Decay rates of interactive hyperbolic-parabolic PDE models with thermal effects on the interface

Applied Mathematics and Optimization

Volumn 42, Issue 2, 2000, Pages 127-167

  Lasiecka, I ^a   Lebiedzik, C ^a  

^a University of Virginia   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; INTERFACES (MATERIALS); NONLINEAR EQUATIONS; OPTIMIZATION; THERMAL EFFECTS; THERMOELASTICITY; WAVE EQUATIONS;

DECAY RATES; HYPERBOLIC MODELS; LOPATINSKI CONDITION; NEUMANN BOUNDARY CONDITIONS; PARABOLIC MODELS; STRUCTURAL ACOUSTIC MODELS; THERMOELASTIC PLATES; UNIFORM STABILITY;

MATHEMATICAL MODELS;

EID: 0034262502     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002450010010     Document Type: Article

References (50)

1. G. Avalos. The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract Appl. Anal., 1(2):203-217, 1996.
2. G. Avalos and I. Lasiecka. Differential Riccati equation for the active control of a problem in structural acoustics. J. Optim. Theory Appl., 91(3):695-728, 1996.
3. G. Avalos and I. Lasiecka. Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation. SIAM J. Math. Anal., 29(1):155-182, 1998.
4. G. Avalos and I. Lasiecka. The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system. Semigroup Forum, 57:278-292, 1998.
5. A.V. Balakrishman. Applied Functional Analysis. Springer-Verlag, New York, 1975.
6. V. Barbu. Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden, 1976.
7. J. Beale. Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J., 9:895-917, 1976.
8. A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter. Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston, 1992.
9. M. Camurdan. Uniform stability of a coupled structural acoustic system by boundary dissipation. Abstract Appl. Anal., 3(3-4):377-400, 1999.
10. M. Camurdan and G. Ji. Uniform feedback stabilization via boundary moments in a three dimensional structural acoustic model. 37 IEEE CDC Proceedings, 3:2058-2064, 1998.
11. E.F. Crawley and J. de Luis. Use of piezoceramic actuators as elements of intelligent structures. AIAA J., 25(10):1373-1385, 1987.
12. F. Fahroo and C. Wang. A new model for acoustic-structure interaction and its exponential stability. Quart. Appl. Math., 57(1):157-179, 1999.
13. C.R. Fuller, G.P. Gibbs, and R.J.R.J. Silcox. Simultaneous active control of flexural and extensional power flow in beams. J. Intel. Materials Systems Struct., 1, 1990.
14. V. Isakov. Inverse Problems for Partial Differential Equations. Springer-Verlag, New York, 1998.
15. V. Isakov and M. Yamamoto. Carleman estimate with Neumann boundary condition and its application to exact observability. UTMS 99, 41, 1999.
16. V. Komornik. Exact Controllability and Stabilization - the Multipliers Method. Masson, Paris, 1994.
17. J. Lagnese. Boundary stabilization of linear elastodynamic systems. SIAM J. Control, 21:968-984, 1983.
18. J. Lagnese. Boundary Stabilization of Thin Plates. SIAM, Philadelphia, PA, 1989.
19. J. Lagnese. Modeling and stabilization of nonlinear plates. Internat. Ser. Num. Math., 100:247-264, 1991.
20. J. Lagnese. Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary. Nonlinear Anal., 16:35-54, 1991.
21. J. Lagnese and G. Leugering. Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Differential Equations, 91:355-388, 1991.
22. J. Lagnese and J.-L. Lions. Modeling Analysis and Control of Thin Plates. Volume 6 of Research in Applied Mathematics. Masson, Paris, 1988.
23. I. Lasiecka. Existence and uniqueness of the solutions to second order abstract equations with nonlinear and non-monotone boundary conditions. J. Nonlinear Anal., Methods Appl., 23:797-823, 1994.
24. I. Lasiecka. Boundary stabilization of a 3-dimensional structural acoustic model. J. Math. Pure Appl., 78:203-232, 1999.
