Active suppression of noise in a 3-D structural acoustic chamber with curved walls

The mathematical model which describes the active boundary point control of acoustic pressure in a 3-D chamber is considered, in which the inherent control operator has a high degree of unboundedness. This unboundedness of the control operator is far from an academic contrivance, inasmuch as this operator quantity mathematically describes the point control actions employed in current smart material technology. The modeling partial differential equation (PDE) is a coupling of the wave equation with a system of shallow shell equations, and as such the PDE manifests both hyperbolic and parabolic-like dynamics. Furthermore, the coupling is accomplished through boundary “traces”, a circumstance which greatly complicates the analysis. The present work culminates in the construction and rigorous justification of an optimal feedback control which minimizes a given quadratic index, and which can be represented by a suitable Riccati equation. Despite the unboundedness of the controls, it is shown that the resulting “gain” operator is bounded. In developing the theory, a careful analysis of the hyperbolic component of the dynamics is undertaken, an analysis which considers the propagation of singularities and optimal trace behavior of the solutions. This scrutiny of the hyperbolic part is followed by an invocation of those techniques derived for the treatment of analytic systems, those techniques applied here to the shell component of the PDE