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.

Keywords: 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, Volume 11 (2005) no. 3, pp. 487-507. doi : 10.1051/cocv:2005016. http://archive.numdam.org/articles/10.1051/cocv:2005016/

[1] Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837-852. | Zbl

and ,[2] Analyse fonctionnelle. Théorie et applications. Masson, Paris (1983). | MR | Zbl

,[3] General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries. Wave Motion 11 (1989) 441-451. | Zbl

, , and ,[4] An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications 13 (1998). | MR | Zbl

and ,[5] Stabilization of second order evolution equations by unbounded nonlinear feedback. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11 (1994) 485-515. | Numdam | Zbl

and ,[6] An introduction to infinite-dimensional linear systems theory. Texts Appl. Math. 21 (1995). | MR | Zbl

and ,[7] Extended diffusive representations and application to non-standard oscillators, in Proc. of Math. Theory on Network Systems (MTNS), Perpignan, France (2000).

, and ,[8] Mathematical analysis and numerical methods for science and technology, Vol. 5. Springer, New York (1984). | MR | Zbl

and ,[9] Direct and inverse scattering problem in porous material having a rigid frame by fractional calculus based method. J. Sound Vibration 244 (2001) 3659-3666.

, and ,[10] 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

, and ,[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] Fundamental solutions of the wave equation with retarded time. Dokl. Akad. Nauk SSSR 239 (1978) 1305-1308 (in Russian). | Zbl

and ,[16] Stability and stabilization of infinite dimensional systems and applications. Comm. Control Engrg. Springer-Verlag, New York (1999). | MR | Zbl

, and ,[17] Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88 (1988) 37-42. | EuDML | Zbl

and ,[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] Well-posed linear systems - a survey with emphasis on conservative systems. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 7-33. | EuDML | Zbl

, and ,[25] 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

and ,*Cited by Sources: *