December  2019, 9(4): 759-791. doi: 10.3934/mcrf.2019049

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

1. 

POEMS (CNRS-INRIA-ENSTA ParisTech), Palaiseau, France

2. 

ISAE-SUPAERO, Université de Toulouse, France

* Corresponding author: florian.monteghetti@inria.fr

Received  August 2018 Revised  July 2019 Published  November 2019

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)).

Citation: Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions. Mathematical Control and Related Fields, 2019, 9 (4) : 759-791. doi: 10.3934/mcrf.2019049
References:
[1]

Z. Abbas and S. Nicaise, Polynomial decay rate for a wave equation with general acoustic boundary feedback laws, SeMA Journal, 61 (2013), 19-47.  doi: 10.1007/s40324-013-0002-5.

[2]

Z. Abbas and S. Nicaise, The multidimensional wave equation with generalized acoustic boundary conditions I: Strong stability, SIAM Journal on Control and Optimization, 53 (2015), 2558-2581.  doi: 10.1137/140971336.

[3] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[4]

F. Alabau-BoussouiraJ. Prüss and R. Zacher, Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels, Comptes Rendus Mathematique, 347 (2009), 277-282.  doi: 10.1016/j.crma.2009.01.005.

[5]

B. D. O. Anderson, A system theory criterion for positive real matrices, SIAM Journal on Control, 5 (1967), 171-182.  doi: 10.1137/0305011.

[6]

W. Arendt and C. J. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[7] E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions, Academic Press, New York, 1966. 
[8]

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

[9] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, Oxford, 1998. 
[10]

G. Chen, A note on the boundary stabilization of the wave equation, SIAM Journal on Control and Optimization, 19 (1981), 106-113.  doi: 10.1137/0319008.

[11]

P. Cornilleau and S. Nicaise, Energy decay for solutions of the wave equation with general memory boundary conditions, Differential and Integral Equations, 22 (2009), 1173-1192. 

[12]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Mathematical Methods in the Applied Sciences, 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[13]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[14]

C. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, Journal of Functional Analysis, 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[15]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.

[16]

W. Desch, E. Fašangová, J. Milota and G. Propst, Stabilization through viscoelastic boundary damping: A semigroup approach, in Semigroup Forum, 80 (2010), 405-415. doi: 10.1007/s00233-009-9197-2.

[17]

W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, The Journal of Integral Equations and Applications, 1 (1988), 397-433.  doi: 10.1216/JIE-1988-1-3-397.

[18]

Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proceedings of the American Mathematical Society, 124 (1996), 591-600.  doi: 10.1090/S0002-9939-96-03132-2.

[19] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC Press, Boca Raton, FL, 1994. 
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K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

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R. GarrappaF. Mainardi and M. Guido, Models of dielectric relaxation based on completely monotone functions, Fractional Calculus and Applied Analysis, 19 (2016), 1105-1160.  doi: 10.1515/fca-2016-0060.

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 2001.

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V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

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P. Grabowski, Stabilization of wave equation using standard/fractional derivative in boundary damping, in Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland (eds. W. Mitkowski, J. Kacprzyk and J. Baranowski), Springer, Cham, 257 (2013), 101–121. doi: 10.1007/978-3-319-00933-9_9.

[25] G. GripenbergS.-O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511662805.
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P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.

[27]

J. K. Hale, Dynamical systems and stability, Journal of Mathematical Analysis and Applications, 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.

[28]

T. Hélie and D. Matignon, Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses, Mathematical Models and Methods in Applied Sciences, 16 (2006), 503-536.  doi: 10.1142/S0218202506001248.

[29]

R. HiptmairM. López-Fernández and A. Paganini, Fast convolution quadrature based impedance boundary conditions, Journal of Computational and Applied Mathematics, 263 (2014), 500-517.  doi: 10.1016/j.cam.2013.12.025.

[30]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.

[31]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, Berlin, 1995.

[32]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, Journal de Mathématiques Pures et Appliquées, 69 (1990), 33-54. 

[33]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6.

[34]

P. D. Lax, Functional Analysis, John Wiley & Sons, New York, 2002.

[35]

C. LiJ. Liang and T.-J. Xiao, Polynomial stability for wave equations with acoustic boundary conditions and boundary memory damping, Applied Mathematics and Computation, 321 (2018), 593-601.  doi: 10.1016/j.amc.2017.11.019.

