.
{"title":"Control sets of linear control systems on R2. The complex case.","authors":"V. Ayala, A. Silva, Erik Mamani","doi":"10.1051/cocv/2023043","DOIUrl":"https://doi.org/10.1051/cocv/2023043","url":null,"abstract":"This paper explicitly computes the unique control set D with non-empty interior of a linear control system on R2, when the associated matrix has complex eigenvalues. It turns out that the closure of D coincides with the region delimited by a computable periodic orbit O of the system.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"14 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79229585","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 investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.
{"title":"Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations","authors":"A. D. Corella, Nicolai Jork, V. Veliov","doi":"10.1051/cocv/2022075","DOIUrl":"https://doi.org/10.1051/cocv/2022075","url":null,"abstract":"This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"34 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81103766","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}
The Schrödinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.
{"title":"Regularity properties of the Schrödinger cost.","authors":"Gauthier Clerc","doi":"10.1051/cocv/2022033","DOIUrl":"https://doi.org/10.1051/cocv/2022033","url":null,"abstract":"The Schrödinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"129 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78325011","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 recent years there have been many in-depth researches on the boundary controllability and boundary synchronization for coupled systems of wave equations with various types of boundary conditions. In order to extend the study of synchronization from wave equations to a much larger range of hyperbolic systems, in this paper we will define and establish the exact boundary synchronization for the first order linear hyperbolic system based on previous work on its exact boundary controllability. The determination and estimate of exactly synchronizable states and some related problems are also discussed. This work can be applied to a great deal of diverse systems, and a new perspective to study the synchronization problem for the coupled system of wave equations can be also provided.
{"title":"Exact boundary synchronization for a kind of first order hyperbolic system","authors":"Tatsien Li, Xing Lu","doi":"10.1051/cocv/2022031","DOIUrl":"https://doi.org/10.1051/cocv/2022031","url":null,"abstract":"In recent years there have been many in-depth researches on the boundary controllability and boundary synchronization for coupled systems of wave equations with various types of boundary conditions. In order to extend the study of synchronization from wave equations to a much larger range of hyperbolic systems, in this paper we will define and establish the exact boundary synchronization for the first order linear hyperbolic system based on previous work on its exact boundary controllability. The determination and estimate of exactly synchronizable states and some related problems are also discussed. This work can be applied to a great deal of diverse systems, and a new perspective to study the synchronization problem for the coupled system of wave equations can be also provided.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"128 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76160808","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 studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc.
{"title":"Structure of optimal control for planetary landing with control and state constraints","authors":"Clara Leparoux, Bruno H'eriss'e, F. Jean","doi":"10.1051/cocv/2022065","DOIUrl":"https://doi.org/10.1051/cocv/2022065","url":null,"abstract":"This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"68 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79878036","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 provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of emph{nonautonomous symmetric joint convexity}, which extends the analogous definition devised for autonomous integrands in cite{FPS} where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in cite{ACDD}, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.
{"title":"Lower semicontinuity in $GSBD$ for nonautonomous surface integrals","authors":"V. Cicco, G. Scilla","doi":"10.1051/cocv/2023001","DOIUrl":"https://doi.org/10.1051/cocv/2023001","url":null,"abstract":"We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of emph{nonautonomous symmetric joint convexity}, which extends the analogous definition devised for autonomous integrands in cite{FPS} where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in cite{ACDD}, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89563067","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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