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 & Related Fields, 2019, 9 (4) : 759-791. doi: 10.3934/mcrf.2019049
References:
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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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

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

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[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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[15]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.   Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[50]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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