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Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface. (English. Abridged French version) Zbl 0982.93062

The coupling of acoustic waves with vibrating membranes or other structural elements has been the subject of several engineering papers, for example [G. A. Baker, Kriegsmann and E. L. Reiss, J. Acoust. Soc. Am. 83, 2 (1988)], or of mathematical studies including [G. Avalos, Abstr. Appl. Anal. 1, 203-217 (1996)] and several publications of the first author, listed in the references of the paper under review. Here, the authors study the three-dimensional acoustic wave \(z_{tt}= c^2\Delta z\) in a spatial domain \(\Omega\) coupled with a thermoelastic plate. \(t\) is time and \(z\) is a variable whose physical interpretation can be deciphered from the fact that \(\rho z_t\) is the acoustic pressure, where \(\rho\) is the density of the fluid. On a part of the boundary (we omit the details) the authors apply the control \(g(\rho z_t)\), and assume validity of the dynamic equation of the plate wall \(w_{tt}- \gamma\Delta w_{tt}+\Delta^2 w=- \Delta\theta- \rho z_t+ [F(w), w]\), \(\gamma>0\), with Fourier’s law \(\theta_t- \Delta\theta=\Delta w_t\), and with appropriate boundary conditions, implying among other things that the plate is held rigidly on its boundary. The readers should observe the importance of the term \(\gamma\Delta w_{tt}\), even though it is supposed to be “small”. With it the plate equation is hyperbolic, but, when \(\gamma= 0\), solutions may become analytic. \([A,B]\) is the von Kármán product given by \([A,B]= A_{xx} B_{yy}+ B_{xx} A_{yy}- 2A_{xy} B_{xy}\). The reviewer dislikes this use of the symbol \([A,B]\), which commonly denotes the commutator. In the old-fashioned engineering papers, it was denoted by \(\blacklozenge(A,B)\). It arises in theories of plates and shells. The derivation of a “natural” Hilbert space for vibrating thin plates containing the von Kármán product can be found in the reviewer’s article [SIAM J. Control 8, 273-304 (1970; Zbl 0198.20003)].
The authors prove the existence of a unique finite energy solution, which is continuous in time, and of uniform stabilizability, that is decaying to zero for every weak solution.

MSC:

93D15 Stabilization of systems by feedback
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
93C20 Control/observation systems governed by partial differential equations
74H20 Existence of solutions of dynamical problems in solid mechanics
74K20 Plates

Citations:

Zbl 0198.20003
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