Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T14:53:17.030Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic stabilization of systems via decoupling techniques

Published online by Cambridge University Press:  15 August 2002

Farid Ammar-Khodja ,
Ahmed Bader  and
Assia Benabdallah
Farid Ammar-Khodja
Affiliation:
Université de Franche-Comté, Laboratoire de Calcul Scientifique et UMR 6623 du CNRS, 16 route de Gray, 25000 Besançon, France.
Ahmed Bader
Affiliation:
Université de Franche-Comté, Laboratoire de Calcul Scientifique et UMR 6623 du CNRS, 16 route de Gray, 25000 Besançon, France.
Assia Benabdallah
Affiliation:
Université de Franche-Comté, Laboratoire de Calcul Scientifique et UMR 6623 du CNRS, 16 route de Gray, 25000 Besançon, France.
Get access

Abstract

We give sufficient conditions which allow the study of the exponential stability of systems closely related to the linear thermoelasticity systems by a decoupling technique. Our approach is based on the multipliers technique and our result generalizes (from the exponential stability point of view) the earlier one obtained by Henry et al.

Keywords

Stabilizationthermoelasticity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 577 - 593
Copyright
© EDP Sciences, SMAI, 1999

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

F. Ammar-Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamic controllers. Dynamics of Continuous, Discrete and Impulsive Systems, to appear.
Ammar-Khodja, F. and Benabdallah, A., Conditions suffisantes pour la stabilisation uniforme d'équations du second ordre par des contrôleurs dynamiques. C.R. Acad. Sci. Sér. I Math. 323 (1996) 615-620.
Ammar Khodja, F., Benabdallah, A. and Teniou, D., Coupled systems. Abstract and Appl. Anal. 1 (1996) 327-340. CrossRef
Atkinson, F.V., Langer, H., Mennicken, R. and Shkalikov, A.A., The essential spectrum of some matrix operators. Math. Nachr. 167 (1994) 5-20. CrossRef
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. CrossRef
A.V. Balakrishnan, Applied functional analysis, Springer-Verlag, New-York, Heidelberg (1976).
Bourgeat A., Simulating gas-liquid flow in a well-reservoir system. Numerical Methods in Engineering and Applied Sciences, H. Adler, J.C. Heinrich, S. Lavanchy, E. Onate and B. Suarez, Eds., CIMNE, Barcelona (1992).
T. Cazenave and Dickstein F., On the initial value problem for a linear model of well-reservoir coupling (1996) preprint.
Chen, S. and Triggiani, R., Proof of extensions of two conjectures on structural damping for elastic systems. Pacific J. Math. 136 (1989) 15-55. CrossRef
R. Dautray and J.L. Lions, Analyse Mathématique et Calcul Numérique 2, Masson (1987).
Curtain, R.F. and Weiss, G., Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Contr. 42 (1997) 4-21.
R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, Texts in Applied Mathematics 21 (1995).
K.-J. Engel, Operator matrices and systems of evolution equations (1998).
Haraux, A., Une remarque sur la stabilisation de certains systemes du deuxième ordre en temps. Portugal Math. 46 (1989) 245-258.
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, RMA 17, Masson (1991).
F. Huang, Characteristic conditions for exponential stability of linear dynamic systems in Hilbert spaces. Ann. Diff. Eqs. 1 (1985).
D.B. Henry, O. Lopes and A. Perissinitto Jr., On the essential spectrum of a semigroup of thermoelasticity. Nonlinear Anal., TMA 21 (1993) 65-75.
J.E. Lagnese and J.L. Lions, Modelling Analysis and Control of Thin Plates, RMA 6, Masson (1988).
Lasiecka I. and Triggiani R., Differential and Algebraic Riccati Equations ..., Springer-Verlag, Lecture Notes in Control and Information Sciences 164 (1991).
Z. Liu and J. Yong, Qualitative properties of certain C0 semigroups arising in elastic systems with various dampings (1998) preprint.
G. Lebeau and E. Zuazua E., Null controllability of a system of linear thermoelasticity (1995) preprint.
R. Nagel, Towards a ``Matrix Theory'' for unbounded operators matrices. Math. Z. 201 (1989) 57-68.
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Applied Mathematical Sciences 44 (1983).
J.E. Munoz Rivera and R. Racke, Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelastic type. SIAM J. Math. Anal. (1995) 1547-1563.
D.L. Russell, A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques. Quart. Appl. Math. 49 (1991) 373-396.
D.L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl. 173 (1993) 339-358.
L. De Teresa and E. Zuzua, Controllability for the linear system of thermoelastic plates. Adv. Diff. Eqs. (1996) 369-402.
J. Zabcyk, Mathematical Control Theory: An Introduction, Birkhauser (1995).
Zuzua, E., Controllability of the linear system of thermoelasticity. J. Math. Pures Appl. 74 (1995) 303-346.