doi: 10.3934/mcrf.2021057
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Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations

Université Polytechnique Hauts-de-France, CÉRAMATHS/DEMAV and FR CNRS 2037, F-59313 - Valenciennes Cedex 9, France

* Corresponding author: Serge Nicaise

Received  May 2021 Revised  October 2021 Early access December 2021

Fund Project: This work was supported by the project ISDEEC ANR-16-CE40-0013, and by the COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology)

In this paper, we obtain some stability results of systems corresponding to the coupling between a dissipative evolution equation (set in an infinite dimensional space) and an ordinary differential equation. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of ODE-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators that garantee stability properties of the system, i.e., strong stability, exponential stability or polynomial one. We also illustrate our abstract statements for different concrete examples, where new results are achieved.

Citation: Serge Nicaise. Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021057
References:
[1]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[2]

F. Ali Mehmeti, A characterisation of generalized $c^{\infty}$ notion on nets, Integral Equations Operator Theory, 9 (1986), 753-766.  doi: 10.1007/BF01202515.

[3]

F. Ali Mehmeti, Nonlinear Waves in Networks, volume 80 of Math. Res., Akademie Verlag, 1994.

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[6]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[7]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

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A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[10]

A. Bressan, Hyperbolic conservation laws: An illustrated tutorial, In Modelling and Optimisation of Flows on Networks, volume 2062 of Lecture Notes in Math., pages 157–245. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32160-3_2.

[11]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

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S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), 1161-1186.  doi: 10.1002/mma.407.

[14]

R. Carlson, Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.  doi: 10.3934/nhm.2011.6.257.

[15]

M. CassierP. Joly and M. Kachanovska, Mathematical models for dispersive electromagnetic waves: An overview, Comput. Math. Appl., 74 (2017), 2792-2830.  doi: 10.1016/j.camwa.2017.07.025.

[16]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

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C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

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C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

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A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[20]

H. Engler, On some parabolic integro-differential equations: Existence and asymptotics of solutions, In Equadiff 82 (Würzburg, 1982), volume 1017 of Lecture Notes in Math., pages 161–167. Springer, Berlin, 1983. doi: 10.1007/BFb0103248.

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M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity, 19 (1988), 63-75.  doi: 10.1007/BF00041695.

[22]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[23]

M. K. Fijavž, D. Mugnolo and S. Nicaise, Linear hyperbolic systems on networks: Well-posedness and qualitative properties, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 7, 46 pp. doi: 10.1051/cocv/2020091.

[24]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, volume 5 of Springer Series in Computational Mathematics, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[25]

M. GugatM. Dick and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.  doi: 10.1137/100799824.

[26]

A. HayekS. NicaiseZ. Salloum and A. Wehbe, Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped networks, Acta Appl. Math., 170 (2020), 823-851.  doi: 10.1007/s10440-020-00360-8.

[27]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[28]

S. Imperiale and P. Joly, Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section, Adv. Appl. Math. Mech., 4 (2012), 647-664.  doi: 10.4208/aamm.12-12S06.

[29]

S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, Appl. Numer. Math., 79 (2014), 42-61.  doi: 10.1016/j.apnum.2013.03.011.

[30]

M. Krstic, Compensating a string PDE in the actuation or sensing path of an unstable ODE, IEEE Trans. Automat. Control, 54 (2009), 1362-1368.  doi: 10.1109/TAC.2009.2015557.

[31]

S. Labrunie and I. Zaafrani, Linearised electrodynamics and stabilisation of a cold magnetised plasma, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 60, 46 pp. doi: 10.1051/cocv/2021056.

[32]

J. Li, Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell's equations in dispersive media, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3081-3094.  doi: 10.1016/j.cma.2006.12.009.

[33]

J. Li, Error analysis of mixed finite element methods for wave propagation in double negative metamaterials, J. Comput. Appl. Math., 209 (2007), 81-96.  doi: 10.1016/j.cam.2006.10.031.

[34]

J. Li, Unified analysis of leap-frog methods for solving time-domain Maxwell's equations in dispersive media, J. Sci. Comput., 47 (2011), 1-26.  doi: 10.1007/s10915-010-9417-7.

[35]

L. LiX. Zhou and H. Gao, The stability and exponential stabilization of the heat equation with memory, J. Math. Anal. Appl., 466 (2018), 199-214.  doi: 10.1016/j.jmaa.2018.05.078.

[36]

J.-J. Liu and J.-M. Wang, Boundary stabilization of a cascade of ODE-wave systems subject to boundary control matched disturbance, Internat. J. Robust Nonlinear Control, 27 (2017), 252-280.  doi: 10.1002/rnc.3572.

