Applied Mathematics & Optimization

, Volume 62, Issue 3, pp 381–410 | Cite as

Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

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Abstract

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.

Keywords

Nonlinear wave equation Westervelt equation Hyperbolic equations Boundary feedback control Absorbing boundary conditions Stabilization 
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References

  1. 1.
    Alabau-Boussouira, F.: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51, 61–105 (2005) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Arosio, A., Spagnolo, S.: Global existence of abstract evolution equations of weakly hyperbolic type. J. Math. Pure Appl. 65, 263–305 (1986) MATHMathSciNetGoogle Scholar
  3. 3.
    Bociu, L., Radu, P.: Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Discrete Contin. Dyn. Syst. (DCDS), Supplements, 60–71 (2009) Google Scholar
  4. 4.
    Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. Electron. J. Differ. Equ. 1998(08), 1–21 (1998) MathSciNetGoogle Scholar
  5. 5.
    Chen, G.: Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain. J. Math. Pures Appl. 58, 249–274 (1979) MATHMathSciNetGoogle Scholar
  6. 6.
    Christov, I., Christov, C.I., Jordan, P.M.: Modeling weakly nonlinear wave propagation. Q. J. Mech. Appl. Math. 60, 473–495 (2007) MATHCrossRefGoogle Scholar
  7. 7.
    Clason, C., Kaltenbacher, B., Veljovic, S.: Boundary optimal control of the Westervelt and the Kuznetsov equation. J. Math. Anal. Appl. 356, 738–751 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dreyer, T., Kraus, W., Bauer, E., Riedlinger, R.E.: Investigations of compact focusing transducers using stacked piezoelectric elements for strong sound pulses in therapy. In: Proceedings of the IEEE Ultrasonics Symposium, pp. 1239–1242 (2000) Google Scholar
  9. 9.
    Farahi, M.H., Rubio, J.E., Wilson, D.A.: The global control of a nonlinear wave equation. Int. J. Control 65(1), 1–15 (1996) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hamilton, M.F., Blackstock, D.T.: Nonlinear Acoustics. Academic Press, New York (1997) Google Scholar
  11. 11.
    Jordan, P.M.: An analytical study of Kuznetsov’s equation: diffusive solitons, shock formation, and solution bifurcation. Phys. Lett. A 326, 77–84 (2004) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kaltenbacher, M.: Numerical Simulations of Mechatronic Sensors and Actuators. Springer, Berlin (2004) Google Scholar
  13. 13.
    Kaltenbacher, B., Lasiecka, I.: Global existence and exponential decay rates for the Westervelt equation. Discrete Contin. Dyn. Syst. (DCDS), Ser. S 2, 503–525 (2009) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaltenbacher, B., Lasiecka, I.: An analysis of nonhomogeneous Kuznetsov’s equation: local and global well-posedness; exponential decay (submitted) Google Scholar
  15. 15.
    Kaltenbacher, B., Lasiecka, I., Veljović, S.: Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data (submitted) Google Scholar
  16. 16.
    Komornik, V.: Contrabilité en un temps minimal. C.R. Acad. Sci. Paris Sér. I Math. 304, 223–225 (1987) MATHMathSciNetGoogle Scholar
  17. 17.
    Komornik, V.: Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29, 197–208 (1991) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method. Masson, Paris, Wiley, Chichester (1994) Google Scholar
  19. 19.
    Komornik, V.: Observability, controllability and stabilizability of linear distributed systems, (lecture series given in September 1996 in the Istituto per le Applicazioni del Calcolo “Mauro Picone” of the Consiglio Nazionale delle Ricerche), private communication Google Scholar
  20. 20.
    Kuznetsov, V.P.: Equations of nonlinear acoustics. Sov. Phys. Acoust. 16(4), 467–470 (1971) Google Scholar
  21. 21.
    Lagnese, J.: Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50, 163–182 (1983) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lasiecka, I.: Mathematical Control Theory of Coupled PDEs. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 75. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002) MATHGoogle Scholar
  23. 23.
    Lasiecka, I., Ong, J.: Global solvability and uniform decays of solutions to quasilinear equations with nonlinear boundary dissipation. Commun. Partial Differ. Equ. 24, 2069–2107 (1999) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lasiecka, I., Triggiani, R.: Regularity of hyperbolic equations under L 2(0,T,L 2(Γ)) boundary terms. Appl. Math. Optim. 10, 275–286 (1983) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lasiecka, I., Triggiani, R.: Exponential uniform energy decay rates of the wave equation in a bounded region with L 2(0,T;L 2(Ω))-boundary feedback in the Dirichlet B.C. J. Differ. Equ. 66, 340–390 (1987) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lasiecka, I., Triggiani, R.: Uniform stabilization of the wave equation with Dirichlet or Neumann-feedback control without geometrical conditions. Appl. Math. Optim. 25, 189–224 (1992) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Encyclopedia of Mathematics and its Applications Series. Cambridge University Press, Cambridge (2000) Google Scholar
  28. 28.
    Lasiecka, I., Lions, J.L., Triggiani, R.: Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65, 149–192 (1986) MATHMathSciNetGoogle Scholar
  29. 29.
    Mordukhovich, B.S., Raymond, J.-P.: Optimal boundary control of hyperbolic equations with pointwise state constraints. Nonlinear Anal. 63(5–7), 823–830 (2005) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Nakao, M.: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equation. Math. Z. 206, 265–276 (1991) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Pazy, A.: Semigroups of Operators and Applications to Partial Differential Equations. Springer, New York (1983) MATHGoogle Scholar
  32. 32.
    Rellich, F.: Darstellung der Eigenwerte von Δu+λ u durch ein Randintegral. Math. Z. 18, 635–636 (1940) CrossRefMathSciNetGoogle Scholar
  33. 33.
    Slemrod, M.: Existence of optimal controls for control systems governed by nonlinear partial differential equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4(1), 229–246 (1974) MathSciNetGoogle Scholar
  34. 34.
    Slemrod, M.: Boundary feedback stabilization for a quasi-linear wave equation. In: Control Theory for Distributed Parameter Systems and Applications. Lecture Notes in Control and Information Sciences vol. 54, pp. 221–237 (1983) Google Scholar
  35. 35.
    Taraldsen, G.: A generalized Westervelt equation for nonlinear medical ultrasound. J. Acoust. Soc. Am. 109, 1329–1333 (2001) CrossRefGoogle Scholar
  36. 36.
    Tartar, L.: Existence globale pour un système hyperbolique semi linéaire de la théorie cinétique des gaz, Séminaire Goulaouic-Schwartz (1975/1976), Equations aux dérivées partielles et analyse fonctionnelle, Exp. No. 1, 11 pp. Centre Math., Ecole Polytech., Palaiseau (1976) Google Scholar
  37. 37.
    Triggiani, R.: Exact boundary controllability of L 2(Ω)×H −1(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems. Appl. Math. Optim. 18, 241–277 (1988) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Triggiani, R.: Wave equation on a bounded domain with boundary dissipation: an operator approach. J. Math. Anal. Appl. 137, 438–461 (1989) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Westervelt, P.J.: Parametric acoustic array. J. Acoust. Soc. Am. 35, 535–537 (1963) CrossRefGoogle Scholar
  40. 40.
    Yamada, Y.: On some quasilinear wave equations with dissipative terms. Nagoya Math. 87, 17–39 (1982) MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of GrazGrazAustria

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