Asymptotic stability of linear conservative systems when coupled with diffusive systems

Denis Matignon; Christophe Prieur

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 3, page 487-507
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Matignon, Denis, and Prieur, Christophe. "Asymptotic stability of linear conservative systems when coupled with diffusive systems." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2010): 487-507. <http://eudml.org/doc/90774>.

@article{Matignon2010,
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. },
author = {Matignon, Denis, Prieur, Christophe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Asymptotic stability; well-posed systems; Lyapunov functional; diffusive representation; fractional calculus.; diffusive representation; fractional calculus},
language = {eng},
month = {3},
number = {3},
pages = {487-507},
publisher = {EDP Sciences},
title = {Asymptotic stability of linear conservative systems when coupled with diffusive systems},
url = {http://eudml.org/doc/90774},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 3
SP - 487
EP - 507
AB - 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.
LA - eng
KW - Asymptotic stability; well-posed systems; Lyapunov functional; diffusive representation; fractional calculus.; diffusive representation; fractional calculus
UR - http://eudml.org/doc/90774
ER -

References

top
  1. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc.306 (1988) 837–852.  
  2. H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson, Paris (1983).  
  3. 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 Motion11 (1989) 441–451.  
  4. T. Cazenave and A. Haraux, An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications13 (1998).  
  5. F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback. Ann. Inst. Henri Poincaré, Anal. Non Linéaire11 (1994) 485–515.  
  6. R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory. Texts Appl. Math.21 (1995).  
  7. 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).  
  8. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 5. Springer, New York (1984).  
  9. Z.E.A. Fellah, C. Depollier and M. Fellah, Direct and inverse scattering problem in porous material having a rigid frame by fractional calculus based method. J. Sound Vibration244 (2001) 3659–3666.  
  10. 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.  
  11. Th. Hélie, Unidimensional models of acoustic propagation in axisymmetric waveguides. J. Acoust. Soc. Am.114 (2003) 2633–2647.  
  12. A.E. Ingham, On Wiener's method in Tauberian theorems, in Proc. London Math. Soc. II38 (1935) 458–480.  
  13. J. Korevaar, On Newman's quick way to the prime number theorem. Math. Intell.4 (1982) 108–115.  
  14. A.A. Lokshin, Wave equation with singular retarded time. Dokl. Akad. Nauk SSSR240 (1978) 43–46 (in Russian).  
  15. A.A. Lokshin and V.E. Rok, Fundamental solutions of the wave equation with retarded time. Dokl. Akad. Nauk SSSR239 (1978) 1305–1308 (in Russian).  
  16. 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).  
  17. Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Stud. Math.88 (1988) 37–42.  
  18. D. Matignon, Stability properties for generalized fractional differential systems. ESAIM: Proc.5 (1998) 145–158.  
  19. G. Montseny, Diffusive representation of pseudo-differential time-operators. ESAIM: Proc.5 (1998) 159–175.  
  20. D.J. Newman, Simple analytic proof of the prime number theorem. Am. Math. Mon.87 (1980) 693–696.  
  21. J.-D. Polack, Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process. J. Acoustique4 (1991) 47–67.  
  22. O.J. Staffans, Well-posedness and stabilizability of a viscoelastic equation in energy space. Trans. Am. Math. Soc.345 (1994) 527–575.  
  23. O.J. Staffans, Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Sig. Syst.15 (2002) 291–315.  
  24. G. Weiss, O.J. Staffans and M. Tucsnak, Well-posed linear systems – a survey with emphasis on conservative systems. Internat. J. Appl. Math. Comput. Sci.11 (2001) 7–33.  
  25. G. Weiss and M. Tucsnak, How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance. ESAIM: COCV9 (2003) 247–273.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.