The exact distributed controllability of the semilinear heat equation begin{document}$ partial_{t}y-Delta y + g(y) = f ,1_{omega} $end{document} posed over multi-dimensional and bounded domains, assuming that begin{document}$ gin C^1(mathbb{R}) $end{document} satisfies the growth condition begin{document}$ limsup_{rto infty} g(r)/ (vert rvert ln^{3/2}vert rvert) = 0 $end{document} has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that begin{document}$ g^prime $end{document} does not grow faster than begin{document}$ beta ln^{3/2}vert rvert $end{document} at infinity for begin{document}$ beta>0 $end{document} small enough and that begin{document}$ g^prime $end{document} is uniformly Hölder continuous on begin{document}$ mathbb{R} $end{document} with exponent begin{document}$ pin [0,1] $end{document}, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order begin{document}$ 1+p $end{document} after a finite number of iterations.
The exact distributed controllability of the semilinear heat equation begin{document}$ partial_{t}y-Delta y + g(y) = f ,1_{omega} $end{document} posed over multi-dimensional and bounded domains, assuming that begin{document}$ gin C^1(mathbb{R}) $end{document} satisfies the growth condition begin{document}$ limsup_{rto infty} g(r)/ (vert rvert ln^{3/2}vert rvert) = 0 $end{document} has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that begin{document}$ g^prime $end{document} does not grow faster than begin{document}$ beta ln^{3/2}vert rvert $end{document} at infinity for begin{document}$ beta>0 $end{document} small enough and that begin{document}$ g^prime $end{document} is uniformly Hölder continuous on begin{document}$ mathbb{R} $end{document} with exponent begin{document}$ pin [0,1] $end{document}, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order begin{document}$ 1+p $end{document} after a finite number of iterations.
{"title":"Constructive exact control of semilinear 1D heat equations","authors":"Jérôme Lemoine, A. Munch","doi":"10.3934/mcrf.2022001","DOIUrl":"https://doi.org/10.3934/mcrf.2022001","url":null,"abstract":"<p style='text-indent:20px;'>The exact distributed controllability of the semilinear heat equation <inline-formula><tex-math id=\"M1\">begin{document}$ partial_{t}y-Delta y + g(y) = f ,1_{omega} $end{document}</tex-math></inline-formula> posed over multi-dimensional and bounded domains, assuming that <inline-formula><tex-math id=\"M2\">begin{document}$ gin C^1(mathbb{R}) $end{document}</tex-math></inline-formula> satisfies the growth condition <inline-formula><tex-math id=\"M3\">begin{document}$ limsup_{rto infty} g(r)/ (vert rvert ln^{3/2}vert rvert) = 0 $end{document}</tex-math></inline-formula> has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that <inline-formula><tex-math id=\"M4\">begin{document}$ g^prime $end{document}</tex-math></inline-formula> does not grow faster than <inline-formula><tex-math id=\"M5\">begin{document}$ beta ln^{3/2}vert rvert $end{document}</tex-math></inline-formula> at infinity for <inline-formula><tex-math id=\"M6\">begin{document}$ beta>0 $end{document}</tex-math></inline-formula> small enough and that <inline-formula><tex-math id=\"M7\">begin{document}$ g^prime $end{document}</tex-math></inline-formula> is uniformly Hölder continuous on <inline-formula><tex-math id=\"M8\">begin{document}$ mathbb{R} $end{document}</tex-math></inline-formula> with exponent <inline-formula><tex-math id=\"M9\">begin{document}$ pin [0,1] $end{document}</tex-math></inline-formula>, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order <inline-formula><tex-math id=\"M10\">begin{document}$ 1+p $end{document}</tex-math></inline-formula> after a finite number of iterations.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90479283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we study the dependence of solutions of the continuous-time finite state mean field game on initial distribution of players. Our approach relies on the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the reformulation of the finite state mean field game as a control problem with mixed constraints, we give the sufficient condition on a given multifunction to be a value multifunction in the terms of the viability theory. The maximal multifunction (i.e., the mapping assigning to an initial time and an initial distribution the whole set of values corresponding to solutions of the mean field game) is characterized via the backward attainability set for the certain control system.
{"title":"Control theory approach to continuous-time finite state mean field games","authors":"Y. Averboukh","doi":"10.3934/mcrf.2022029","DOIUrl":"https://doi.org/10.3934/mcrf.2022029","url":null,"abstract":"In the paper, we study the dependence of solutions of the continuous-time finite state mean field game on initial distribution of players. Our approach relies on the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the reformulation of the finite state mean field game as a control problem with mixed constraints, we give the sufficient condition on a given multifunction to be a value multifunction in the terms of the viability theory. The maximal multifunction (i.e., the mapping assigning to an initial time and an initial distribution the whole set of values corresponding to solutions of the mean field game) is characterized via the backward attainability set for the certain control system.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83145078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We characterize strict dissipativity properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelty of these results is the possibility to encompass the presence of state and input constraints.
{"title":"On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems","authors":"L. Grüne, Roberto Guglielmi","doi":"10.3934/MCRF.2020032","DOIUrl":"https://doi.org/10.3934/MCRF.2020032","url":null,"abstract":"The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We characterize strict dissipativity properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelty of these results is the possibility to encompass the presence of state and input constraints.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81244794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related begin{document}$ epsilon $end{document} -Nash equilibrium property is verified.
This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related begin{document}$ epsilon $end{document} -Nash equilibrium property is verified.
