In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping begin{document}$ |u|^{r-1}u, rin[1, infty) $end{document} in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any begin{document}$ rgeq 1, $end{document} the existence and uniqueness of a weak solution is discussed when the domain begin{document}$ Omega $end{document} is periodic/bounded in begin{document}$ mathbb R^3 $end{document} while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.
In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping begin{document}$ |u|^{r-1}u, rin[1, infty) $end{document} in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any begin{document}$ rgeq 1, $end{document} the existence and uniqueness of a weak solution is discussed when the domain begin{document}$ Omega $end{document} is periodic/bounded in begin{document}$ mathbb R^3 $end{document} while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.
{"title":"Optimal control of the 3D damped Navier-Stokes-Voigt equations with control constraints","authors":"Sakthivel Kumarasamy","doi":"10.3934/eect.2022030","DOIUrl":"https://doi.org/10.3934/eect.2022030","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping <inline-formula><tex-math id=\"M1\">begin{document}$ |u|^{r-1}u, rin[1, infty) $end{document}</tex-math></inline-formula> in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any <inline-formula><tex-math id=\"M2\">begin{document}$ rgeq 1, $end{document}</tex-math></inline-formula> the existence and uniqueness of a weak solution is discussed when the domain <inline-formula><tex-math id=\"M3\">begin{document}$ Omega $end{document}</tex-math></inline-formula> is periodic/bounded in <inline-formula><tex-math id=\"M4\">begin{document}$ mathbb R^3 $end{document}</tex-math></inline-formula> while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"24 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85199681","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}
We study a~quasilinear coupled magneto-quasistatic model from a~systems theoretic perspective.} First, by taking the injected voltages as input and the associated currents as output, we prove that the magneto-quasistatic system is passive. Moreover, by defining suitable Dirac and resistive structures, we show that it admits a~representation as a~port-Hamiltonian system. Thereafter, we consider dependence on initial and input data. We show that the current and the magnetic vector potential can be estimated by means of the initial magnetic vector potential and the voltage. We also analyse the free dynamics of the system and study the asymptotic behavior of the solutions for $ttoinfty$.
{"title":"Passivity, port-hamiltonian formulation and solution estimates for a coupled magneto-quasistatic system","authors":"Timo Reis, T. Stykel","doi":"10.3934/eect.2023008","DOIUrl":"https://doi.org/10.3934/eect.2023008","url":null,"abstract":"We study a~quasilinear coupled magneto-quasistatic model from a~systems theoretic perspective.} First, by taking the injected voltages as input and the associated currents as output, we prove that the magneto-quasistatic system is passive. Moreover, by defining suitable Dirac and resistive structures, we show that it admits a~representation as a~port-Hamiltonian system. Thereafter, we consider dependence on initial and input data. We show that the current and the magnetic vector potential can be estimated by means of the initial magnetic vector potential and the voltage. We also analyse the free dynamics of the system and study the asymptotic behavior of the solutions for $ttoinfty$.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"29 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89876161","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}
An oblique projections based feedback stabilizability result in the literature is extended to a larger class of reaction-convection terms. A discussion is presented including a comparison between explicit oblique projections based feedback controls and Riccati based feedback controls. Advantages and limitations of each type of feedback are addressed as well as their finite-elements implementation. Results of numerical simulations are presented comparing their stabilizing performances for the case of time-periodic dynamics. Estimates are presented on the convergence rate of a proposed iterative algorithm to compute the time-periodic Riccati feedback.
{"title":"Stabilization of nonautonomous linear parabolic-like equations: Oblique projections versus Riccati feedbacks","authors":"S. Rodrigues","doi":"10.3934/eect.2022045","DOIUrl":"https://doi.org/10.3934/eect.2022045","url":null,"abstract":"An oblique projections based feedback stabilizability result in the literature is extended to a larger class of reaction-convection terms. A discussion is presented including a comparison between explicit oblique projections based feedback controls and Riccati based feedback controls. Advantages and limitations of each type of feedback are addressed as well as their finite-elements implementation. Results of numerical simulations are presented comparing their stabilizing performances for the case of time-periodic dynamics. Estimates are presented on the convergence rate of a proposed iterative algorithm to compute the time-periodic Riccati feedback.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"3 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85066566","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 generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asymptotically sharp estimate for the observation constant and which is explicit in the model parameters. We give two types of applications where the extension of the growth/decay rates naturally appear. By exploiting subordination techniques we show how the dissipation estimate of a semigroup transfers to subordinated semigroups. Furthermore, we apply our results to semigroups related to L{'e}vy processes.
