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 & Related Fields, doi: 10.3934/mcrf.2021057
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

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Figure 1.  The cascade of the ODE-wave system
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