Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-26T09:37:52.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic stability of linear conservative systems when coupledwith diffusive systems

Published online by Cambridge University Press:  15 July 2005

Denis Matignon  and
Christophe Prieur
Denis Matignon
Affiliation:
Télécom Paris, dépt TSI & CNRS, UMR 5141, 37-39 rue Dareau, 75 014 Paris, France; matignon@tsi.enst.fr
Christophe Prieur
Affiliation:
LAAS - CNRS, 7 avenue du Colonel Roche 31077 Toulouse, France; prieur@laas.fr
Get access

Abstract

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 stabilitywell-posed systemsLyapunov functionaldiffusive representationfractional calculus.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 487 - 507
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arendt, W. and Batty, C.J.K., Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837–852. CrossRef
H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson, Paris (1983).
Bruneau, M., Herzog, Ph., Kergomard, J. and Polack, J.-D., General formulation of the dispersion equation in bounded visco-thermal fluid, and application to some simple geometries. Wave Motion 11 (1989) 441–451. CrossRef
T. Cazenave and A. Haraux, An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications 13 (1998).
F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11 (1994) 485–515.
R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory. Texts Appl. Math. 21 (1995).
G. Dauphin, D. Heleschewitz and D. Matignon, Extended diffusive representations and application to non-standard oscillators, in Proc. of Math. Theory on Network Systems (MTNS), Perpignan, France (2000).
R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 5. Springer, New York (1984).
Fellah, Z.E.A., Depollier, C. and Fellah, M., Direct and inverse scattering problem in porous material having a rigid frame by fractional calculus based method. J. Sound Vibration 244 (2001) 3659–3666. CrossRef
H. Haddar, T. Hélie and D. Matignon, 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.
Hélie, Th., Unidimensional models of acoustic propagation in axisymmetric waveguides. J. Acoust. Soc. Am. 114 (2003) 2633–2647. CrossRef
A.E. Ingham, On Wiener's method in Tauberian theorems, in Proc. London Math. Soc. II 38 (1935) 458–480.
Korevaar, J., Newman's, On quick way to the prime number theorem. Math. Intell. 4 (1982) 108–115. CrossRef
Lokshin, A.A., Wave equation with singular retarded time. Dokl. Akad. Nauk SSSR 240 (1978) 43–46 (in Russian).
Lokshin, A.A. and Rok, V.E., Fundamental solutions of the wave equation with retarded time. Dokl. Akad. Nauk SSSR 239 (1978) 1305–1308 (in Russian).
Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems and applications. Comm. Control Engrg. Springer-Verlag, New York (1999).
Yu.I. Lyubich, V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88 (1988) 37–42.
Matignon, D., Stability properties for generalized fractional differential systems. ESAIM: Proc. 5 (1998) 145–158. CrossRef
Montseny, G., Diffusive representation of pseudo-differential time-operators. ESAIM: Proc. 5 (1998) 159–175. CrossRef
Newman, D.J., Simple analytic proof of the prime number theorem. Am. Math. Mon. 87 (1980) 693–696. CrossRef
Polack, J.-D., Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process. J. Acoustique 4 (1991) 47–67.
Staffans, O.J., Well-posedness and stabilizability of a viscoelastic equation in energy space. Trans. Am. Math. Soc. 345 (1994) 527–575. CrossRef
Staffans, O.J., Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Sig. Syst. 15 (2002) 291–315. CrossRef
Weiss, G., Staffans, O.J. and Tucsnak, M., Well-posed linear systems – a survey with emphasis on conservative systems. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 7–33.
Weiss, G. and Tucsnak, M., 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. CrossRef