Evolution Equations and Control Theory最新文献

{"title":"The Heavy ball method regularized by Tikhonov term. Simultaneous convergence of values and trajectories","authors":"Akram Chahid Bagy, Z. Chbani, H. Riahi","doi":"10.3934/eect.2022046","DOIUrl":"https://doi.org/10.3934/eect.2022046","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ f: {mathcal H} rightarrow mathbb{R} $end{document}</tex-math></inline-formula> be a convex differentiable function whose solution set <inline-formula><tex-math id=\"M2\">begin{document}$ {{rm{argmin}}}; f $end{document}</tex-math></inline-formula> is nonempty. To attain a solution of the problem <inline-formula><tex-math id=\"M3\">begin{document}$ min_{mathcal H}f $end{document}</tex-math></inline-formula>, we consider the second order dynamic system <inline-formula><tex-math id=\"M4\">begin{document}$ ;ddot{x}(t) + alpha , dot{x}(t) + beta (t) nabla f(x(t)) + c x(t) = 0 $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M5\">begin{document}$ beta $end{document}</tex-math></inline-formula> is a positive function such that <inline-formula><tex-math id=\"M6\">begin{document}$ lim_{trightarrow +infty}beta(t) = +infty $end{document}</tex-math></inline-formula>. By imposing adequate hypothesis on first and second order derivatives of <inline-formula><tex-math id=\"M7\">begin{document}$ beta $end{document}</tex-math></inline-formula>, we simultaneously prove that the value of the objective function in a generated trajectory converges in order <inline-formula><tex-math id=\"M8\">begin{document}$ {mathcal O}big(frac{1}{beta(t)}big) $end{document}</tex-math></inline-formula> to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of <inline-formula><tex-math id=\"M9\">begin{document}$ {{rm{argmin}}}; f $end{document}</tex-math></inline-formula> and that <inline-formula><tex-math id=\"M10\">begin{document}$ Vert dot{x}(t)Vert $end{document}</tex-math></inline-formula> converges to zero in order <inline-formula><tex-math id=\"M11\">begin{document}$ mathcal{O} big( sqrt{frac{dot{beta}(t)}{beta (t)}}+ e^{-mu t} big) $end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\"M12\">begin{document}$ mu<frac{alpha}2 $end{document}</tex-math></inline-formula>. We then present two choices of <inline-formula><tex-math id=\"M13\">begin{document}$ beta $end{document}</tex-math></inline-formula> to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87095812","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":"Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement","authors":"M. Alahyane, M. Al‐Gharabli, Adel M. Al-Mahdi","doi":"10.3934/eect.2022027","DOIUrl":"https://doi.org/10.3934/eect.2022027","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [<xref ref-type=\"bibr\" rid=\"b27\">27</xref>], [<xref ref-type=\"bibr\" rid=\"b4\">4</xref>], [<xref ref-type=\"bibr\" rid=\"b33\">33</xref>], [<xref ref-type=\"bibr\" rid=\"b58\">58</xref>] and [<xref ref-type=\"bibr\" rid=\"b34\">34</xref>]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78301442","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":"Stability properties for a problem of light scattering in a dispersive metallic domain","authors":"S. Nicaise, C. Scheid","doi":"10.3934/eect.2022020","DOIUrl":"https://doi.org/10.3934/eect.2022020","url":null,"abstract":"In this work, we study the well-posedness and some stability properties of a PDE system that models the propagation of light in a metallic domain with a hole. This model takes into account the dispersive properties of the metal. It consists of a linear coupling between Maxwell's equations and a wave type system. We prove that the problem is well posed for several types of boundary conditions. Furthermore, we show that it is polynomially stable and that the exponential stability is conditional on the exponential stability of the Maxwell system.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76183014","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":"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":null,"pages":null},"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}

{"title":"On the exact controllability for the Benney-Luke equation in a bounded domain","authors":"Jose R. Quintero","doi":"10.3934/eect.2022052","DOIUrl":"https://doi.org/10.3934/eect.2022052","url":null,"abstract":"","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72855362","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":"Exact controllability of semilinear heat equations through a constructive approach","authors":"S. Ervedoza, Jérôme Lemoine, A. Münch","doi":"10.3934/eect.2022042","DOIUrl":"https://doi.org/10.3934/eect.2022042","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 + f(y) = v , 1_{omega} $end{document}</tex-math></inline-formula> posed over multi-dimensional and bounded domains, assuming that <inline-formula><tex-math id=\"M2\">begin{document}$ f $end{document}</tex-math></inline-formula> is locally Lipschitz continuous and satisfies the growth condition <inline-formula><tex-math id=\"M3\">begin{document}$ limsup_{| r|to infty} | f(r)| /(| r| ln^{3/2}| r|)leq beta $end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id=\"M4\">begin{document}$ beta $end{document}</tex-math></inline-formula> small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that <inline-formula><tex-math id=\"M5\">begin{document}$ f $end{document}</tex-math></inline-formula> is locally Lipschitz continuous and satisfies the growth condition <inline-formula><tex-math id=\"M6\">begin{document}$ limsup_{| r|to infty} | f^prime(r)|/ln^{3/2}| r|leq beta $end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id=\"M7\">begin{document}$ beta $end{document}</tex-math></inline-formula> small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80071459","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":"Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle","authors":"Arzu Ahmadova, N. Mahmudov, J. Nieto","doi":"10.3934/eect.2022008","DOIUrl":"https://doi.org/10.3934/eect.2022008","url":null,"abstract":"In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83857875","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":"Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control","authors":"Zhiling Guo, Shugen Chai","doi":"10.3934/eect.2022001","DOIUrl":"https://doi.org/10.3934/eect.2022001","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we address exponential stabilization of transmission problem of the wave equation with linear dynamical feedback control. Using the classical energy method and multiplier technique, we prove that the energy of system exponentially decays.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82415458","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":"Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources","authors":"T. Tuan","doi":"10.3934/eect.2022032","DOIUrl":"https://doi.org/10.3934/eect.2022032","url":null,"abstract":"In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90287504","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":"Mathematical analysis of an abstract model and its applications to structured populations (I)","authors":"M. Boulanouar","doi":"10.3934/eect.2022021","DOIUrl":"https://doi.org/10.3934/eect.2022021","url":null,"abstract":"The first part of this works deals with an integro–differential operator with boundary condition related to the interior solution. We prove that the model is governed by a strongly continuous semigroup and we precise its growth inequality. In the second part of this works, we model the proliferation-quiescence phases through a system of first order equations. We also prove that the proliferation-quiescence model is governed by a strongly continuous semigroup and we precise its growth inequality. In the last part, we give some applications in Demography and Biology.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76992902","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}