[36]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer-Verlag, 1972.

[37]

B. Lombard and D. Matignon, Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics, SIAM Journal on Applied Mathematics, 76 (2016), 1765-1791.  doi: 10.1137/16M1062491.

[38]

R. Lozano, B. Brogliato, O. Egeland and B. Maschke, Dissipative Systems Analysis and Control: Theory and Applications, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-3668-2.

[39]

Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3.

[40]

Y. Lyubich and P. Vũ, Asymptotic stability of linear differential equations in Banach spaces, Studia Mathematica, 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[41]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), 291–348, CISM Courses and Lect., 378, Springer, Vienna, 1997. doi: 10.1007/978-3-7091-2664-6_7.

[42]

D. Matignon and H. Zwart, Standard diffusive systems as well-posed linear systems, International Journal of Control.

[43]

D. Matignon, An introduction to fractional calculus, in Scaling, Fractals and Wavelets (eds. P. Abry, P. Gonçalvès and J. Levy-Vehel), ISTE–Wiley, London–Hoboken, 2009, 237–277. doi: 10.1002/9780470611562.ch7.

[44]

D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 487-507.  doi: 10.1051/cocv:2005016.

[45]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Mathematical Control and Related Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

[46]

F. Monteghetti, G. Haine and D. Matignon, Stability of linear fractional differential equations with delays: A coupled parabolic-hyperbolic PDEs formulation, in 20th World Congress of the International Federation of Automatic Control (IFAC), 2017.

[47]

F. MonteghettiD. Matignon and E. Piot, Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized euler equations, Journal of Computational Physics, 375 (2018), 393-426.  doi: 10.1016/j.jcp.2018.08.037.

[48]

F. MonteghettiD. MatignonE. Piot and L. Pascal, Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models, The Journal of the Acoustical Society of America, 140 (2016), 1663-1674.  doi: 10.1121/1.4962277.

[49]

F. Monteghetti, Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics, PhD thesis, ISAE-SUPAERO, Université de Toulouse, Toulouse, France, 2018.

[50]

G. Montseny, Diffusive representation of pseudo-differential time-operators, in ESAIM: Proceedings, 5 (1998), 159-175. doi: 10.1051/proc:1998005.

[51]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[52]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[53]

G. R. Peralta, Stabilization of viscoelastic wave equations with distributed or boundary delay, Zeitschrift Für Analysis und Ihre Anwendungen, 35 (2016), 359-381.  doi: 10.4171/ZAA/1569.

[54]

G. R. Peralta, Stabilization of the wave equation with acoustic and delay boundary conditions, Semigroup Forum, 96 (2018), 357-376.  doi: 10.1007/s00233-018-9930-9.

[55] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[56]

F. Rellich, Darstellung der Eigenwerte von ${\Delta}u+\lambda u = 0$ durch ein Randintegral, Mathematische Zeitschrift, 46 (1940), 635-636.  doi: 10.1007/BF01181459.

[57]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993.

[58]

S. Sauter and M. Schanz, Convolution quadrature for the wave equation with impedance boundary conditions, Journal of Computational Physics, 334 (2017), 442-459.  doi: 10.1016/j.jcp.2017.01.013.

[59]

L. Schwartz, Mathematics for the Physical Sciences, Hermann, Paris, 1966.

[60]

O. J. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space, Transactions of the American Mathematical Society, 345 (1994), 527-575.  doi: 10.1090/S0002-9947-1994-1264153-X.

[61]

O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. part I: Well-posed systems, Mathematics of Control, Signals and Systems, 15 (2002), 291-315.  doi: 10.1007/s004980200012.

[62] O. J. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.
[63]

R. Stahn, On the decay rate for the wave equation with viscoelastic boundary damping, Journal of Differential Equations, 265 (2018), 2793-2824.  doi: 10.1016/j.jde.2018.04.048.

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M. Tucsnak and G. Weiss, Well-posed systems – the LTI case and beyond, Automatica, 50 (2014), 1757-1779.  doi: 10.1016/j.automatica.2014.04.016.

