{"title":"Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour","authors":"J. Banasiak, Adam Bloch","doi":"10.3934/eect.2022016","DOIUrl":"https://doi.org/10.3934/eect.2022016","url":null,"abstract":"In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boundary matrix.","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"173 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134462190","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}
{"title":"Controllability for Schrödinger type system with mixed dispersion on compact star graphs","authors":"R. A. Capistrano-Filho, M. Cavalcante, F. Gallego","doi":"10.3934/eect.2022019","DOIUrl":"https://doi.org/10.3934/eect.2022019","url":null,"abstract":"<p style='text-indent:20px;'>In this work we are concerned with solutions to the linear Schrödinger type system with mixed dispersion, the so-called biharmonic Schrödinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure <inline-formula><tex-math id=\"M1\">begin{document}$ mathcal{G} $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M2\">begin{document}$ T>T_{min} $end{document}</tex-math></inline-formula>, with</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ T_{min} = sqrt{ frac{ overline{L} (L^2+pi^2)}{pi^2varepsilon(1- overline{L} varepsilon)}}, $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the couplings and the controls appear only on the Neumann boundary conditions.</p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127403236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data","authors":"J. Djida, G. Mophou, M. Warma","doi":"10.3934/eect.2022015","DOIUrl":"https://doi.org/10.3934/eect.2022015","url":null,"abstract":"<p style='text-indent:20px;'>We consider parabolic equations on bounded smooth open sets <inline-formula><tex-math id=\"M1\">begin{document}$ {Omega}subset mathbb{R}^N $end{document}</tex-math></inline-formula> (<inline-formula><tex-math id=\"M2\">begin{document}$ Nge 1 $end{document}</tex-math></inline-formula>) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator <inline-formula><tex-math id=\"M3\">begin{document}$ mathscr{L} : = - Delta + (-Delta)^{s} $end{document}</tex-math></inline-formula> (<inline-formula><tex-math id=\"M4\">begin{document}$ 0<s<1 $end{document}</tex-math></inline-formula>). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.</p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123691716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production","authors":"Runlin Hu, Pan Zheng, Zhangqin Gao","doi":"10.3934/eect.2022018","DOIUrl":"https://doi.org/10.3934/eect.2022018","url":null,"abstract":"<p style='text-indent:20px;'>This paper deals with a quasilinear parabolic-elliptic chemo-repulsion system with nonlinear signal production</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{eqnarray*} label{1a} left{ begin{split}{} & u_t = nablacdot(phi(u)nabla u)+chinablacdot(u(u+1)^{alpha-1}nabla v)+f(u), & (x,t)in Omegatimes (0,infty), & 0 = Delta v-v+u^{beta}, & (x,t)in Omegatimes (0,infty), end{split} right. end{eqnarray*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id=\"M1\">begin{document}$ Omega subset mathbb{R}^{n}(ngeq1), $end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=\"M2\">begin{document}$ chi,beta>0,alphainmathbb{R}, $end{document}</tex-math></inline-formula> the nonlinear diffusion <inline-formula><tex-math id=\"M3\">begin{document}$ phiin C^{2}([0,infty)) $end{document}</tex-math></inline-formula> satisfies <inline-formula><tex-math id=\"M4\">begin{document}$ phi(u)geq(u+1)^{m} $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M5\">begin{document}$ minmathbb{R}, $end{document}</tex-math></inline-formula> and the function <inline-formula><tex-math id=\"M6\">begin{document}$ fin C^{1}([0,infty)) $end{document}</tex-math></inline-formula> is a generalized growth term.</p><p style='text-indent:20px;'><inline-formula><tex-math id=\"M7\">begin{document}$ bullet $end{document}</tex-math></inline-formula> When <inline-formula><tex-math id=\"M8\">begin{document}$ fequiv0, $end{document}</tex-math></inline-formula> it is shown that the solution of the above system is global and uniformly bounded for all <inline-formula><tex-math id=\"M9\">begin{document}$ chi,beta>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M10\">begin{document}$ m,alphainmathbb{R} $end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'><inline-formula><tex-math id=\"M11\">begin{document}$ bullet $end{document}</tex-math></inline-formula> When <inline-formula><tex-math id=\"M12\">begin{document}$ fnotequiv0 $end{document}</tex-math></inline-formula> and assume that <inline-formula><tex-math id=\"M13\">begin{document}$ f(u)leq ku-bu^{gamma+1} $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M14\">begin{document}$ k,b,gamma>0, $end{document}</tex-math></inline-formula> it is proved that the solution of the above system is also global and uniformly bounded for all <inline-formula><tex-math id=\"M15\">begin{document}$ chi,beta>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M16\">begin{document}$ m,alphainmathbb{R}. $end{document}</tex-math></inline-formula></p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"6 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":"123039453","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}
