ABSTRACT
A first mathematical (PDE) model for the underlying control problem was provided in [3-4], where the structural acoustic problem is formulated in the setting of C0 semigroups with unbounded control operators. However, relatively little was first known about the mathematical structure and PDE theory of these problems. It quickly became apparent that the properties of the acoustic system and thus of the resulting control depend heavily on the coupling inherent in the model. The unbounded control theory developed for the hyperbolic and parabolic equations separately is of paramount importance here, but the coupling between the equations introduces new phenomena. Though the coupling increases the level of mathematical difficulty, it also serves to propagate stability from the parabolic to the hyperbolic component and allows for control.
2. Background and Literature In the case that the flexible (active) wall is assumed to be structurally damped (Kelvin-Voight damping), we have the situation alluded to above. The coupling of the parabolic (wall) and hyperbolic (wave) equations yields a system whose dynamics are related to an analytic semigroup [1]. Because of the regularizing effect of this analyticity, the gain operator is in fact shown to be bounded. Since the coupling allows for the propagation of some of this regularizing effect onto the wave equation, optimal control with bounded gains for this system is achievable even though the control operator is intrinsically unbounded. This result was proved in [1]. Other possible models for the flexible wall include a higher-order Kirchoff model with boundary (mechanical) dissipation [7], and a model which includes thermal effects on the wall [14]. Each of these systems has different mathematical properties which affect the formulation and resolution of the stabilization and control problems.
