Rapid boundary stabilization of linear distributed systems

SIAM Journal on Control and Optimization

Volumn 35, Issue 5, 1997, Pages 1591-1613

  Komornik, Vilmos ^a  

^a UNIVERSITÉ DE STRASBOURG   (France)

Author keywords

Controllability; Observability; Partial differential equation; Petrovsky system; Stabilizability by feedback; Wave equation

Indexed keywords

CONTROLLABILITY; FEEDBACK; LINEAR CONTROL SYSTEMS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; RICCATI EQUATIONS; SYSTEM STABILITY;

PETROVSKY SYSTEM; WAVE EQUATION;

DISTRIBUTED PARAMETER CONTROL SYSTEMS;

EID: 0031234633     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012996301609     Document Type: Article

References (42)

1. A. V. BALAKRISHNAN, Applied Functional Analysis, Springer-Verlag, New York, 1976.
2. C. BARDOS, G. LEBEAU, AND J. RAUCH, Contrôle et stabilisation dans les problèmes hyperboliques, in Contrôlabilité exacte et stabilisation de systèmes distribuées, Vol. 1, J. L. Lions, Masson, Paris, 1988, Appendix II, pp. 492-537.
3. C. BARDOS, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024-1065.
4. A. BENABDALLAH AND M. LENCZNER, Stabilisation de l'équation des ondes par un contrôle optimal distribué: Estimation du taux de décroissance, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), pp. 691-696.
5. A. BENSOUSSAN, G. DA PRATO, M. C. DELFOUR, AND S. K. MITTER, Representation and Control of Infinite Dimensional Systems I-II, Birkhäuser, Boston, 1992, 1993.
6. G. CHEN, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), pp. 249-274.
7. S. DOLECKI AND D. L. RUSSELL, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), pp. 185-220.
8. F. FLANDOLI, A New Approach to the L-Q-R Problem for Hyperbolic Dynamics with Boundary Control, Lecture Notes in Control and Information Sciences 102, Springer-Verlag, Berlin, Heidelberg, New York, 1987, pp. 89-111.
9. K. D. GRAHAM AND D. L. RUSSELL, Boundary value control of the wave equation in a spherical region, SIAM J. Control Optim., 13 (1975), pp. 174-196.
10. A. HARAUX, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), pp. 457-465.
11. A. HARAUX, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal Math., 46 (1989), pp. 246-257.
12. L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), pp. 443-446.
13. S. JAFFARD, Contrôle interne exact des vibrations d'une plaque rectangulaire, Portugal Math., 47 (1990), pp. 423-429.
14. H. KOCH AND D. TATARU, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations, 20 (1995), pp. 901-937.
15. V. KOMORNIK, Contrôlabilité exacte en un temps minimal, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), pp. 223-225.
16. V. KOMORNIK, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), pp. 197-208.
17. V. KOMORNIK, On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl., 71 (1992), pp. 331-342.
18. V. KOMORNIK, On the zeros of Bessel type functions and applications to exact controllability problems, Asymptotic Analy., 5 (1991), pp. 115-128.
19. V. KOMORNIK, Exact Controllability and Stabilization: The Multiplier Method, Masson-John Wiley, Paris, 1994.
20. V. KOMORNIK, Boundary stabilization, observation and control of Maxwell's equations, Panamer. Math. J., 4 (1994), pp. 47-61.
21. V. KOMORNIK, Boundary Stabilization of Linear Elasticity Systems, Lecture Notes in Pure and Appl. Math. 174, Marcel Dekker, New York, 1995, pp. 135-146.
22. V. KOMORNIK, Stabilisation frontière rapide de systèmes distribués linéaires, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), pp. 433-437.
23. V. KOMORNIK, Stabilisation rapide de problèmes d'évolution linéaires, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), pp. 581-586.
24. V. KOMORNIK AND E. ZUAZUA, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), pp. 33-54.
25. J. LAGNESE, Boundary Stabilization of Thin Plates, SIAM Studies in Appl. Math., SIAM, Philadelphia, 1989.
26. 2(Γ)) boundary terms, Appl. Math. Optim., 10 (1983), pp. 275-286.
27. I. LASIECKA AND R. TRIGGIANI, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Inform. Sci. 164, Springer-Verlag, Berlin, Heidelberg, New York, 1991.
28. G. LEBEAU, Contrôle de l'équation de Schrödinger, J. Math. Pures Appl., 71 (1992), pp. 267-291.
29. E. B. LEE AND L. MARKUS, Foundations of Optimal Control Theory, John Wiley, New York, 1967.
30. J.-L. LIONS, Contrôle des systèmes distribués singuliers, Gauthiers-Villars, Paris, 1983.
31. J.-L. LIONS, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1-68.
32. J.-L. LIONS, Contrôlabilité exacte et stabilisation de systèmes distribués, Vol. 1, Masson, Paris, 1988.
33. J.-L. LIONS AND E. MAGENES, Problèmes aux limites non homogènes et applications I-III, Dunod, Paris, 1968-1970.
34. D. L. LUKES, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), pp. 39-50.
35. P. MARTINEZ, Stabilisation frontière de l'équation des ondes dans des domaines polygonaux, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), pp. 365-370.
36. D. L. RUSSELL, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), pp. 639-739.
37. T. I. SEIDMAN, Private communication, 1995.
38. M. SLEMROD, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), pp. 500-508.
39. T. L. R. TCHEUGOUÉ, Deux remarques sur la stabilisation de l'équation des ondes en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), pp. 585-588.
40. R. TRIGGIANI, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl., 137 (1989), pp. 438-461.
41. E. ZUAZUA, Contrôlabilité exacte en un temps arbitrairement petit de quelques modèles de plaques, in Contrôlabilité exacte et stabilisation de systèmes distribuées, Vol. 1, J. L. Lions, Masson, Paris, 1988, Appendix I, pp. 465-491.
42. E. ZUAZUA, Sur l'optimalité des feedbacks de stabilisation, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), pp. 547-552.