{"title":"Final state observability in Banach spaces with applications to subordination and semigroups induced by Lévy processes","authors":"Dennis Gallaun, J. Meichsner, C. Seifert","doi":"10.3934/eect.2023002","DOIUrl":"https://doi.org/10.3934/eect.2023002","url":null,"abstract":"This paper generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asymptotically sharp estimate for the observation constant and which is explicit in the model parameters. We give two types of applications where the extension of the growth/decay rates naturally appear. By exploiting subordination techniques we show how the dissipation estimate of a semigroup transfers to subordinated semigroups. Furthermore, we apply our results to semigroups related to L{'e}vy processes.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"142 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80150306","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}
{"title":"Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system","authors":"","doi":"10.3934/eect.2022054","DOIUrl":"https://doi.org/10.3934/eect.2022054","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"4 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90907171","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, the approximate controllability for Sobolev type begin{document}$ psi - $end{document} Hilfer fractional backward perturbed integro-differential equations with begin{document}$ psi - $end{document} fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, begin{document}$ psi - $end{document}Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward begin{document}$ psi - $end{document} fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.
In this paper, the approximate controllability for Sobolev type begin{document}$ psi - $end{document} Hilfer fractional backward perturbed integro-differential equations with begin{document}$ psi - $end{document} fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, begin{document}$ psi - $end{document}Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward begin{document}$ psi - $end{document} fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.
{"title":"Controllability results for Sobolev type $ psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space","authors":"Ichrak Bouacida, Mourad Kerboua, S. Segni","doi":"10.3934/eect.2022028","DOIUrl":"https://doi.org/10.3934/eect.2022028","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, the approximate controllability for Sobolev type <inline-formula><tex-math id=\"M2\">begin{document}$ psi - $end{document}</tex-math></inline-formula> Hilfer fractional backward perturbed integro-differential equations with <inline-formula><tex-math id=\"M3\">begin{document}$ psi - $end{document}</tex-math></inline-formula> fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, <inline-formula><tex-math id=\"M4\">begin{document}$ psi - $end{document}</tex-math></inline-formula>Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward <inline-formula><tex-math id=\"M5\">begin{document}$ psi - $end{document}</tex-math></inline-formula> fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78634973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control begin{document}$ g $end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when begin{document}$ g in L^2(0, T;L^2(Gamma)) equiv L^2(Sigma) $end{document}, which, moreover, in the canonical case begin{document}$ gamma = 0 $end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether begin{document}$ gamma = 0 $end{document} or begin{document}$ 0 neq gamma in L^{infty}(Omega) $end{document}, since begin{document}$ gamma neq 0 $end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with begin{document}$ g $end{document} "smoother" than begin{document}$ L^2(Sigma) $end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than begin{document}$ L^2(0, T;L^2(Gamma)) $end{document}, and [44] for control less regular in space than begin{document}$ L^2(Gamma) $end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].
We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control begin{document}$ g $end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when begin{document}$ g in L^2(0, T;L^2(Gamma)) equiv L^2(Sigma) $end{document}, which, moreover, in the canonical case begin{document}$ gamma = 0 $end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether begin{document}$ gamma = 0 $end{document} or begin{document}$ 0 neq gamma in L^{infty}(Omega) $end{document}, since begin{document}$ gamma neq 0 $end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with begin{document}$ g $end{document} "smoother" than begin{document}$ L^2(Sigma) $end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than begin{document}$ L^2(0, T;L^2(Gamma)) $end{document}, and [44] for control less regular in space than begin{document}$ L^2(Gamma) $end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].
