Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 7
Number of page(s) 46
DOI https://doi.org/10.1051/cocv/2020091
Published online 03 March 2021
  1. F. Ali Mehmeti Regular solutions of transmission and interaction problems for wave equations. Math. Meth. Appl. Sci. 11 (1989) 665–685. [Google Scholar]
  2. F. Ali Mehmeti Nonlinear Waves in Networks, Mathematical Research, vol. 80. Akademie Verlag, Berlin (1994). [Google Scholar]
  3. K. Ammari and M. Mehrenberger, Study of the nodal feedback stabilization of a string–beams network. J. Appl. Math. Comput. 36 (2011) 441–458. [Google Scholar]
  4. K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback. Vol. 2124 of Lecture Notes in Mathematics. Springer, Cham (2015). [Google Scholar]
  5. K. Ammari, D. Mercier, V. Régnier and J. Valein, Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Comm. Pure Appl. Anal. 11 (2012) 785–807. [Google Scholar]
  6. G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Vol. 88 of Progress in Nonlinear Differential Equations. Birkhäuser, Basel (2016). [Google Scholar]
  7. A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups. Vol. 257 of Operator Theory: Advances and Applications. Birkhäuser, Cham (2017). [Google Scholar]
  8. S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations – First-order Systems and Applications. Clarendon Press, Oxford (2007). [Google Scholar]
  9. J. Bolte and J. Harrison, Spectral statistics for the Dirac operator on graphs, J. Phys. A 36 (2003) 2747–2769. [Google Scholar]
  10. J. Bolte and H.-M. Stiepan, The Selberg trace formula for Dirac operators. J. Math. Phys. 47 (2007) 112104. [Google Scholar]
  11. A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478. [Google Scholar]
  12. A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111. [Google Scholar]
  13. A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111. [Google Scholar]
  14. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer-Verlag, Berlin (2010). [Google Scholar]
  15. S. Cardanobile and D. Mugnolo, Parabolic systems with coupled boundary conditions. J. Differ. Equ. 247 (2009) 1229–1248. [Google Scholar]
  16. R. Carlson, Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc. 351 (1999) 4069–4088. [Google Scholar]
  17. R. Carlson, Nonclassical Sturm–Liouville problems and Schrödinger operators on radial trees. Electr. J. Differ. Equ. 71 (2000) 1–24. [Google Scholar]
  18. R. Carlson, Spectral theory for nonconservative transmission line networks. Netw. Heterog. Media 6 (2011) 257–277. [Google Scholar]
  19. G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58 (1979) 249–273. [Google Scholar]
  20. R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures. Vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006). [Google Scholar]
  21. B. Dorn, Semigroups for flows in infinite networks. Semigroup Forum 76 (2008) 341–356. [Google Scholar]
  22. B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks. Physica D 239 (2010) 1416–1421. [Google Scholar]
  23. S. Endres and F. Steiner, The Berry–Keating operator on L2(ℝ>, dx) and on compact quantum graphs with general self-adjoint realizations. J. Phys. A 43 (2010) 095204. [Google Scholar]
  24. K.-J. Engel, Generator property and stability for generalized difference operators. J. Evol. Equ. 13 (2013) 311–334. [Google Scholar]
  25. K.-J. Engel and M. Kramar Fijavž Exact and positive controllability of boundary control systems. Netw. Heterog. Media 12 (2017) 319–337. [Google Scholar]
  26. K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709–722. [Google Scholar]
  27. K.-J. Engel, M. Kramar Fijavž, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems. Appl. Math. Optim. 62 (2010) 205–227. [Google Scholar]
  28. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Vol. 194 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000). [Google Scholar]
  29. L. Evans, Partial differential Equations – second edition. Vol. 19 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI (2010). [Google Scholar]
  30. P. Exner, Momentum operators on graphs. In H. Holden, B. Simon, and G. Teschl, editors, Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday. Vol. 87 of Proc. Symp. Pure Math. Amer. Math. Soc., Providence, RI (2013) 105–118. [Google Scholar]
  31. S. Huberman, R. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. Maznev, G. Chen and K. Nelson, Observation of second sound in graphite at temperatures above 100 k. Science 364 (2019) 375–379. [Google Scholar]
  32. A. Hussein and D. Mugnolo, Quantum graphs with mixed dynamics: the transport/diffusion case. J. Phys. A 46 (2013) 235202. [Google Scholar]
  33. S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section. Appl. Num. Math. 79 (2014) 42–61. [Google Scholar]
  34. B. Jacob, K. Morris and H. Zwart, C0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. J. Evol. Equ. 15 (2015) 493–502. [Google Scholar]
  35. B. Jacoband H. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces. Vol. 223 of Oper. Theory Adv. Appl. Birkhäuser, Basel (2012). [Google Scholar]