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J.-M. WangB.-Z. Guo and M. Krstic, Wave equation stabilization by delays equal to even multiples of the wave propagation time, SIAM Journal on Control and Optimization, 49 (2011), 517-554.  doi: 10.1137/100796261.

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G. WeissO. J. Staffans and M. Tucsnak, Well-posed linear systems–a survey with emphasis on conservative systems, International Journal of Applied Mathematics and Computer Science, 11 (2001), 7-33. 

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[68] S. V. Yuferev and N. Ida, Surface Impedance Boundary Conditions: A Comprehensive Approach, CRC Press, Boca Raton, 2010. 

show all references

References:
[1]

Z. Abbas and S. Nicaise, Polynomial decay rate for a wave equation with general acoustic boundary feedback laws, SeMA Journal, 61 (2013), 19-47.  doi: 10.1007/s40324-013-0002-5.

[2]

Z. Abbas and S. Nicaise, The multidimensional wave equation with generalized acoustic boundary conditions I: Strong stability, SIAM Journal on Control and Optimization, 53 (2015), 2558-2581.  doi: 10.1137/140971336.

[3] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[4]

F. Alabau-BoussouiraJ. Prüss and R. Zacher, Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels, Comptes Rendus Mathematique, 347 (2009), 277-282.  doi: 10.1016/j.crma.2009.01.005.

[5]

B. D. O. Anderson, A system theory criterion for positive real matrices, SIAM Journal on Control, 5 (1967), 171-182.  doi: 10.1137/0305011.

[6]

W. Arendt and C. J. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[7] E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions, Academic Press, New York, 1966. 
[8]

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

[9] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford University Press, Oxford, 1998. 
[10]

G. Chen, A note on the boundary stabilization of the wave equation, SIAM Journal on Control and Optimization, 19 (1981), 106-113.  doi: 10.1137/0319008.

[11]

P. Cornilleau and S. Nicaise, Energy decay for solutions of the wave equation with general memory boundary conditions, Differential and Integral Equations, 22 (2009), 1173-1192. 

[12]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Mathematical Methods in the Applied Sciences, 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[13]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[14]

C. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, Journal of Functional Analysis, 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[15]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.

[16]

W. Desch, E. Fašangová, J. Milota and G. Propst, Stabilization through viscoelastic boundary damping: A semigroup approach, in Semigroup Forum, 80 (2010), 405-415. doi: 10.1007/s00233-009-9197-2.

[17]

W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, The Journal of Integral Equations and Applications, 1 (1988), 397-433.  doi: 10.1216/JIE-1988-1-3-397.

[18]

Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proceedings of the American Mathematical Society, 124 (1996), 591-600.  doi: 10.1090/S0002-9939-96-03132-2.

[19] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC Press, Boca Raton, FL, 1994. 
[20]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[21]

R. GarrappaF. Mainardi and M. Guido, Models of dielectric relaxation based on completely monotone functions, Fractional Calculus and Applied Analysis, 19 (2016), 1105-1160.  doi: 10.1515/fca-2016-0060.

[22]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 2001.

[23]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[24]

P. Grabowski, Stabilization of wave equation using standard/fractional derivative in boundary damping, in Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland (eds. W. Mitkowski, J. Kacprzyk and J. Baranowski), Springer, Cham, 257 (2013), 101–121. doi: 10.1007/978-3-319-00933-9_9.

[25] G. GripenbergS.-O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511662805.
[26]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.

[27]

J. K. Hale, Dynamical systems and stability, Journal of Mathematical Analysis and Applications, 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.

[28]

T. Hélie and D. Matignon, Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses, Mathematical Models and Methods in Applied Sciences, 16 (2006), 503-536.  doi: 10.1142/S0218202506001248.

[29]

R. HiptmairM. López-Fernández and A. Paganini, Fast convolution quadrature based impedance boundary conditions, Journal of Computational and Applied Mathematics, 263 (2014), 500-517.  doi: 10.1016/j.cam.2013.12.025.

[30]

L. Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.

[31]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, Berlin, 1995.

[32]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, Journal de Mathématiques Pures et Appliquées, 69 (1990), 33-54. 

[33]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6.

