Resumen
We consider a controlled and observed partial differential equation (PDE) which describes a structural acoustics interaction. Physically, this PDE describes an acoustic chamber with a flexible chamber wall. The control is applied to this flexible wall, and the class of controls under consideration includes those generated by piezoceramic patches. The observation we consider is point measurements of acoustic pressure inside the cavity. Mathematically, the model consists of a wave equation coupled, through boundary trace terms, to a structurally damped plate (or beam) equation, and the point controls and observations for this system are modeled by highly unbounded operators. We analyze the map from the control to the observation, since the properties of this map are central to any control design which is based upon this observation. We also show there exists an appropriate state space , so that if the initial state is in and the control is in L2, then the state evolves continuously in and the observation is in L2. The analysis of this system entails a microlocal analysis of the wave component of the system, and the use of pseudodifferential machinery