25. I. Lasiecka. Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation. Comm. Partial Differential Equations, 24(9), 1999.
26. I. Lasiecka. Mathematical Control Theory of Coupled PDE Systems. NSF-CMBS Lecture Notes. SIAM, Philadelphia, PA, 2000.
27. I. Lasiecka and C. Lebiedzik. Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping. Control Cybernet., 28(3):557-581, 1999.
28. I. Lasiecka and C. Lebiedzik. Boundary stabilizibility of nonlinear acoustic models with thermal effects on the interface. C. R. Acad. Sci. Paris, 328(2):187-192, 2000.
29. I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations, 6(3):507-533, 1993.
30. I. Lasiecka and R. Triggiani. Sharp regularity theory for second order hyperbolic equations of Neumann type. Part I. Nonhomogenous data. Ann. Mat. Pura Appl. IV, CLVII:285-367, 1990.
31. I. Lasiecka and R. Triggiani. Sharp regularity theory for second order hyperbolic equations of Neumann type. Part II: General boundary data. J. Differential Equations 94(1): 112-164, 1991.
32. I. Lasiecka and R. Triggiani. Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometric conditions. Appl. Math. Optim., 25:189-224, 1992.
33. I. Lasiecka and R. Triggiani. Sharp trace estimates of solutions to Kirchoff and Euler-Bernoulli equations. Appl. Math. Optim., 28:277-306, 1993.
34. I. Lasiecka and R. Triggiani. Analyticity of thermoelastic semigroups with free boundary conditions. Ann. Scuola Norm. Sup., XXVII:457-487, 1998.
35. I. Lasiecka and R. Triggiani. Control Theory for Partial Differential Equations. Volumes I and II of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
36. I. Lasiecka and R. Triggiani. Structural decomposition of thermoelastic semigroups with rotational forces. Semi-group Forum, 60:16-66, 2000.
37. I. Lasiecka and R. Triggiani. Sharp regularity theory for elastic and thermoelastic Kirchoff equations wih free boundary conditions. Rocky Mountain J. Math., to appear.
38. W. Littman and B. Liu. On the spectral properties and stabilization of acoustic flow. SIAM J. Appl. Math., 59(1): 17-34, 1999.
39. W. Littman and L. Markus. Stabilization of hybrid system of elasticity by feedback boundary damping. Ann. Math. Pura Appl., 152:281-330, 1988.
40. P.M. Morse and K.U. Ingard. Theoretical Acoustics. McGraw-Hill, New York, 1968.
41. D.L. Russell. Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev., 20:639-739, 1978.
42. D.L. Russell. Mathematical models for the elastic beam and their control-theoretic properties. Semigroups Theory and Applications. Pitman Research Notes, 152:177-217, 1986.
43. D.L. Russell. On the positive square root of the fourth order operator. Quart. Appl. Math., 6(7):751-773, 1988.
44. D. Tataru. On the regularity of boundary traces for the wave equations. Ann. Scuola Norm. Sup. (4), 26(1): 189-200, 1998.
45. D. Tataru. Carleman estimates, unique continuation and applications. Preprint, 1999.
46. H.T. Banks, R.J. Silcox, and R.C. Smith. The modeling and control of acoustic/structure interaction problems with piezoceramic actuators: 2d numerical examples. ASME J. Vibrations Acoustics, 2:343-390.
47. I. Lasiecka, R. Triggiani, and X. Zhang. Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot. AMS Contemporary Mathematics, to appear in 2000.
48. C. Lebiedzik. Uniform stability of a coupled structural acoustic system with thermoelastic effects and weak structural damping. AMS Contemporary Mathematics, to appear.
49. C. Lebiedzik. Uniform stability of a coupled structural acoustic system. Dynamics Contin. Discrete Impulsive Systems, 7(3):369-385, 2000.
50. S. Myatake. Mixed problems for hyperbolic equations of second order. J. Math. Kyoto Univ., 130(3):435-487, 1973.