{"title":"Linear-quadratic-Gaussian mean-field-game with partial observation and common noise","authors":"A. Bensoussan, Xinwei Feng, Jianhui Huang","doi":"10.3934/mcrf.2020025","DOIUrl":"https://doi.org/10.3934/mcrf.2020025","url":null,"abstract":"This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related begin{document}$ epsilon $end{document} -Nash equilibrium property is verified.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88251737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper performance indices for economic model predictive controllers (MPC) are considered. Since existing relative performance measures, designed for stabilizing controllers, fail in the economic setting, we propose alternative absolute quantities. We show that these can be applied to assess the performance of the closed loop trajectories on-line while the controller is running. The advantages of our approach are demonstrated by simulations involving a convection-diffusion-system. The method is also combined with proper orthogonal decomposition, thus demonstrating the possibility for both efficient and performant MPC for systems governed by partial differential equations.
{"title":"Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations","authors":"L. Grüne, L. Mechelli, S. Pirkelmann, S. Volkwein","doi":"10.3934/MCRF.2021013","DOIUrl":"https://doi.org/10.3934/MCRF.2021013","url":null,"abstract":"In this paper performance indices for economic model predictive controllers (MPC) are considered. Since existing relative performance measures, designed for stabilizing controllers, fail in the economic setting, we propose alternative absolute quantities. We show that these can be applied to assess the performance of the closed loop trajectories on-line while the controller is running. The advantages of our approach are demonstrated by simulations involving a convection-diffusion-system. The method is also combined with proper orthogonal decomposition, thus demonstrating the possibility for both efficient and performant MPC for systems governed by partial differential equations.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88332319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.
{"title":"On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback","authors":"J. Valein","doi":"10.3934/mcrf.2021039","DOIUrl":"https://doi.org/10.3934/mcrf.2021039","url":null,"abstract":"The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79273502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.
{"title":"A concise introduction to control theory for stochastic partial differential equations","authors":"Qi Lu, Xu Zhang","doi":"10.3934/MCRF.2021020","DOIUrl":"https://doi.org/10.3934/MCRF.2021020","url":null,"abstract":"The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89403140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [12] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues begin{document}$ {lambda_m} $end{document} and construct corresponding eigenfunctions. Moreover, the order of growth for these begin{document}$ {lambda_m} $end{document} are obtained: begin{document}$ lambda_msim m^2 $end{document}, as begin{document}$ mrightarrow +infty $end{document}. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.
In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [12] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues begin{document}$ {lambda_m} $end{document} and construct corresponding eigenfunctions. Moreover, the order of growth for these begin{document}$ {lambda_m} $end{document} are obtained: begin{document}$ lambda_msim m^2 $end{document}, as begin{document}$ mrightarrow +infty $end{document}. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.
{"title":"Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application","authors":"Guangdong Jing, Penghui Wang","doi":"10.3934/mcrf.2021055","DOIUrl":"https://doi.org/10.3934/mcrf.2021055","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we solve the eigenvalue problem of stochastic Hamiltonian system with boundary conditions. Firstly, we extend the results in Peng [<xref ref-type=\"bibr\" rid=\"b12\">12</xref>] from time-invariant case to time-dependent case, proving the existence of a series of eigenvalues <inline-formula><tex-math id=\"M1\">begin{document}$ {lambda_m} $end{document}</tex-math></inline-formula> and construct corresponding eigenfunctions. Moreover, the order of growth for these <inline-formula><tex-math id=\"M2\">begin{document}$ {lambda_m} $end{document}</tex-math></inline-formula> are obtained: <inline-formula><tex-math id=\"M3\">begin{document}$ lambda_msim m^2 $end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id=\"M4\">begin{document}$ mrightarrow +infty $end{document}</tex-math></inline-formula>. As applications, we give an explicit estimation formula about the statistic period of solutions of Forward-Backward SDEs. Besides, by a meticulous example we show the subtle situation in time-dependent case that some eigenvalues appear when the solution of the associated Riccati equation does not blow-up, which does not happen in time-invariant case.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84262482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 abstract thermoelastic system in Hilbert space with infinite memory and time delay. Under some suitable conditions, we prove the well-posedness by invoking semigroup theory. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay term, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. By establishing suitable Lyapunov functionals which are equivalent to the energy of system we also establish the general energy decay results for abstract thermoelastic system.
{"title":"General stability of abstract thermoelastic system with infinite memory and delay","authors":"Jianghao Hao, Junna Zhang","doi":"10.3934/mcrf.2020040","DOIUrl":"https://doi.org/10.3934/mcrf.2020040","url":null,"abstract":"In this paper we study an abstract thermoelastic system in Hilbert space with infinite memory and time delay. Under some suitable conditions, we prove the well-posedness by invoking semigroup theory. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay term, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. By establishing suitable Lyapunov functionals which are equivalent to the energy of system we also establish the general energy decay results for abstract thermoelastic system.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74520104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Kovtanyuk, A. Chebotarev, N. Botkin, V. Turova, I. Sidorenko, R. Lampe
. A boundary value problem for the Poisson’s equation with unknown intensities of sources is studied in context of mathematical modeling the pressure distribution in cerebral capillary networks. The problem is formulated as an inverse problem with finite-dimensional overdetermination. The unique solvability of the problem is proven. A numerical algorithm is proposed and implemented.
{"title":"Modeling the pressure distribution in a spatially averaged cerebral capillary network","authors":"A. Kovtanyuk, A. Chebotarev, N. Botkin, V. Turova, I. Sidorenko, R. Lampe","doi":"10.3934/MCRF.2021016","DOIUrl":"https://doi.org/10.3934/MCRF.2021016","url":null,"abstract":". A boundary value problem for the Poisson’s equation with unknown intensities of sources is studied in context of mathematical modeling the pressure distribution in cerebral capillary networks. The problem is formulated as an inverse problem with finite-dimensional overdetermination. The unique solvability of the problem is proven. A numerical algorithm is proposed and implemented.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72370719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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