We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.
{"title":"Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation","authors":"Evelyn Herberg, M. Hinze","doi":"10.3934/mcrf.2022013","DOIUrl":"https://doi.org/10.3934/mcrf.2022013","url":null,"abstract":"We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133868477","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}
In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.
{"title":"Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph","authors":"G. Leugering, G. Mophou, M. Moutamal, M. Warma","doi":"10.3934/mcrf.2022015","DOIUrl":"https://doi.org/10.3934/mcrf.2022015","url":null,"abstract":"In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125305885","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}
This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain begin{document}$ X $end{document} can be controlled exactly from incoming boundary conditions for begin{document}$ X $end{document} under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.
This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain begin{document}$ X $end{document} can be controlled exactly from incoming boundary conditions for begin{document}$ X $end{document} under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.
{"title":"Boundary control for transport equations","authors":"G. Bal, A. Jollivet","doi":"10.3934/mcrf.2022014","DOIUrl":"https://doi.org/10.3934/mcrf.2022014","url":null,"abstract":"<p style='text-indent:20px;'>This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain <inline-formula><tex-math id=\"M1\">begin{document}$ X $end{document}</tex-math></inline-formula> can be controlled exactly from incoming boundary conditions for <inline-formula><tex-math id=\"M2\">begin{document}$ X $end{document}</tex-math></inline-formula> under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.</p>","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114221757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value begin{document}$ partial_t^{alpha} u(x, t) = -Au(x, t) $end{document}, where begin{document}$ -A = sum_{i, j = 1}^d partial_i(a_{ij}(x) partial_j) + sum_{j = 1}^d b_j(x) partial_j + c(x) $end{document}. We establish the uniqueness for an inverse problem of determining an order begin{document}$ alpha $end{document} of fractional derivatives by data begin{document}$ u(x_0, t) $end{document} for begin{document}$ 0 at one point begin{document}$ x_0 $end{document} in a spatial domain begin{document}$ Omega $end{document}. The uniqueness holds even under assumption that begin{document}$ Omega $end{document} and begin{document}$ A $end{document} are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.
{"title":"Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations","authors":"Masahiro Yamamoto","doi":"10.3934/mcrf.2022017","DOIUrl":"https://doi.org/10.3934/mcrf.2022017","url":null,"abstract":"<p style='text-indent:20px;'>We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value <inline-formula><tex-math id=\"M1\">begin{document}$ partial_t^{alpha} u(x, t) = -Au(x, t) $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M2\">begin{document}$ -A = sum_{i, j = 1}^d partial_i(a_{ij}(x) partial_j) + sum_{j = 1}^d b_j(x) partial_j + c(x) $end{document}</tex-math></inline-formula>. We establish the uniqueness for an inverse problem of determining an order <inline-formula><tex-math id=\"M3\">begin{document}$ alpha $end{document}</tex-math></inline-formula> of fractional derivatives by data <inline-formula><tex-math id=\"M4\">begin{document}$ u(x_0, t) $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M5\">begin{document}$ 0<t<T $end{document}</tex-math></inline-formula> at one point <inline-formula><tex-math id=\"M6\">begin{document}$ x_0 $end{document}</tex-math></inline-formula> in a spatial domain <inline-formula><tex-math id=\"M7\">begin{document}$ Omega $end{document}</tex-math></inline-formula>. The uniqueness holds even under assumption that <inline-formula><tex-math id=\"M8\">begin{document}$ Omega $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M9\">begin{document}$ A $end{document}</tex-math></inline-formula> are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.</p>","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131369398","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}
In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.
{"title":"Carleman estimates for a magnetohydrodynamics system and application to inverse source problems","authors":"Xinchi Huang, Masahiro Yamamoto","doi":"10.3934/mcrf.2022005","DOIUrl":"https://doi.org/10.3934/mcrf.2022005","url":null,"abstract":"In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128022414","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}
In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.
{"title":"Controllability under positivity constraints of semilinear heat equations","authors":"Dario Pighin, E. Zuazua","doi":"10.3934/mcrf.2018041","DOIUrl":"https://doi.org/10.3934/mcrf.2018041","url":null,"abstract":"In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. \u0000In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. \u0000More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. \u0000We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116762022","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}
This paper studies the integral turnpike and turnpike in average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon.
{"title":"Averaged turnpike property for differential equations with random constant coefficients","authors":"M. Hernández, R. Lecaros, S. Zamorano","doi":"10.3934/mcrf.2022016","DOIUrl":"https://doi.org/10.3934/mcrf.2022016","url":null,"abstract":"This paper studies the integral turnpike and turnpike in average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127115480","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}
In this paper we are concerned with a viscoelastic wave equation with infinite memory and nonlinear frictional damping of variable-exponent type. First, we establish explicit and general decay results with a very general assumption on the relaxation function. Then, we remove the constraint imposed on the boundedness condition on the initial data used in the earlier results in the literature. Finally, we perform several numerical tests to illustrate our theoretical findings. This study generalizes and improves previous literature outcomes.