[34]

P. D. Lax, Functional Analysis, John Wiley & Sons, New York, 2002.

[35]

C. LiJ. Liang and T.-J. Xiao, Polynomial stability for wave equations with acoustic boundary conditions and boundary memory damping, Applied Mathematics and Computation, 321 (2018), 593-601.  doi: 10.1016/j.amc.2017.11.019.

[36]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer-Verlag, 1972.

[37]

B. Lombard and D. Matignon, Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics, SIAM Journal on Applied Mathematics, 76 (2016), 1765-1791.  doi: 10.1137/16M1062491.

[38]

R. Lozano, B. Brogliato, O. Egeland and B. Maschke, Dissipative Systems Analysis and Control: Theory and Applications, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-3668-2.

[39]

Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3.

[40]

Y. Lyubich and P. Vũ, Asymptotic stability of linear differential equations in Banach spaces, Studia Mathematica, 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[41]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), 291–348, CISM Courses and Lect., 378, Springer, Vienna, 1997. doi: 10.1007/978-3-7091-2664-6_7.

[42]

D. Matignon and H. Zwart, Standard diffusive systems as well-posed linear systems, International Journal of Control.

[43]

D. Matignon, An introduction to fractional calculus, in Scaling, Fractals and Wavelets (eds. P. Abry, P. Gonçalvès and J. Levy-Vehel), ISTE–Wiley, London–Hoboken, 2009, 237–277. doi: 10.1002/9780470611562.ch7.

[44]

D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 487-507.  doi: 10.1051/cocv:2005016.

[45]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Mathematical Control and Related Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

[46]

F. Monteghetti, G. Haine and D. Matignon, Stability of linear fractional differential equations with delays: A coupled parabolic-hyperbolic PDEs formulation, in 20th World Congress of the International Federation of Automatic Control (IFAC), 2017.

[47]

F. MonteghettiD. Matignon and E. Piot, Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized euler equations, Journal of Computational Physics, 375 (2018), 393-426.  doi: 10.1016/j.jcp.2018.08.037.

[48]

F. MonteghettiD. MatignonE. Piot and L. Pascal, Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models, The Journal of the Acoustical Society of America, 140 (2016), 1663-1674.  doi: 10.1121/1.4962277.

[49]

F. Monteghetti, Analysis and Discretization of Time-Domain Impedance Boundary Conditions in Aeroacoustics, PhD thesis, ISAE-SUPAERO, Université de Toulouse, Toulouse, France, 2018.

[50]

G. Montseny, Diffusive representation of pseudo-differential time-operators, in ESAIM: Proceedings, 5 (1998), 159-175. doi: 10.1051/proc:1998005.

[51]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[52]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[53]

G. R. Peralta, Stabilization of viscoelastic wave equations with distributed or boundary delay, Zeitschrift Für Analysis und Ihre Anwendungen, 35 (2016), 359-381.  doi: 10.4171/ZAA/1569.

[54]

G. R. Peralta, Stabilization of the wave equation with acoustic and delay boundary conditions, Semigroup Forum, 96 (2018), 357-376.  doi: 10.1007/s00233-018-9930-9.

[55] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
[56]

F. Rellich, Darstellung der Eigenwerte von ${\Delta}u+\lambda u = 0$ durch ein Randintegral, Mathematische Zeitschrift, 46 (1940), 635-636.  doi: 10.1007/BF01181459.

[57]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993.

[58]

S. Sauter and M. Schanz, Convolution quadrature for the wave equation with impedance boundary conditions, Journal of Computational Physics, 334 (2017), 442-459.  doi: 10.1016/j.jcp.2017.01.013.

[59]

L. Schwartz, Mathematics for the Physical Sciences, Hermann, Paris, 1966.

[60]

O. J. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space, Transactions of the American Mathematical Society, 345 (1994), 527-575.  doi: 10.1090/S0002-9947-1994-1264153-X.

[61]

O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. part I: Well-posed systems, Mathematics of Control, Signals and Systems, 15 (2002), 291-315.  doi: 10.1007/s004980200012.

[62] O. J. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.
[63]

R. Stahn, On the decay rate for the wave equation with viscoelastic boundary damping, Journal of Differential Equations, 265 (2018), 2793-2824.  doi: 10.1016/j.jde.2018.04.048.

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