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 , coupled to a “parabolic–like” beam equation holding on , 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.
"The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics." Abstr. Appl. Anal. 1 (2) 203 - 217, 1996. https://doi.org/10.1155/S1085337596000103