Stabilization of the wave equation by on-off and positive-negative feedbacks

Patrick Martinez; Judith Vancostenoble

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 7, page 335-377
  • ISSN: 1292-8119

Abstract

top
Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback a ( t ) u t . We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0,T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: a ( t ) = a 0 > 0 on (0,T), and a ( t ) = - b 0 < 0 on (T,qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual “optimal time condition” in the locally distributed case. These new inequalities provide also new exact controllability results.

How to cite

top

Martinez, Patrick, and Vancostenoble, Judith. "Stabilization of the wave equation by on-off and positive-negative feedbacks." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 335-377. <http://eudml.org/doc/90626>.

@article{Martinez2010,
abstract = { Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a(t)u_t$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0,T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: $a(t) = a_0 >0$ on (0,T), and $a(t) = -b_0 <0 $ on (T,qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual “optimal time condition” in the locally distributed case. These new inequalities provide also new exact controllability results. },
author = {Martinez, Patrick, Vancostenoble, Judith},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Damped wave equation; asymptotic behavior; on-off feedback; congruences; observability inequalities.; damped wave equation; congruence properties; on-off feedback; observability inequalities; exact controllability},
language = {eng},
month = {3},
pages = {335-377},
publisher = {EDP Sciences},
title = {Stabilization of the wave equation by on-off and positive-negative feedbacks},
url = {http://eudml.org/doc/90626},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Martinez, Patrick
AU - Vancostenoble, Judith
TI - Stabilization of the wave equation by on-off and positive-negative feedbacks
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 335
EP - 377
AB - Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a(t)u_t$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0,T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: $a(t) = a_0 >0$ on (0,T), and $a(t) = -b_0 <0 $ on (T,qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual “optimal time condition” in the locally distributed case. These new inequalities provide also new exact controllability results.
LA - eng
KW - Damped wave equation; asymptotic behavior; on-off feedback; congruences; observability inequalities.; damped wave equation; congruence properties; on-off feedback; observability inequalities; exact controllability
UR - http://eudml.org/doc/90626
ER -

References

top
  1. Z. Artstein and E.F. Infante, On the asymptotic stability of oscillators with unbounded damping. Quart. Appl. Math.34 (1976) 195-199.  Zbl0336.34048
  2. C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim.30 (1992) 1024-1065.  Zbl0786.93009
  3. C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue (1989) 11-31.  Zbl0673.93037
  4. A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations. Comm. Pure Appl. Math.33 (1980) 707-725.  Zbl0438.35043
  5. A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping. J. Differential Equations161 (2000) 337-357.  Zbl0960.35058
  6. C. Castro and S.J. Cox, Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim.39 (2001) 1748-1755.  Zbl0983.35095
  7. P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping. J. Differential Equations132 (1996) 338-352.  Zbl0878.35067
  8. A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Publications du Laboratoire d'Analyse numérique. Université Pierre et Marie Curie (1988).  
  9. A. Haraux, A generalized internal control for the wave equation in a rectangle. J. Math. Anal. Appl.153 (1990) 190-216.  Zbl0719.49008
  10. W.A. Harris Jr., P. Pucci and J. Serrin, Asymptotic behavior of solutions of a nonstandard second order differential equation. Differential Integral Equations6 (1993) 1201-1215.  Zbl0780.34038
  11. L. Hatvani, On the stability of the zero solution of second order nonlinear differential equations. Acta Sci. Math.32 (1971) 1-9.  Zbl0216.11704
  12. L. Hatvani and V. Totik, Asymptotic stability of the equilibrium of the damped oscillator. Differential Integral Equation6 (1993) 835-848.  Zbl0777.34036
  13. L. Hatvani, T. Krisztin, V. Totik and Vilmos, A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. Differential Equations119 (1995) 209-223.  Zbl0831.34052
  14. S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations145 (1998) 184-215.  Zbl0920.35029
  15. V. Komornik and E. Zuazua A direct method for boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54.  
  16. V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. John Wiley, Chicester and Masson, Paris (1994).  Zbl0937.93003
  17. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations50 (1983) 163-182.  Zbl0536.35043
  18. J. Lagnese, Note on boundary stabilization of wave equation. SIAM J. Control Optim.26 (1988) 1250-1256.  Zbl0657.93052
  19. I. Lasiecka and R. Triggiani, Uniform exponential decay in a bounded region with L 2 ( 0 , T ; L 2 ( Σ ) ) -feedback control in the Dirichlet boundary condition. J. Differential Equations 66 (1987) 340-390.  Zbl0629.93047
  20. I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim.25 (1992) 189-224.  Zbl0780.93082
  21. J.-L. Lions, Contrôlabilité exacte de systèmes distribués. C. R. Acad. Sci. Paris302 (1986) 471-475.  Zbl0589.49022
  22. J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. Masson, RMA 8 (1988).  Zbl0653.93002
  23. J.-L. Lions, Exact controllability, stabilization adn perturbations for distributd systems. SIAM Rev.30 (1988) 1-68.  
  24. S. Marakuni, Asymptotic behavior of solutions of one-dimensional damped wave equations. Comm. Appl. Nonlin. Anal.1 (1999) 99-116.  
  25. P. Martinez, Precise decay rate estimates for time-dependent dissipative systems. Israël J. Math.119 (2000) 291-324.  Zbl0963.35031
  26. P. Martinez and J. Vancostenoble, Exact controllability in ``arbitrarily short time" of the semilinear wave equation. Discrete Contin. Dynam. Systems (to appear).  Zbl1026.93008
  27. M. Nakao, On the time decay of solutions of the wave equation with a local time-dependent nonlinear dissipation. Adv. Math. Sci. Appl.7 (1997) 317-331.  Zbl0880.35076
  28. K. Petersen, Ergodic Theory. Cambridge University Press, Cambridge, Studies in Adv. Math. 2 (1983).  
  29. P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math.170 (1993) 275-307.  Zbl0797.34059
  30. P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, II. J. Differential Equations113 (1994) 505-534.  Zbl0814.34033
  31. P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. Anal.25 (1994) 815-835.  Zbl0809.34067
  32. P. Pucci and J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems. Comm. Pure Appl. Math. XLIX (1996) 177-216.  Zbl0865.35089
  33. P. Pucci and J. Serrin, Local asymptotic stability for dissipative wave systems. Israël J. Math.104 (1998) 29-50.  Zbl0924.35085
  34. R.A. Smith, Asymptotic stability of x''+a(t)x'+x=0. Quart. J. Math. Oxford (2)12 (1961) 123-126.  Zbl0103.05604
  35. L.H. Thurston and J.W. Wong, On global asymptotic stability of certain second order differential equations with integrable forcing terms. SIAM J. Appl. Math.24 (1973) 50-61.  Zbl0279.34041
  36. J. Vancostenoble and P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks. SIAM J. Control Optim.39 (2000) 776-797.  Zbl0984.35029
  37. E. Zuazua, An introduction to the exact controllability for distributed systems, Textos et Notas 44, CMAF. Universidades de Lisboa (1990).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.