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

Abstract

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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 . 


How to cite

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Tchousso, 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

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  1. F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés. C.R. Acad. Sci. Paris Série I328 (1999) 1015–1020.  
  2. F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Eq.2 (2002) 127–150.  Zbl1011.35018
  3. J. Baillieul and M. Levi, Rotational elastic dynamics. Physica D27 (1987) 43–62.  Zbl0644.73054
  4. 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.  Zbl1152.35009
  5. H. Brezis, Analyse fonctionnelle : théorie et applications. Masson, Paris (1983).  Zbl0511.46001
  6. N. Burq and G. Lebeau, Mesure de défaut de compacité, application au système de Lamé. Ann. Sci. École Norm. Sup.34 (2001) 817–870.  Zbl1043.35009
  7. A. Chapelon and C.Z. Xu, Boundary control of a class of hyperbolic systems. Eur. J. Control9 (2003) 589–604.  Zbl1293.93101
  8. 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).  
  9. 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.  
  10. R.F. Curtain and H.J. Zwart, An introduction to infinite-dimensional linear systems theory. Springer-Verlag, New York (1995).  Zbl0839.93001
  11. B. d'Andréa-Novel, Commande non linéaire des robots. Hermès (1988).  
  12. P. Freitas, Stability results for the wave equation with indefinite damping. J. Diff. Eq.132 (1996) 338–353.  Zbl0878.35067
  13. J.M. Greenberg and T.T. Li, The effect of boundary damping for the quasilinear wave equation. J. Diff. Eq.52 (1984) 66–75.  Zbl0576.35080
  14. 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.  
  15. B. Kalitine, Sur la stabilité des ensembles compacts positivement invariants des systèmes dynamiques. RAIRO-Automatique16 (1982) 275–286.  Zbl0493.93039
  16. L.V. Kantorovich and G.P. Akilov, Functional analysis in normed spaces. Pergamon Press, Oxford (1964).  Zbl0127.06104
  17. 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).  Zbl0937.93003
  18. V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl.69 (1990) 33–54.  Zbl0636.93064
  19. J.P. LaSalle and S. Lefschetz, Stability by Liapunov's direct method with applications. Academic Press, New York (1961).  Zbl0098.06102
  20. P.D. Lax and R.S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math.13 (1960) 427–455.  Zbl0094.07502
  21. T.-T. Li, Global classical solutions for quasilinear hyperbolic systems, Research in Applied Mathematics. John Wiley & Sons, New York (1994).  
  22. A. Liapunov, Problème général de la stabilité du mouvement. Princeton University Press, Princeton, New Jersey (1947).  
  23. J. Liéto, Génie chimique à l'usage des chimistes. Lavoisier, Paris (1998).  
  24. Z.H. Luo, B.Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems with applications. Springer, London (1999).  Zbl0922.93001
  25. 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).  
  26. R. Outbib and G. Sallet, Stabilizability of the angular velocity of a rigid body revisited. Systems Control Lett.18 (1992) 93–98.  Zbl0743.93082
  27. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).  Zbl0516.47023
  28. 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.  
  29. D. Ramkrishna and A.W. Mahoney, Population balancemodeling. Promise for the future. Chem. Engrg. Sci.57 (2002) 595–606.  
  30. 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.  
  31. J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J.24 (1974) 79–86.  Zbl0281.35012
  32. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev.20 (1978) 639–739.  Zbl0397.93001
  33. D. Serre, Solvability of hyperbolic IBVPS through filtering. Methods Appl. Anal.12 (2005) 253–266.  Zbl1114.35118
  34. 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.  Zbl0675.93064
  35. 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.  
  36. 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).  
  37. 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.  
  38. A.J. van der Schaft, Stabilization of Hamiltonian systems. Nonlinear Anal. Methods Appl.10 (1986) 1021–1035.  Zbl0613.93049
  39. 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.  
  40. C.Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM: COCV7 (2002) 421–442.  Zbl1040.93031
  41. 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.  
  42. E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire10 (1993) 109–129.  Zbl0769.93017

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