Pointwise pressure observations of a canonical structural acoustics model

In this paper, we study regularity properties of a physically motivated observation associated with a controlled partial differential equation (PDE) model on a rectangular domain. This particular PDE describes the active control of acoustic pressure in a chamber by means of piezoelectric activation; and involves a wave coupled to a structurally damped beam equation, with the coupling being accomplished by boundary trace terms. The observation here consists of pointwise (in space) measurements of the velocity of the wave component, with this velocity representing the acoustic pressure of the system. As defined then, the observation of the system, like the control imput, is modeled by an unbounded operator. The main result of this paper is the discovery of an energy space such that for initial data taken from this space, the desired observation of the acoustic pressure is well defined.