# American Institute of Mathematical Sciences

December  2014, 4(4): 481-500. doi: 10.3934/mcrf.2014.4.481

## Asymptotic stability of Webster-Lokshin equation

 1 Université de Toulouse; ISAE, DMIA, 10, avenue E. Belin, B.P. 54032, F-31055 Toulouse cedex 4, France 2 GIPSA-lab, CNRS, 11 rue des Mathématiques, Grenoble Campus, 38402 Grenoble Cedex, France

Received  May 2013 Revised  February 2014 Published  September 2014

The Webster-Lokshin equation is a partial differential equation considered in this paper. It models the sound velocity in an acoustic domain. The dynamics contains linear fractional derivatives which can admit an infinite dimensional representation of diffusive type. The boundary conditions are described by impedance condition, which can be represented by two finite dimensional systems. Under the physical assumptions, there is a natural energy inequality. However, due to a lack of precompactness of the solutions, the LaSalle invariance principle can not be applied. The asymptotic stability of the system is proved by studying the resolvent equation, and by using the Arendt-Batty stability condition.
Citation: Denis Matignon, Christophe Prieur. Asymptotic stability of Webster-Lokshin equation. Mathematical Control & Related Fields, 2014, 4 (4) : 481-500. doi: 10.3934/mcrf.2014.4.481
##### References:
 [1] B. d'Andréa-Novel, F. Boustany, F. Conrad and B. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane,, Mathematics of Control, 7 (1994), 1.  doi: 10.1007/BF01211483.  Google Scholar [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Am. Math. Soc., 306 (1988), 837.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar [3] D. Z. Arov and O. J. Staffans, The infinite-dimensional continuous time Kalman-Yakubovich-Popov inequality,, The extended field of operator theory, 171 (2007), 37.  doi: 10.1007/978-3-7643-7980-3_3.  Google Scholar [4] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evolution Equation, 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar [5] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Mathematische Annalen, 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar [6] S. Boyd, L. El Ghaoui, E. Féron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory,, volume 15 of {Studies in Applied Mathematics}. SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar [7] M. Bruneau, Ph. Herzog, J. Kergomard and J.-D. Polack, General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries,, Wave Motion, 11 (1989), 441.  doi: 10.1016/0165-2125(89)90018-8.  Google Scholar [8] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [9] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback,, Ann. Inst. Henri Poincaré, 11 (1994), 485.   Google Scholar [10] J.-M. Coron, Control and Nonlinearity,, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, (2007).   Google Scholar [11] R. F. Curtain, Old and new perspectives on the positive-real lemma in systems and control theory,, Z. Angew. Math. Mech., 79 (1999), 579.  doi: 10.1002/(SICI)1521-4001(199909)79:9<579::AID-ZAMM579>3.0.CO;2-8.  Google Scholar [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar [13] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, vol. 5, (1984).  doi: 10.1007/978-3-642-58090-1.  Google Scholar [14] H. Haddar and D. Matignon, Theoretical and Numerical Analysis of the Webster-Lokshin Model,, Research. Report, (2008).   Google Scholar [15] H. Haddar, J.-R. Li and D. Matignon, Efficient solution of a wave equation with fractional order dissipative terms,, Journal of Computational and Applied Mathematics, 234 (2010), 2003.  doi: 10.1016/j.cam.2009.08.051.  Google Scholar [16] Th. Hélie, Unidimensional models of the acoustic propagation in axisymmetric waveguides,, J. Acoust. Soc. Amer., 114 (2003), 2633.   Google Scholar [17] A. A. Lokshin, Wave equation with singular retarded time,, Dokl. Akad. Nauk SSSR, 240 (1978), 43.   Google Scholar [18] A. A. Lokshin and V. E. Rok, Fundamental solutions of the wave equation with retarded time,, Dokl. Akad. Nauk SSSR, 239 (1978), 1305.   Google Scholar [19] Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications,, Communications and Control Engineering. Springer Verlag, (1999).  doi: 10.1007/978-1-4471-0419-3.  Google Scholar [20] Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces,, Stud. Math., 88 (1988), 37.   Google Scholar [21] D. Matignon, Stability properties for generalized fractional differential systems,, ESAIM: Proc., 5 (1998), 145.  doi: 10.1051/proc:1998004.  Google Scholar [22] D. Matignon, Asymptotic stability of the Webster-Lokshin model,, in Mathematical Theory of Networks and Systems, (2006).   Google Scholar [23] D. Matignon, An introduction to fractional calculus,, in Scaling, 7 (2009), 237.  doi: 10.1002/9780470611562.ch7.  Google Scholar [24] D. Matignon and B. d'Andréa-Novel, Spectral and time-domain consequences of an integro-differential perturbation of the wave PDE,, in Third int. conf. on math. and num. aspects of wave propagation phenomena, (1995), 769.   Google Scholar [25] D. Matignon, J. Audounet and G. Montseny, Energy decay for wave equations with damping of fractional order,, in Fourth int. conf. on math. and num. aspects of wave propagation phenomena, (1998), 638.   Google Scholar [26] D. Matignon and Ch. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM: Control, 11 (2005), 487.  doi: 10.1051/cocv:2005016.  Google Scholar [27] G. Montseny, Diffusive representation of pseudo-differential time-operators,, ESAIM: Proc., 5 (1998), 159.  doi: 10.1051/proc:1998005.  Google Scholar [28] J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations,, Advances in Mathematics, 22 (1976), 278.  doi: 10.1016/0001-8708(76)90096-7.  Google Scholar [29] J.-D. Polack, Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process,, J. Acoustique, 4 (1991), 47.   Google Scholar [30] A. Rantzer, On the Kalman-Yakubovich-Popov lemma,, Systems & Control Letters, 28 (1996), 7.  doi: 10.1016/0167-6911(95)00063-1.  Google Scholar [31] O. J. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space,, Trans. Amer. Math. Soc., 345 (1994), 527.  doi: 10.1090/S0002-9947-1994-1264153-X.  Google Scholar [32] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar

