In this paper, we are concerned with the stabilization problem of the game-based control system. In particular, two players are involved in the system where one is to minimize the related cost function and the other is to stabilize the system. Different from the previous works, the new contribution is to derive the necessary and sufficient condition for the stabilization of the game-based control system by applying Stackelberg game method. The key technique is to explicitly solve the forward and backward difference equations (FBDEs) from the Stackelberg game and give the optimal feedback gain matrix of the leader by using the matrix maximum principle.
{"title":"Stackelberg method to stabilize game-based control system","authors":"Yue Sun, Juanjuan Xu, Huanshui Zhang, Renren Zhang","doi":"10.1051/cocv/2023060","DOIUrl":"https://doi.org/10.1051/cocv/2023060","url":null,"abstract":"In this paper, we are concerned with the stabilization problem of the game-based control system. In particular, two players are involved in the system where one is to minimize the related cost function and the other is to stabilize the system. Different from the previous works, the new contribution is to derive the necessary and sufficient condition for the stabilization of the game-based control system by applying Stackelberg game method. The key technique is to explicitly solve the forward and backward difference equations (FBDEs) from the Stackelberg game and give the optimal feedback gain matrix of the leader by using the matrix maximum principle.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"15 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74482326","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":"The Hinfinity - Control problem for parabolic systems. Applications to systems with singular hardy potentials","authors":"G. Marinoschi","doi":"10.1051/cocv/2023059","DOIUrl":"https://doi.org/10.1051/cocv/2023059","url":null,"abstract":"<div class=\"abstract\"> <p style=\"margin: 0px;\"> <div>We solve the $H^{infty }$-control problem with state</div> </div> <p style=\"margin: 0px;\"> <div>feedback for infinite dimensional boundary control systems of parabolic type</div> </div> <p style=\"margin: 0px;\"> <div>with distributed disturbances and provide some applications of this result</div> </div> <p style=\"margin: 0px;\"> <div>to equations with Hardy potentials with the singularity inside or on the</div> </div> <p style=\"margin: 0px;\"> <div>boundary, in the cases of a distributed control and of a boundary control.</div> </div> <div class=\"sectionWrapper\"> <h1 class=\"heading\" xslelement=\"title\"> <div>2020 Mathematics Subject Classification</div> </h1> <div class=\"containerGroup\"> <div class=\"paragraphWrapper\"> <div><div>93B36, 93B52, 93B35, 35K90.</div> </div> </div> </div> </div></div>","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"48 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73594577","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 consider uniformly exponential approximation for a vibrating cable with tip mass under a non-collocated output stabilizing feedback control. By designing an observer-based output feedback control, the closed-loop system is composed of the coupled same type of PDEs and ODEs. By order reduction method, we find a global Lyapunov functional for the closed-loop system. The closed-loop system is then semi-discretized by the finite difference method. For the discrete systems, we also construct the Lyapunov functions. The uniform exponential stability of the semi-discretized systems is then established analogously as the proof for the continuous counterpart via an indirect Lyapunov functional approach.
{"title":"Uniformly exponential stability of semi-discrete scheme for a vibration cable with a tip mass under observer-based feedback control","authors":"B. Guo, Xi Zhao","doi":"10.1051/cocv/2023058","DOIUrl":"https://doi.org/10.1051/cocv/2023058","url":null,"abstract":"In this paper, we consider uniformly exponential approximation for a vibrating cable with tip mass under a non-collocated output stabilizing feedback control. By designing an observer-based output feedback control, the closed-loop system is composed of the coupled same type of PDEs and ODEs. By order reduction method, we find a global Lyapunov functional for the closed-loop system. The closed-loop system is then semi-discretized by the finite difference method. For the discrete systems, we also construct the Lyapunov functions. The uniform exponential stability of the semi-discretized systems is then established analogously as the proof for the continuous counterpart via an indirect Lyapunov functional approach.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"109 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74748813","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}
For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.
{"title":"Directional differentiability for shape optimization with variational inequalities as constraints","authors":"V. A. Kovtunenko, K. Kunisch","doi":"10.1051/cocv/2023056","DOIUrl":"https://doi.org/10.1051/cocv/2023056","url":null,"abstract":"For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"58 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81378603","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 the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [6]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [1] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present. AMS Subject Classification Primary: 93B07, 74K10; Secondary: 42A16, 35M10, 35A25.
{"title":"Observability of a string-beams network with many beams","authors":"Anna Chiara Lai, P. Loreti, M. Mehrenberger","doi":"10.1051/cocv/2023054","DOIUrl":"https://doi.org/10.1051/cocv/2023054","url":null,"abstract":"We prove the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [6]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [1] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present.\u0000\u0000AMS Subject Classification\u0000\u0000Primary: 93B07, 74K10; Secondary: 42A16, 35M10, 35A25.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"15 5","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72411432","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 historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.