{"title":"From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae","authors":"R. Triggiani, X. Wan","doi":"10.3934/eect.2022007","DOIUrl":"https://doi.org/10.3934/eect.2022007","url":null,"abstract":"<p style='text-indent:20px;'>We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control <inline-formula><tex-math id=\"M1\">begin{document}$ g $end{document}</tex-math></inline-formula>. Optimal interior and boundary regularity results were given in [<xref ref-type=\"bibr\" rid=\"b1\">1</xref>], after [<xref ref-type=\"bibr\" rid=\"b41\">41</xref>], when <inline-formula><tex-math id=\"M2\">begin{document}$ g in L^2(0, T;L^2(Gamma)) equiv L^2(Sigma) $end{document}</tex-math></inline-formula>, which, moreover, in the canonical case <inline-formula><tex-math id=\"M3\">begin{document}$ gamma = 0 $end{document}</tex-math></inline-formula>, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [<xref ref-type=\"bibr\" rid=\"b19\">19</xref>], [<xref ref-type=\"bibr\" rid=\"b17\">17</xref>], [<xref ref-type=\"bibr\" rid=\"b24\">24</xref>,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether <inline-formula><tex-math id=\"M4\">begin{document}$ gamma = 0 $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=\"M5\">begin{document}$ 0 neq gamma in L^{infty}(Omega) $end{document}</tex-math></inline-formula>, since <inline-formula><tex-math id=\"M6\">begin{document}$ gamma neq 0 $end{document}</tex-math></inline-formula> is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with <inline-formula><tex-math id=\"M7\">begin{document}$ g $end{document}</tex-math></inline-formula> \"smoother\" than <inline-formula><tex-math id=\"M8\">begin{document}$ L^2(Sigma) $end{document}</tex-math></inline-formula>, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [<xref ref-type=\"bibr\" rid=\"b17\">17</xref>]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [<xref ref-type=\"bibr\" rid=\"b22\">22</xref>], [<xref ref-type=\"bibr\" rid=\"b23\">23</xref>], [<xref ref-type=\"bibr\" rid=\"b37\">37</xref>] for control smoother than <inline-formula><tex-math id=\"M9\">begin{document}$ L^2(0, T;L^2(Gamma)) $end{document}</tex-math></inline-formula>, and [<xref ref-type=\"bibr\" rid=\"b44\">44</xref>] for control less regular in space than <inline-formula><tex-math id=\"M10\">begin{document}$ L^2(Gamma) $end{document}</tex-math></inline-formula>. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [<xref ref-type=\"bibr\" rid=\"b42\">42</xref>], [<xref ref-type=\"bibr\" rid=\"b24\">24</xref>,Section 9.8.2].</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86316633","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}
{"title":"Optimal tubes for non-cylindrical Navier-Stokes flows with Navier boundary condition","authors":"R. Dziri","doi":"10.3934/eect.2022058","DOIUrl":"https://doi.org/10.3934/eect.2022058","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"103 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74639900","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 work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration.
{"title":"On the approximate boundary controllability of some partial functional integrodifferential equations with finite delay in Banach spaces","authors":"P. Ndambomve, Shuqin Che","doi":"10.3934/eect.2022050","DOIUrl":"https://doi.org/10.3934/eect.2022050","url":null,"abstract":"This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"12 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76882990","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 work addresses the optimal birth control problem for invasive species in a spatial environment. We apply the method of semigroups to qualitatively analyze a size-structured population model in which individuals occupy a position in a spatial environment. With insect population in mind, we study the optimal control problem which takes fertility rate as a control variable. With the help of adjoint system, we derive optimality conditions. We obtain the optimality conditions by fixing the birth rate on three different sets. Using Ekeland's variational principle, the existence, and uniqueness of optimal birth controller to the given population model which minimizes a given cost functional is shown. A concrete example is also given to see the behaviour of population density. Outcomes of our article are new and complement the existing ones.
{"title":"Analysis of diffusive size-structured population model and optimal birth control","authors":"Manoj Kumar, Syed Abbas, R. Sakthivel","doi":"10.3934/eect.2022036","DOIUrl":"https://doi.org/10.3934/eect.2022036","url":null,"abstract":"This work addresses the optimal birth control problem for invasive species in a spatial environment. We apply the method of semigroups to qualitatively analyze a size-structured population model in which individuals occupy a position in a spatial environment. With insect population in mind, we study the optimal control problem which takes fertility rate as a control variable. With the help of adjoint system, we derive optimality conditions. We obtain the optimality conditions by fixing the birth rate on three different sets. Using Ekeland's variational principle, the existence, and uniqueness of optimal birth controller to the given population model which minimizes a given cost functional is shown. A concrete example is also given to see the behaviour of population density. Outcomes of our article are new and complement the existing ones.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"88 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73987647","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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