[37]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[38]

G. Lumer, Connecting of local operators and evolution equations on networks, In Potential Theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), volume 787 of Lecture Notes in Math., pages 219–234. Springer, Berlin, 1980.

[39]

Yu. I. Lyubich and Q. P. V u, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[40]

M. M. Martínez, Decay estimates of functions through singular extensions of vector-valued Laplace transforms, J. Math. Anal. Appl., 375 (2011), 196-206.  doi: 10.1016/j.jmaa.2010.08.077.

[41]

S. Nicaise, Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413. 

[42]

S. Nicaise, Exact boundary controllability of Maxwell's equations in heterogeneous media and an application to an inverse source problem, SIAM J. Control Optim., 38 (2000), 1145-1170.  doi: 10.1137/S0363012998344373.

[43]

S. Nicaise, Stabilization and asymptotic behavior of dispersive medium models, Systems Control Lett., 61 (2012), 638-648.  doi: 10.1016/j.sysconle.2012.03.001.

[44]

S. Nicaise, Stabilization and asymptotic behavior of a generalized telegraph equation, Z. Angew. Math. Phys., 66 (2015), 3221-3247.  doi: 10.1007/s00033-015-0568-0.

[45]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.

[46]

S. Nicaise and C. Pignotti, Asymptotic behavior of dispersive electromagnetic waves in bounded domains, Z. Angew. Math. Phys., 71 (2020), Paper No. 76, 26 pp. doi: 10.1007/s00033-020-01297-6.

[47]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[48]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[49]

J. RozendaalD. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007.

[50]

K. SakthivelK. Balachandran and B. R. Nagaraj, On a class of non-linear parabolic control systems with memory effects, Internat. J. Control, 81 (2008), 764-777.  doi: 10.1080/00207170701447114.

[51]

S. J. SherwinV. FrankeJ. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.  doi: 10.1023/B:ENGI.0000007979.32871.e2.

[52]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Linear Algebra Appl., 71 (1985), 309-325.  doi: 10.1016/0024-3795(85)90258-7.

[53]

J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.

[54]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[55]

J. L. Young, Propagation in linear dispersive media: Finite difference time-domain methodologies, IEEE Trans. Antennas Propag., 43 (1995), 422-426. 

[56]

R. W. Ziolkowski, Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs, Opt. Exp., 11 (2003), 662-681.  doi: 10.1364/OE.11.000662.

[57]

R. W. Ziolkowski and E. Heyman, Wave propagation in media having negative permittivity and permeability, Phys. Rev. E, 64 (2001), 056625.  doi: 10.1103/PhysRevE.64.056625.

show all references

References:
[1]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[2]

F. Ali Mehmeti, A characterisation of generalized $c^{\infty}$ notion on nets, Integral Equations Operator Theory, 9 (1986), 753-766.  doi: 10.1007/BF01202515.

[3]

F. Ali Mehmeti, Nonlinear Waves in Networks, volume 80 of Math. Res., Akademie Verlag, 1994.

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[6]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[7]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

[9]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[10]

A. Bressan, Hyperbolic conservation laws: An illustrated tutorial, In Modelling and Optimisation of Flows on Networks, volume 2062 of Lecture Notes in Math., pages 157–245. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32160-3_2.

[11]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[13]

S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), 1161-1186.  doi: 10.1002/mma.407.

[14]

R. Carlson, Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.  doi: 10.3934/nhm.2011.6.257.

[15]

M. CassierP. Joly and M. Kachanovska, Mathematical models for dispersive electromagnetic waves: An overview, Comput. Math. Appl., 74 (2017), 2792-2830.  doi: 10.1016/j.camwa.2017.07.025.

[16]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[17]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4.

[18]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[19]

A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[20]

H. Engler, On some parabolic integro-differential equations: Existence and asymptotics of solutions, In Equadiff 82 (Würzburg, 1982), volume 1017 of Lecture Notes in Math., pages 161–167. Springer, Berlin, 1983. doi: 10.1007/BFb0103248.

[21]

M. Fabrizio and A. Morro, Viscoelastic relaxation functions compatible with thermodynamics, J. Elasticity, 19 (1988), 63-75.  doi: 10.1007/BF00041695.

[22]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[23]

M. K. Fijavž, D. Mugnolo and S. Nicaise, Linear hyperbolic systems on networks: Well-posedness and qualitative properties, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 7, 46 pp. doi: 10.1051/cocv/2020091.

[24]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, volume 5 of Springer Series in Computational Mathematics, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[25]

M. GugatM. Dick and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.  doi: 10.1137/100799824.

[26]

A. HayekS. NicaiseZ. Salloum and A. Wehbe, Existence, uniqueness and stabilization of solutions of a generalized telegraph equation on star shaped networks, Acta Appl. Math., 170 (2020), 823-851.  doi: 10.1007/s10440-020-00360-8.