show all references

##### References:
 [1] B. d'Andréa-Novel, F. Boustany, F. Conrad and B. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane,, Mathematics of Control, 7 (1994), 1.  doi: 10.1007/BF01211483.  Google Scholar [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Am. Math. Soc., 306 (1988), 837.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar [3] D. Z. Arov and O. J. Staffans, The infinite-dimensional continuous time Kalman-Yakubovich-Popov inequality,, The extended field of operator theory, 171 (2007), 37.  doi: 10.1007/978-3-7643-7980-3_3.  Google Scholar [4] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evolution Equation, 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar [5] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Mathematische Annalen, 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar [6] S. Boyd, L. El Ghaoui, E. Féron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory,, volume 15 of {Studies in Applied Mathematics}. SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar [7] M. Bruneau, Ph. Herzog, J. Kergomard and J.-D. Polack, General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries,, Wave Motion, 11 (1989), 441.  doi: 10.1016/0165-2125(89)90018-8.  Google Scholar [8] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998).   Google Scholar [9] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback,, Ann. Inst. Henri Poincaré, 11 (1994), 485.   Google Scholar [10] J.-M. Coron, Control and Nonlinearity,, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, (2007).   Google Scholar [11] R. F. Curtain, Old and new perspectives on the positive-real lemma in systems and control theory,, Z. Angew. Math. Mech., 79 (1999), 579.  doi: 10.1002/(SICI)1521-4001(199909)79:9<579::AID-ZAMM579>3.0.CO;2-8.  Google Scholar [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Texts in Applied Mathematics, (1995).  doi: 10.1007/978-1-4612-4224-6.  Google Scholar [13] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, vol. 5, (1984).  doi: 10.1007/978-3-642-58090-1.  Google Scholar [14] H. Haddar and D. Matignon, Theoretical and Numerical Analysis of the Webster-Lokshin Model,, Research. Report, (2008).   Google Scholar [15] H. Haddar, J.-R. Li and D. Matignon, Efficient solution of a wave equation with fractional order dissipative terms,, Journal of Computational and Applied Mathematics, 234 (2010), 2003.  doi: 10.1016/j.cam.2009.08.051.  Google Scholar [16] Th. Hélie, Unidimensional models of the acoustic propagation in axisymmetric waveguides,, J. Acoust. Soc. Amer., 114 (2003), 2633.   Google Scholar [17] A. A. Lokshin, Wave equation with singular retarded time,, Dokl. Akad. Nauk SSSR, 240 (1978), 43.   Google Scholar [18] A. A. Lokshin and V. E. Rok, Fundamental solutions of the wave equation with retarded time,, Dokl. Akad. Nauk SSSR, 239 (1978), 1305.   Google Scholar [19] Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications,, Communications and Control Engineering. Springer Verlag, (1999).  doi: 10.1007/978-1-4471-0419-3.  Google Scholar [20] Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces,, Stud. Math., 88 (1988), 37.   Google Scholar [21] D. Matignon, Stability properties for generalized fractional differential systems,, ESAIM: Proc., 5 (1998), 145.  doi: 10.1051/proc:1998004.  Google Scholar [22] D. Matignon, Asymptotic stability of the Webster-Lokshin model,, in Mathematical Theory of Networks and Systems, (2006).   Google Scholar [23] D. Matignon, An introduction to fractional calculus,, in Scaling, 7 (2009), 237.  doi: 10.1002/9780470611562.ch7.  Google Scholar [24] D. Matignon and B. d'Andréa-Novel, Spectral and time-domain consequences of an integro-differential perturbation of the wave PDE,, in Third int. conf. on math. and num. aspects of wave propagation phenomena, (1995), 769.   Google Scholar [25] D. Matignon, J. Audounet and G. Montseny, Energy decay for wave equations with damping of fractional order,, in Fourth int. conf. on math. and num. aspects of wave propagation phenomena, (1998), 638.   Google Scholar [26] D. Matignon and Ch. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM: Control, 11 (2005), 487.  doi: 10.1051/cocv:2005016.  Google Scholar [27] G. Montseny, Diffusive representation of pseudo-differential time-operators,, ESAIM: Proc., 5 (1998), 159.  doi: 10.1051/proc:1998005.  Google Scholar [28] J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations,, Advances in Mathematics, 22 (1976), 278.  doi: 10.1016/0001-8708(76)90096-7.  Google Scholar [29] J.-D. Polack, Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process,, J. Acoustique, 4 (1991), 47.   Google Scholar [30] A. Rantzer, On the Kalman-Yakubovich-Popov lemma,, Systems & Control Letters, 28 (1996), 7.  doi: 10.1016/0167-6911(95)00063-1.  Google Scholar [31] O. J. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space,, Trans. Amer. Math. Soc., 345 (1994), 527.  doi: 10.1090/S0002-9947-1994-1264153-X.  Google Scholar [32] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar
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