  36. P. Jorgensen, S. Pedersen and F. Tiang, Momentum operators in two intervals: Spectra and phase transition. Compl. Anal. Oper. Theory 7 (2013) 1735–1773. [Google Scholar]
  37. B. Klöss, Difference operators as semigroup generators. Semigroup Forum 81 (2010) 461–482. [Google Scholar]
  38. B. Klöss, The flow approach for waves in networks. Oper. Matrices 6 (2012) 107–128. [Google Scholar]
  39. V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires. J. Phys. A 32 (1999) 595–630. [Google Scholar]
  40. M. Kramar Fijavž and A. Puchalska, Semigroups for dynamical processes on metric graphs. Phil. Trans. R. Soc. A 378 (2020) 20190619. [Google Scholar]
  41. M. Kramar Fijavž, D. Mugnolo and S. Nicaise, Hyperbolic systems with dynamic boundary conditions. Inpreparation (2020). [Google Scholar]
  42. M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249 (2005) 139–162. [Google Scholar]
  43. P. Kuchment, Graph models of wave propagation in thin structures. Waves Random Media 12 (2002) 1–24. [Google Scholar]
  44. P. Kurasov, D. Mugnolo and V. Wolf, Analytic solutions for stochastic hybrid models of gene regulatory networks. Preprint arXiv:1812.07788 (2021). [Google Scholar]
  45. J. Lagnese, G. Leugering, and E. Schmidt, Modeling, Analysis, and Control of dynamic Elastic Multi-Link Structures, Systems and Control: Foundations and Applications. Birkhäuser, Basel (1994). [Google Scholar]
  46. T.Y. Lam, Introduction to quadratic forms over fields. Vol. 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2005). [Google Scholar]
  47. P. Lax and R. Phillips, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13 (1960) 427–455. [Google Scholar]
  48. G. Leugering and E. Schmidt, On the control of networks of vibrating strings and beams. In IEEE Conference on Decision and Control, IEEE, Providence, RI (1989) 2287–2290. [Google Scholar]
  49. H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15 (1967) 299–309. [Google Scholar]
  50. G. Lumer, Connecting of local operators and evolution equations on networks. In Potential Theory (Proc. Copenhagen 1979), edited by F. Hirsch. Springer-Verlag, Berlin (1980) 230–243. [Google Scholar]
  51. A. Maffucci and G. Miano, A unified approach for the analysis of networks composed of transmission lines and lumped circuits. In Scientific computing in electrical engineering. Vol. 9 of Mathematics in industry. Springer-Verlag, Berlin (2006) 3–11. [Google Scholar]
  52. T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks. Forum Math. 19 (2007) 429–461. [Google Scholar]
  53. J. Milnor and D. Husemoller, Symmetric bilinear forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Springer-Verlag, New York-Heidelberg (1973). [Google Scholar]
  54. D. Mugnolo, Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer-Verlag, Berlin (2014). [Google Scholar]
  55. S. Nicaise, Polygonal Interface Problems. Vol. 39 of Methoden und Verfahren der mathematischen Physik. Peter Lang GmbH, Europäischer Verlag der Wissenschaften, Frankfurt/M (1993). [Google Scholar]
  56. S. Nicaise, Control and stabilization of 2 × 2 hyperbolic systems on graphs. Math. Control Relat. Fields 7 (2017) 53–72. [Google Scholar]
  57. S. Nicaise and J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks Het. Media 3 (2007) 425–479. [Google Scholar]
  58. E. Ouhabaz, Analysis of Heat Equations on domains. Vol. 30 of Lond. Math. Soc. Monograph Series. Princeton Univ. Press, Princeton, NJ (2005). [Google Scholar]
  59. B. Pavlov and M.D. Faddeev, Model of free electrons and the scattering problem. Theor. Math. Phys. 55 (1983) 485–492. [Google Scholar]
  60. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). [Google Scholar]
  61. J. Prüss, On the spectrum of C0-semigroups. Trans. Am. Math. Soc. 284 (1984) 847–857. [Google Scholar]
  62. R. Racke, Thermoelasticity with second sound – exponential stability in linear and nonlinear 1D. Math. Methods Appl. Sci. 25 (2002) 409–441. [Google Scholar]
  63. J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Am. Math. Soc. 291 (1985) 167–187. [Google Scholar]
  64. J. Rauch, Hyperbolic partial differential equations and geometric optics. Vol. 133 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI (2012). [Google Scholar]
  65. C. Schubert, C. Seifert, J. Voigt and M. Waurick, Boundary systems and (skew-) self-adjoint operators on infinite metric graphs. Math. Nachr. 288 (2015) 1776–1785. [Google Scholar]
  66. B. Thaller, The Dirac Equation. Springer-Verlag, New York (1992). [Google Scholar]
  67. M. Waurick and S.-A. Wegner, Dissipative extensions and port-hamiltonian operators onnetwork (2019). [Google Scholar]
  68. T. Yokota, Invariance of closed convex sets under semigroups of nonlinear operators in complex Hilbert spaces. SUT J. Math. 37 (2001) 91–104. [Google Scholar]
  69. H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16 (2010) 1077–1093. [EDP Sciences] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.