{"title":"Pullback attractors for weak solution to modified Kelvin-Voigt model","authors":"M. Turbin, A. Ustiuzhaninova","doi":"10.3934/eect.2022011","DOIUrl":"https://doi.org/10.3934/eect.2022011","url":null,"abstract":"The paper is devoted to the investigation of the qualitative dynamics of weak solutions for the modified Kelvin-Voigt model on the base of the theory of pullback attractors for trajectory spaces. At first, for the studied model, an auxiliary problem is considered, its solvability in the weak sense is proved, and some solution estimates are established. Then, on the base of these estimates, a family of trajectory spaces is determined and the existence of trajectory and minimal pullback attractors for the considered trajectory spaces is proved.","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"40 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":"132451534","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}
{"title":"Attractors for a class of extensible beams with strong nonlinear damping","authors":"E. Tavares, V. Narciso","doi":"10.3934/eect.2022013","DOIUrl":"https://doi.org/10.3934/eect.2022013","url":null,"abstract":"<p style='text-indent:20px;'>We concern to stablish the existence and qualitative properties of the compact global attractor associate to solutions of a class of extensible beam equations with strong nonlinear damping arising from the wave model proposed by Prestel [<xref ref-type=\"bibr\" rid=\"b18\">18</xref>].</p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"14 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":"131907977","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}
{"title":"Blow-up of solutions to a viscoelastic wave equation with nonlocal damping","authors":"Donghao Li, Hongwei Zhang, Shuo Liu, Qing-Quan Hu","doi":"10.3934/eect.2022009","DOIUrl":"https://doi.org/10.3934/eect.2022009","url":null,"abstract":"<p style='text-indent:20px;'>The viscoelastic wave equation with nonlinear nonlocal weak damping is considered. The local existence of solutions is established. Under arbitrary positive initial energy, a finite-time blow-up result is proved by a new modified concavity method.</p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"77 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":"122927944","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}
{"title":"On analytic semigroup generators involving Caputo fractional derivative","authors":"P. Grabowski","doi":"10.3934/eect.2022014","DOIUrl":"https://doi.org/10.3934/eect.2022014","url":null,"abstract":"<p style='text-indent:20px;'>Our investigations are motivated by the well - posedness problem of some dynamical models with anomalous diffusion described by the Caputo spatial fractional derivative of order <inline-formula><tex-math id=\"M1\">begin{document}$ alpha in (1, 2) $end{document}</tex-math></inline-formula>. We propose a characterization of an exponentially stable analytic semigroup generator using the inverse operator. This characterization enables us to establish the form of a generator involving the Caputo fractional derivative, under various boundary conditions. In particular, the results simplify those known from literature obtained by means of the fractional Sobolev spaces and some perturbation results. Going further, we show how to construct a control system in factor form, having such a generator as the state operator.</p>","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","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":"115256556","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}
{"title":"Well-posedness and stability of non-autonomous semilinear input-output systems","authors":"J. Schmid","doi":"10.3934/eect.2022017","DOIUrl":"https://doi.org/10.3934/eect.2022017","url":null,"abstract":"We establish well-posedness results for non-autonomous semilinear input-output systems, the central assumption being the scattering-passivity of the considered semilinear system. Along the way, we also establish global stability estimates. We consider both systems with distributed control and observation and systems with boundary control and observation, and we treat them in a unified manner. Applications are given to nonlinearly controlled collocated systems and to nonlinearly controlled port-Hamiltonian systems.","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"26 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":"133806639","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}
{"title":"Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations","authors":"Li Song, Yangrong Li, Fengling Wang","doi":"10.3934/eect.2022010","DOIUrl":"https://doi.org/10.3934/eect.2022010","url":null,"abstract":"A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.","PeriodicalId":176362,"journal":{"name":"Evolution Equations & Control Theory","volume":"17 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":"121387702","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}