[27]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[28]

S. Imperiale and P. Joly, Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section, Adv. Appl. Math. Mech., 4 (2012), 647-664.  doi: 10.4208/aamm.12-12S06.

[29]

S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, Appl. Numer. Math., 79 (2014), 42-61.  doi: 10.1016/j.apnum.2013.03.011.

[30]

M. Krstic, Compensating a string PDE in the actuation or sensing path of an unstable ODE, IEEE Trans. Automat. Control, 54 (2009), 1362-1368.  doi: 10.1109/TAC.2009.2015557.

[31]

S. Labrunie and I. Zaafrani, Linearised electrodynamics and stabilisation of a cold magnetised plasma, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 60, 46 pp. doi: 10.1051/cocv/2021056.

[32]

J. Li, Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell's equations in dispersive media, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3081-3094.  doi: 10.1016/j.cma.2006.12.009.

[33]

J. Li, Error analysis of mixed finite element methods for wave propagation in double negative metamaterials, J. Comput. Appl. Math., 209 (2007), 81-96.  doi: 10.1016/j.cam.2006.10.031.

[34]

J. Li, Unified analysis of leap-frog methods for solving time-domain Maxwell's equations in dispersive media, J. Sci. Comput., 47 (2011), 1-26.  doi: 10.1007/s10915-010-9417-7.

[35]

L. LiX. Zhou and H. Gao, The stability and exponential stabilization of the heat equation with memory, J. Math. Anal. Appl., 466 (2018), 199-214.  doi: 10.1016/j.jmaa.2018.05.078.

[36]

J.-J. Liu and J.-M. Wang, Boundary stabilization of a cascade of ODE-wave systems subject to boundary control matched disturbance, Internat. J. Robust Nonlinear Control, 27 (2017), 252-280.  doi: 10.1002/rnc.3572.

[37]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[38]

G. Lumer, Connecting of local operators and evolution equations on networks, In Potential Theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), volume 787 of Lecture Notes in Math., pages 219–234. Springer, Berlin, 1980.

[39]

Yu. I. Lyubich and Q. P. V u, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[40]

M. M. Martínez, Decay estimates of functions through singular extensions of vector-valued Laplace transforms, J. Math. Anal. Appl., 375 (2011), 196-206.  doi: 10.1016/j.jmaa.2010.08.077.

[41]

S. Nicaise, Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413. 

[42]

S. Nicaise, Exact boundary controllability of Maxwell's equations in heterogeneous media and an application to an inverse source problem, SIAM J. Control Optim., 38 (2000), 1145-1170.  doi: 10.1137/S0363012998344373.

[43]

S. Nicaise, Stabilization and asymptotic behavior of dispersive medium models, Systems Control Lett., 61 (2012), 638-648.  doi: 10.1016/j.sysconle.2012.03.001.

[44]

S. Nicaise, Stabilization and asymptotic behavior of a generalized telegraph equation, Z. Angew. Math. Phys., 66 (2015), 3221-3247.  doi: 10.1007/s00033-015-0568-0.

[45]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.

[46]

S. Nicaise and C. Pignotti, Asymptotic behavior of dispersive electromagnetic waves in bounded domains, Z. Angew. Math. Phys., 71 (2020), Paper No. 76, 26 pp. doi: 10.1007/s00033-020-01297-6.

[47]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[48]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[49]

J. RozendaalD. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007.

[50]

K. SakthivelK. Balachandran and B. R. Nagaraj, On a class of non-linear parabolic control systems with memory effects, Internat. J. Control, 81 (2008), 764-777.  doi: 10.1080/00207170701447114.

[51]

S. J. SherwinV. FrankeJ. Peiró and K. Parker, One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.  doi: 10.1023/B:ENGI.0000007979.32871.e2.

[52]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Linear Algebra Appl., 71 (1985), 309-325.  doi: 10.1016/0024-3795(85)90258-7.

[53]

J. von Below, Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.  doi: 10.1016/0022-0396(88)90158-1.

[54]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[55]

J. L. Young, Propagation in linear dispersive media: Finite difference time-domain methodologies, IEEE Trans. Antennas Propag., 43 (1995), 422-426. 

[56]

R. W. Ziolkowski, Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs, Opt. Exp., 11 (2003), 662-681.  doi: 10.1364/OE.11.000662.

[57]

R. W. Ziolkowski and E. Heyman, Wave propagation in media having negative permittivity and permeability, Phys. Rev. E, 64 (2001), 056625.  doi: 10.1103/PhysRevE.64.056625.

Figure 1.  The cascade of the ODE-wave system
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