In this paper we study linear conservative systems of finite dimension coupled with an infinite dimensional system of diffusive type. Computing the time-derivative of an appropriate energy functional along the solutions helps us to prove the well-posedness of the system and a stability property. But in order to prove asymptotic stability we need to apply a sufficient spectral condition. We also illustrate the sharpness of this condition by exhibiting some systems for which we do not have the asymptotic property.
Mots-clés : asymptotic stability, well-posed systems, Lyapunov functional, diffusive representation, fractional calculus
@article{COCV_2005__11_3_487_0,
author = {Matignon, Denis and Prieur, Christophe},
title = {Asymptotic stability of linear conservative systems when coupled with diffusive systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {487--507},
publisher = {EDP-Sciences},
volume = {11},
number = {3},
year = {2005},
doi = {10.1051/cocv:2005016},
mrnumber = {2148855},
zbl = {1125.93030},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2005016/}
}
TY - JOUR AU - Matignon, Denis AU - Prieur, Christophe TI - Asymptotic stability of linear conservative systems when coupled with diffusive systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 487 EP - 507 VL - 11 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005016/ DO - 10.1051/cocv:2005016 LA - en ID - COCV_2005__11_3_487_0 ER -
%0 Journal Article %A Matignon, Denis %A Prieur, Christophe %T Asymptotic stability of linear conservative systems when coupled with diffusive systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 487-507 %V 11 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2005016/ %R 10.1051/cocv:2005016 %G en %F COCV_2005__11_3_487_0
Matignon, Denis; Prieur, Christophe. Asymptotic stability of linear conservative systems when coupled with diffusive systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 3, pp. 487-507. doi : 10.1051/cocv:2005016. http://archive.numdam.org/articles/10.1051/cocv:2005016/
[1] and, Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837-852. | Zbl
[2] , Analyse fonctionnelle. Théorie et applications. Masson, Paris (1983). | MR | Zbl
[3] ,, and, General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries. Wave Motion 11 (1989) 441-451. | Zbl
[4] and, An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications 13 (1998). | MR | Zbl
[5] and, Stabilization of second order evolution equations by unbounded nonlinear feedback. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11 (1994) 485-515. | Numdam | Zbl
[6] and, An introduction to infinite-dimensional linear systems theory. Texts Appl. Math. 21 (1995). | MR | Zbl
[7] , and, Extended diffusive representations and application to non-standard oscillators, in Proc. of Math. Theory on Network Systems (MTNS), Perpignan, France (2000).
[8] and, Mathematical analysis and numerical methods for science and technology, Vol. 5. Springer, New York (1984). | MR | Zbl
[9] , and, Direct and inverse scattering problem in porous material having a rigid frame by fractional calculus based method. J. Sound Vibration 244 (2001) 3659-3666.
[10] , and, A Webster-Lokshin model for waves with viscothermal losses and impedance boundary conditions: strong solutions, in Proc. of Sixth international conference on mathematical and numerical aspects of wave propagation phenomena, Jyväskylä, Finland (2003) 66-71. | Zbl
[11] , Unidimensional models of acoustic propagation in axisymmetric waveguides. J. Acoust. Soc. Am. 114 (2003) 2633-2647.
[12] , On Wiener's method in Tauberian theorems, in Proc. London Math. Soc. II 38 (1935) 458-480. | JFM | Zbl
[13] , On Newman's quick way to the prime number theorem. Math. Intell. 4 (1982) 108-115. | Zbl
[14] , Wave equation with singular retarded time. Dokl. Akad. Nauk SSSR 240 (1978) 43-46 (in Russian). | Zbl
[15] and, Fundamental solutions of the wave equation with retarded time. Dokl. Akad. Nauk SSSR 239 (1978) 1305-1308 (in Russian). | Zbl
[16] , and, Stability and stabilization of infinite dimensional systems and applications. Comm. Control Engrg. Springer-Verlag, New York (1999). | MR | Zbl
[17] and, Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88 (1988) 37-42. | EuDML | Zbl
[18] , Stability properties for generalized fractional differential systems. ESAIM: Proc. 5 (1998) 145-158. | Zbl
[19] , Diffusive representation of pseudo-differential time-operators. ESAIM: Proc. 5 (1998) 159-175. | Zbl
[20] , Simple analytic proof of the prime number theorem. Am. Math. Mon. 87 (1980) 693-696. | Zbl
[21] , Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process. J. Acoustique 4 (1991) 47-67.
[22] , Well-posedness and stabilizability of a viscoelastic equation in energy space. Trans. Am. Math. Soc. 345 (1994) 527-575. | Zbl
[23] , Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Sig. Syst. 15 (2002) 291-315. | Zbl
[24] , and, Well-posed linear systems - a survey with emphasis on conservative systems. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 7-33. | EuDML | Zbl
[25] and, How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance. ESAIM: COCV 9 (2003) 247-273. | EuDML | Numdam | Zbl
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