Uniform stabilization of a nonlinear structural acoustic model with a timoshenko beam interface

Mathematical Methods in the Applied Sciences

Volumn 29, Issue 15, 2006, Pages 1749-1766

  Grobbelaar Van Dalsen, M ^a  

^a UNIVERSITY OF PRETORIA   (South Africa)

Author keywords

Feedback boundary damping; Nonlinear; Shear effects; Structural acoustic model; Timoshenko beam; Uniform stabilization

Indexed keywords

DAMPING; ELASTIC WAVES; FEEDBACK; GASES; LIGHT; LINEAR EQUATIONS; NONLINEAR CONTROL SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS; SHEAR STRESS;

FEEDBACK BOUNDARY DAMPING; SHEAR EFFECTS; STRUCTURAL ACOUSTIC MODEL; TIMOSHENKO BEAM; UNIFORM STABILIZATION;

ACOUSTIC NOISE;

EID: 33748881451     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.737     Document Type: Article

References (32)

1. Grobbelaar-Van Dalsen M. Uniform stabilization of a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam. Journal of Mathematical Analysis and Applications 2006, in press.
2. Reissner E. On the theory of bending of elastic plates. Journal of Mathematics and Physics 1944; 23:184-191.
3. Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine 1921; 41:744-746.
4. Arosio A. On the non-linear Timoshenko-Kirchoff beam equation. Chinese Annals of Mathematics, Series B 1999; 20(4):495-506.
5. Benham PP, Crawford RJ, Armstrong CG. Mechanics of Engineering Materials (2nd edn). Prentice-Hall: London, 1996.
6. Lagnese JE, Leugering G. Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. Journal of Differential Equations 1991; 91:355-388.
7. Lasiecka I. Uniform stabilizability of a full Von Kármán system with nonlinear feedback. SIAM Journal on Control and Optimization 1998; 36:1376-1422.
8. Avalos G. The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis 1996; 1(2):203-217.
9. Avalos G, Lasiecka I. Differential Riccati equation for the active control of a problem in structural acoustics. Journal of Optimization Theory and Applications 1996; 91(3):695-728.
10. Avalos G, Lasiecka I. The strong stability arising from a coupled hyperbolic/parabolic system. Semigroup Forum 1998; 57:278-292.
11. Lasiecka I. Mathematical control theory in structural acoustic problems. Mathematical Methods in the Applied Sciences 1998; 8(7):1119-1153.
12. Lasiecka I. Boundary stabilization of a 3-dimensional structural acoustic model. Journal de Mathematiques Pures et Appliquees 1999; 78:203-232.
13. Lasiecka I, Triggiani R. Feedback noise control in an acoustic chamber: mathematical theory. Stability and Control Theory Methods and Applications, vol. 12. Gordon and Breach Science Publishers: London, 1999.
14. Lasiecka I, Lebiedzik C. Asymptotic behaviour of nonlinear structural acoustic interactions with thermal effects. Nonlinear Analysis 2002; 49:703-735.
15. Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories II. Cambridge University Press: Cambridge, MA, 2000.
16. Howe MS. Acoustics of Fluid-Structure Interaction. Cambridge University Press: Cambridge, MA, 1998.
17. Tucsnak M. On an initial and boundary value problem for the nonlinear Timoshenko beam. Anais da Academia Brasileira de Ciencias 1991; 63:115-125.
18. Aassila M. Stabilization of a nonlinear Timoshenko beam. Zeitschrift für Angewandte Mathematik und Physik 2002; 53:747-768.
19. Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1951; 18:31-38.
20. Lagnese JE, Leugering G, Schmidt EJPG. Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Birkhauser: Boston, 1994.
21. Lagnese JE, Lions J-L. Modelling Analysis and Control of Thin Plates. Masson: Paris, Milan, 1989.
22. Balakrishnan AV. Applied Functional Analysis (2nd edn). Springer: Berlin, 1981.
23. Barbu V. Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff: Leiden, 1976.
24. Lasiecka I. Mathematical Control Theory of Coupled PDE's. NSF-CBMS Lecture Notes. SIAM: Philadelphia, PA, 2002.
25. Lasiecka I, Tataru D. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differential and Integral Equations 1993; 6:507-533.
26. Lions JL, Magenes E. Nonhomogeneous Boundary Value Problems and Applications I. Springer: New York, 1972.
27. Tanabe. Equations of Evolution. Pitman: Melbourne, 1979.
28. Lasiecka I, Triggiani R. Exact controllability of the wave equation with Neumann boundary control. Applied Mathematics and Optimization 1989; 19:243-290.
29. Lasiecka I, Triggiani R. Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Applied Mathematics and Optimization 1992; 25:189-224.
30. p(0, T; B). Annali di Matematica Pura ed Applicata 1987; 146(4):65-96.
31. Isakov I. Inverse Problems for Partial Differential Equations. Springer: New York, 1998.
32. Lasiecka I, Triggiani R. Control Theory for Partial Differential Equations: Continuous and Approximation Theories I. Cambridge University Press: Cambridge, MA, 2000.