{"title":"Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity","authors":"A. Al‐Mahdi, M. Al‐Gharabli, M. Zahri","doi":"10.3934/mcrf.2022010","DOIUrl":"https://doi.org/10.3934/mcrf.2022010","url":null,"abstract":"In this paper we are concerned with a viscoelastic wave equation with infinite memory and nonlinear frictional damping of variable-exponent type. First, we establish explicit and general decay results with a very general assumption on the relaxation function. Then, we remove the constraint imposed on the boundedness condition on the initial data used in the earlier results in the literature. Finally, we perform several numerical tests to illustrate our theoretical findings. This study generalizes and improves previous literature outcomes.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132805708","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}
In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By using the semigroup theory, we show that the system is globally well posed. We give two approaches to obtain the exponential stability. The first one is established by multiplier approach provided the coefficients of delay terms are small. We can also obtain the stability by establishing an equivalence between the stabilization of this system and the observability of the corresponding undamped system. The result is new and is the first result of observability on the Rao-Nakra sandwich beam with with time-varying weight and time-varying delay.
{"title":"Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability","authors":"B. Feng, C. Raposo, C. Nonato, A. Soufyane","doi":"10.3934/mcrf.2022011","DOIUrl":"https://doi.org/10.3934/mcrf.2022011","url":null,"abstract":"In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By using the semigroup theory, we show that the system is globally well posed. We give two approaches to obtain the exponential stability. The first one is established by multiplier approach provided the coefficients of delay terms are small. We can also obtain the stability by establishing an equivalence between the stabilization of this system and the observability of the corresponding undamped system. The result is new and is the first result of observability on the Rao-Nakra sandwich beam with with time-varying weight and time-varying delay.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114077926","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}
In this paper, we investigate Gross-Pitaevskii equations with double power nonlinearities. Firstly, due to the defocusing effect from the lower power order nonlinearity, Gross-Pitaevskii equations still have standing waves when the frequency begin{document}$ omega $end{document} is the negative of the first eigenvalue of the linear operator begin{document}$ - Delta + gamma|x{|^2} $end{document}. The existence of this class of standing waves is proved by the variational method, especially the mountain pass lemma. Secondly, by establishing the relationship to the known standing waves of the classical nonlinear Schrödinger equations, we study the instability of standing waves for begin{document}$ q ge 1 + 4/N $end{document} and begin{document}$ omega $end{document} sufficiently large. Finally, we use the variational argument to prove the stability of standing waves for begin{document}$ q le 1 + 4/N $end{document}.
In this paper, we investigate Gross-Pitaevskii equations with double power nonlinearities. Firstly, due to the defocusing effect from the lower power order nonlinearity, Gross-Pitaevskii equations still have standing waves when the frequency begin{document}$ omega $end{document} is the negative of the first eigenvalue of the linear operator begin{document}$ - Delta + gamma|x{|^2} $end{document}. The existence of this class of standing waves is proved by the variational method, especially the mountain pass lemma. Secondly, by establishing the relationship to the known standing waves of the classical nonlinear Schrödinger equations, we study the instability of standing waves for begin{document}$ q ge 1 + 4/N $end{document} and begin{document}$ omega $end{document} sufficiently large. Finally, we use the variational argument to prove the stability of standing waves for begin{document}$ q le 1 + 4/N $end{document}.
{"title":"Stability and instability of standing waves for Gross-Pitaevskii equations with double power nonlinearities","authors":"Yue Zhang, Jian Zhang","doi":"10.3934/mcrf.2022007","DOIUrl":"https://doi.org/10.3934/mcrf.2022007","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we investigate Gross-Pitaevskii equations with double power nonlinearities. Firstly, due to the defocusing effect from the lower power order nonlinearity, Gross-Pitaevskii equations still have standing waves when the frequency <inline-formula><tex-math id=\"M1\">begin{document}$ omega $end{document}</tex-math></inline-formula> is the negative of the first eigenvalue of the linear operator <inline-formula><tex-math id=\"M2\">begin{document}$ - Delta + gamma|x{|^2} $end{document}</tex-math></inline-formula>. The existence of this class of standing waves is proved by the variational method, especially the mountain pass lemma. Secondly, by establishing the relationship to the known standing waves of the classical nonlinear Schrödinger equations, we study the instability of standing waves for <inline-formula><tex-math id=\"M3\">begin{document}$ q ge 1 + 4/N $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">begin{document}$ omega $end{document}</tex-math></inline-formula> sufficiently large. Finally, we use the variational argument to prove the stability of standing waves for <inline-formula><tex-math id=\"M5\">begin{document}$ q le 1 + 4/N $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127902431","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}
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