{"title":"Zermelo navigation problems on surfaces of revolution and geometric optimal control\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":"B. Bonnard, O. Cots, B. Wembe","doi":"10.1051/cocv/2023052","DOIUrl":"https://doi.org/10.1051/cocv/2023052","url":null,"abstract":"In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"162 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74772148","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 many applications, in systems that are governed by a linear hyperbolic partial differential equations some of the problem parameters are uncertain. If information about the probability distribution of the parametric uncertainty distribution is available, the uncertain state of the system can be described using an intrinsic formulation through a polynomial chaos expansion. This allows to obtain solutions for optimal boundary control problems with random parameters. We show that similar to the deterministic case, a turnpike result holds in the sense that for large time horizons the optimal states for dynamic optimal control problems on a substantial part of the time interval approach the optimal states for the corresponding uncertain static optimal control problem. We show integral turnpike results both for the full uncertain system as well as for a generalized polynomial chaos approximation.
{"title":"Turnpike properties of optimal boundary control problems with random linear hyperbolic systems","authors":"M. Gugat, M. Herty","doi":"10.1051/cocv/2023051","DOIUrl":"https://doi.org/10.1051/cocv/2023051","url":null,"abstract":"In many applications, in systems that are governed by a linear hyperbolic partial differential equations some of the problem parameters are uncertain. If information about the probability distribution of the parametric uncertainty distribution is available, the uncertain state of the system can be described using an intrinsic formulation through a polynomial chaos expansion. \u0000 This allows to obtain solutions for optimal boundary control problems with random parameters. \u0000We show that similar to the deterministic case, a turnpike result holds in the sense that for large time horizons the optimal states for dynamic optimal control problems on a substantial part of the time interval approach the optimal states for the corresponding uncertain static optimal control problem. We show integral turnpike results both for the full uncertain system as well as for a generalized polynomial chaos approximation.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83043518","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 an optimal control problem of partially observed mean-field type stochastic control system with Markov chain in progressive structure. The control variable is allowed to enter the diffusion term of the state process and the drift term of the observation process. The control domain need not be convex. In our model, the cost functional and the observation are also of mean-field type. By virtue of a special spike variation, the related stochastic maximum principle has been obtained. The stochastic maximum principle in progressive structure is essentially different from the classical case.
{"title":"A general maximum principle for progressive optimal control of partially observed mean-field stochastic system with markov chain.","authors":"Tian Chen, Y. Jia","doi":"10.1051/cocv/2023050","DOIUrl":"https://doi.org/10.1051/cocv/2023050","url":null,"abstract":"In this paper, we study an optimal control problem of partially observed mean-field type stochastic control system with Markov chain in progressive structure. The control variable is allowed to enter the diffusion term of the state process and the drift term of the observation process. The control domain need not be convex. In our model, the cost functional and the observation are also of mean-field type. By virtue of a special spike variation, the related stochastic maximum principle has been obtained. The stochastic maximum principle in progressive structure is essentially different from the classical case.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88145713","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 quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in R^n. We first obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally H''older-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set by utilizing a telescoping series method.
{"title":"Analyticity and observability for fractional order parabolic equations in the whole space","authors":"Ming Wang, Can Zhang","doi":"10.1051/cocv/2023053","DOIUrl":"https://doi.org/10.1051/cocv/2023053","url":null,"abstract":"In this paper, we study the quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in R^n. We first obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally H''older-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set by utilizing a telescoping series method.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"11 2 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75528982","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 novel space-discretized Finite Differences-based model reduction introduced in cite{Guo3} is extended to the partial differential equations (PDE) model of a multi- layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE can not be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the "order-reduced'' Finite-Differences technique. This method does not require any type of filtering though the exact observability as $hto 0$ is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.
{"title":"A novel sensor design for a cantilevered Mead-Marcus-type sandwich beam model by the order-reduction technique","authors":"A. Ozer, Ahmet Kaan Aydin","doi":"10.1051/cocv/2023061","DOIUrl":"https://doi.org/10.1051/cocv/2023061","url":null,"abstract":"A novel space-discretized Finite Differences-based model reduction introduced in cite{Guo3} is extended to the partial differential equations (PDE) model of a multi- layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE can not be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the \"order-reduced'' Finite-Differences technique. This method does not require any type of filtering though the exact observability as $hto 0$ is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"136 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79302193","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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