THE EXPONENTIAL STABILITY OF A COUPLED HYPERBOLIC/PARABOLIC SYSTEM ARISING IN STRUCTURAL ACOUSTICS

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a "parabolic-like" beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic equations.

[1]  Goong Chen,et al.  A Note on the Boundary Stabilization of the Wave Equation , 1981 .

[2]  R. Triggiani,et al.  Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions , 1992 .

[3]  R. Triggiani.  Wave equation on a bounded domain with boundary dissipation: An operator approach☆ , 1989 .

[4]  Harvey Thomas Banks,et al.  Approximation methods for control of acoustic/structure models with piezoceramic actuators , 1991 .

[5]  R. Triggiani,et al.  Uniform stabilization of the wave equation with dirichlet-feedback control without geometrical conditions , 1992 .

[6]  P. Grisvard,et al.  Caractérisation de quelques espaces d'interpolation , 1967 .

[7]  Harvey Thomas Banks,et al.  Feedback control of noise in a 2-D nonlinear structural acoustics model , 1994 .

[8]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[9]  J. Ball.  Strongly continuous semigroups, weak solutions, and the variation of constants formula , 1977 .

[10]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[11]  Irena Lasiecka,et al.  The Strong Stability of a Semigroup Arising from a Coupled Hyperbolic/Parabolic System , 1998 .

[12]  David L. Russel.  Representation and Control of Infinite Dimensional Systems, Vols. 1 and 2 (A. Bensonssan, G. Da Prato, M. Delfour, and S. Mitter) , 1995, SIAM Rev..

[13]  Roberto Triggiani,et al.  PROOF OF EXTENSIONS OF TWO CONJECTURES ON STRUCTURAL DAMPING FOR ELASTIC SYSTEMS , 1989 .

[14]  J. Lagnese.  Decay of solutions of wave equations in a bounded region with boundary dissipation , 1983 .