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Exponential stability of distributed parameter systemsgoverned by symmetric hyperbolic partial differential equations usingLyapunov's second method

Published online by Cambridge University Press:  30 May 2008

Abdoua Tchousso ,
Thibaut Besson  and
Cheng-Zhong Xu
Abdoua Tchousso
Affiliation:
LAGEP, Bâtiment CPE, Université Claude Bernard, Lyon I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne, France. Departement de Mathématiques et Informatique, Université Abdou Moumouni de Niamey, BP 10662, Niger; [email protected]
Thibaut Besson
Affiliation:
LAGEP, Bâtiment CPE, Université Claude Bernard, Lyon I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne, France.
Cheng-Zhong Xu
Affiliation:
LAGEP, Bâtiment CPE, Université Claude Bernard, Lyon I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne, France.
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Abstract

In this paper we study asymptotic behaviour of distributed parameter systems governedby partial differential equations (abbreviated to PDE). We first review some recently developed resultson the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionalswe prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDEsystems. Then we apply the result to establish exponential stability of various chemical engineeringprocesses and, in particular, exponential stability of heat exchangers. Through concrete examples weshow how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE.Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp , $1<p\leq \infty$ . 


Keywords

Hyperbolic symmetric systemspartial differential equationsexponential stabilitystrongly continuous semigroupsLyapunov functionalsheat exchangers
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 2 , April 2009 , pp. 403 - 425
Copyright
© EDP Sciences, SMAI, 2008

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References

Alabau, F., Stabilisation frontière indirecte de systèmes faiblement couplés. C.R. Acad. Sci. Paris Série I 328 (1999) 1015–1020. CrossRef
Alabau, F., Cannarsa, P. and Komornik, V., Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Eq. 2 (2002) 127–150. CrossRef
Baillieul, J. and Levi, M., Rotational elastic dynamics. Physica D 27 (1987) 43–62. CrossRef
Bianchini, S., Hanouzet, B. and Natalini, R., Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math. 60 (2007) 1559–1622. CrossRef
H. Brezis, Analyse fonctionnelle : théorie et applications. Masson, Paris (1983).
Burq, N. and Lebeau, G., Mesure de défaut de compacité, application au système de Lamé. Ann. Sci. École Norm. Sup. 34 (2001) 817–870. CrossRef
Chapelon, A. and Boundary, C.Z. Xu control of a class of hyperbolic systems. Eur. J. Control 9 (2003) 589–604. CrossRef
J.M. Coron, B. d'Andréa-Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations. European Control Conference ECC'99, Karlsruhe, September (1999).
Coron, J.M., d'Andréa-Novel, B. and Bastin, G., A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Control 52 (2007) 2–11. CrossRef
R.F. Curtain and H.J. Zwart, An introduction to infinite-dimensional linear systems theory. Springer-Verlag, New York (1995).
B. d'Andréa-Novel, Commande non linéaire des robots. Hermès (1988).
Freitas, P., Stability results for the wave equation with indefinite damping. J. Diff. Eq. 132 (1996) 338–353. CrossRef
Greenberg, J.M. and The, T.T. Li effect of boundary damping for the quasilinear wave equation. J. Diff. Eq. 52 (1984) 66–75. CrossRef
Immanuel, C.D., Cordeiro, C.F., Sundaram, S.S., Meadows, E.S., Crowley, T.J. and Doyle III, F.J., Modeling of particule size distribution in emulsion co-polymerization: comparaison with experimental data and parameter sensitivity studies. Comput. Chem. Eng. 26 (2002) 1133–1152. CrossRef
Kalitine, B., Sur la stabilité des ensembles compacts positivement invariants des systèmes dynamiques. RAIRO-Automatique 16 (1982) 275–286.
L.V. Kantorovich and G.P. Akilov, Functional analysis in normed spaces. Pergamon Press, Oxford (1964).
V. Komornik, Exact controllability and stabilization: the multiplier method, Research in Applied Mathematics. Series Editors: P.G. Ciarlet and J.L. Lions, Masson, Paris (1994).
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33–54.
J.P. LaSalle and S. Lefschetz, Stability by Liapunov's direct method with applications. Academic Press, New York (1961).
Lax, P.D. and Phillips, R.S., Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13 (1960) 427–455. CrossRef
T.-T. Li, Global classical solutions for quasilinear hyperbolic systems, Research in Applied Mathematics. John Wiley & Sons, New York (1994).
A. Liapunov, Problème général de la stabilité du mouvement. Princeton University Press, Princeton, New Jersey (1947).
J. Liéto, Génie chimique à l'usage des chimistes. Lavoisier, Paris (1998).
Z.H. Luo, B.Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems with applications. Springer, London (1999).
Nasa Technical Memorandum, Progress Report No. 8, in Proceedings of the twenty-fourth seminar on space flight and guidance theory, NASA George G. Marshall space flight center, Huntsville, Alabama, June 3 (1966).
Outbib, R. and Sallet, G., Stabilizability of the angular velocity of a rigid body revisited. Systems Control Lett. 18 (1992) 93–98. CrossRef
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
Puel, F., Févotte, G. and Klein, J.P., Simulation and analysis of industrial cristallization processes through multidimensional population balance equation. Part 1: A resolution algorithm based on the method of classes. Chem. Engrg. Sci. 58 (2003) 3715–3727. CrossRef
Ramkrishna, D. and Mahoney, A.W., Population balancemodeling. Promise for the future. Chem. Engrg. Sci. 57 (2002) 595–606. CrossRef
Rao, B., Le taux optimal de décroissance de l'énergie dans l'équation de poutre de Rayleigh. C. R. Acad. Sci. Paris 325 (1997) 737–742. CrossRef
Rauch, J. and Taylor, M., Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J. 24 (1974) 79–86. CrossRef
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. CrossRef
Serre, D., Solvability of hyperbolic IBVPS through filtering. Methods Appl. Anal. 12 (2005) 253–266.
Sontag, E. and Sussmann, H., Further comments on the stabilizability of the angular velocity of a rigid body. Systems Control Lett. 12 (1988) 213–217. CrossRef
Szegö, G., On the application of Zubov's method of constructing Liapunov functions for nonlinear control systems. Transaction of ASME Journal of Basic Eng. Series D 85 (1963) 137–142. CrossRef
A. Tchousso, Étude de la stabilité asymptotique de quelques modèles de transfert de chaleur. Ph.D. thesis, University of Claude Bernard - Lyon 1, France (2004).
A. Tchousso and C.Z. Xu, Exponential stability of symmetric hyperbolic systems using Lyapunov functionals, in Proceedings of the 10th IEEE International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland (2004) 361–364.
van der Schaft, A.J., Stabilization of Hamiltonian systems. Nonlinear Anal. Methods Appl. 10 (1986) 1021–1035. CrossRef
Xu, C.Z. and Sallet, G., Exponential stability and transfer functions of a heat exchanger network. Rapport de Recherche de l'INRIA 3823 (1999) 1–21.
Xu, C.Z. and Sallet, G., Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM: COCV 7 (2002) 421–442. CrossRef
C.Z. Xu, J.P. Gauthier and I. Kupka, Exponential stability of the heat exchanger equation, in Proceedings of the European Control Conference, Groningen, The Netherlands (1993) 303–307.
Zuazua, E., Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 109–129. CrossRef