# Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

Abdoua Tchousso; Thibaut Besson; Cheng-Zhong Xu

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 2, page 403-425
- ISSN: 1292-8119

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topTchousso, Abdoua, Besson, Thibaut, and Xu, Cheng-Zhong. "Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2008): 403-425. <http://eudml.org/doc/90919>.

@article{Tchousso2008,

abstract = {
In this paper we study asymptotic behaviour of distributed parameter systems governed
by partial differential equations (abbreviated to PDE). We first review some recently developed results
on the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionals
we prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDE
systems. Then we apply the result to establish exponential stability of various chemical engineering
processes and, in particular, exponential stability of heat exchangers. Through concrete examples we
show 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,
$1<p\leq \infty$.
},

author = {Tchousso, Abdoua, Besson, Thibaut, Xu, Cheng-Zhong},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hyperbolic symmetric systems; partial differential equations; exponential stability;
strongly continuous semigroups; Lyapunov functionals; heat exchangers; hyperbolic symmetric systems; strongly continuous semigroups},

language = {eng},

month = {5},

number = {2},

pages = {403-425},

publisher = {EDP Sciences},

title = {Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method},

url = {http://eudml.org/doc/90919},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Tchousso, Abdoua

AU - Besson, Thibaut

AU - Xu, Cheng-Zhong

TI - Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/5//

PB - EDP Sciences

VL - 15

IS - 2

SP - 403

EP - 425

AB -
In this paper we study asymptotic behaviour of distributed parameter systems governed
by partial differential equations (abbreviated to PDE). We first review some recently developed results
on the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionals
we prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDE
systems. Then we apply the result to establish exponential stability of various chemical engineering
processes and, in particular, exponential stability of heat exchangers. Through concrete examples we
show 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,
$1<p\leq \infty$.

LA - eng

KW - Hyperbolic symmetric systems; partial differential equations; exponential stability;
strongly continuous semigroups; Lyapunov functionals; heat exchangers; hyperbolic symmetric systems; strongly continuous semigroups

UR - http://eudml.org/doc/90919

ER -

## References

top- F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés. C.R. Acad. Sci. Paris Série I328 (1999) 1015–1020.
- F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Eq.2 (2002) 127–150.
- J. Baillieul and M. Levi, Rotational elastic dynamics. Physica D27 (1987) 43–62.
- S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Comm. Pure Appl. Math.60 (2007) 1559–1622.
- H. Brezis, Analyse fonctionnelle : théorie et applications. Masson, Paris (1983).
- N. Burq and G. Lebeau, Mesure de défaut de compacité, application au système de Lamé. Ann. Sci. École Norm. Sup.34 (2001) 817–870.
- A. Chapelon and C.Z. Xu, Boundary control of a class of hyperbolic systems. Eur. J. Control9 (2003) 589–604.
- 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).
- J.M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Control52 (2007) 2–11.
- 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).
- P. Freitas, Stability results for the wave equation with indefinite damping. J. Diff. Eq.132 (1996) 338–353.
- J.M. Greenberg and T.T. Li, The effect of boundary damping for the quasilinear wave equation. J. Diff. Eq.52 (1984) 66–75.
- C.D. Immanuel, C.F. Cordeiro, S.S. Sundaram, E.S. Meadows, T.J. Crowley and F.J. Doyle III, Modeling of particule size distribution in emulsion co-polymerization: comparaison with experimental data and parameter sensitivity studies. Comput. Chem. Eng.26 (2002) 1133–1152.
- B. Kalitine, Sur la stabilité des ensembles compacts positivement invariants des systèmes dynamiques. RAIRO-Automatique16 (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).
- V. Komornik and E. Zuazua, 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).
- P.D. Lax and R.S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math.13 (1960) 427–455.
- 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).
- R. Outbib and G. Sallet, Stabilizability of the angular velocity of a rigid body revisited. Systems Control Lett.18 (1992) 93–98.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
- F. Puel, G. Févotte and J.P. Klein, 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.
- D. Ramkrishna and A.W. Mahoney, Population balancemodeling. Promise for the future. Chem. Engrg. Sci.57 (2002) 595–606.
- B. Rao, Le taux optimal de décroissance de l'énergie dans l'équation de poutre de Rayleigh. C. R. Acad. Sci. Paris325 (1997) 737–742.
- J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J.24 (1974) 79–86.
- D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev.20 (1978) 639–739.
- D. Serre, Solvability of hyperbolic IBVPS through filtering. Methods Appl. Anal.12 (2005) 253–266.
- E. Sontag and H. Sussmann, Further comments on the stabilizability of the angular velocity of a rigid body. Systems Control Lett.12 (1988) 213–217.
- G. Szegö, On the application of Zubov's method of constructing Liapunov functions for nonlinear control systems. Transaction of ASME Journal of Basic Eng. Series D85 (1963) 137–142.
- 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.
- A.J. van der Schaft, Stabilization of Hamiltonian systems. Nonlinear Anal. Methods Appl.10 (1986) 1021–1035.
- C.Z. Xu and G. Sallet, Exponential stability and transfer functions of a heat exchanger network. Rapport de Recherche de l'INRIA3823 (1999) 1–21.
- C.Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM: COCV7 (2002) 421–442.
- 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.
- E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire10 (1993) 109–129.

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