ABSTRACT
In this article we intend to provide, within the limits of space allowed, a review of a selection of recent results on control theoretic and dynamical properties concerning structurally acoustic (noise reduction) problems in an acoustic chamber. Two greatly distinct models — with, accordingly, vastly different corresponding theories — will be considered: (i) a system with parabolic-hyperbolic coupling; and (ii) a system with hyperbolic-hyperbolic coupling. The article is organized as follows: Sections 7.1 and 7.3 review the two systems of coupled PDE describing an acoustic chamber subject to unwanted noise (acoustic pressure), with a flat elastic wall (The cases involving curved walls, modeled by shell equations, are discussed in [30, 24, 29]). The elastic wall is assumed to be of Euler-Bemoulli or Kirchhoff type, with Kelvin Voight-type of damping in Section 7.1, while it is assumed to be modelled by an undamped Kirchhoff equation (accounting for rotational forces)
in Section 7.3. Mathematically, the acoustic wave (unwanted noise) is modelled by the wave equation within the acoustic chamber; the Kelvin-Voight damped elastic equation is parabolic; while the elastic undamped Kirchhoff equation is hyperbolic. Thus, Section 7.1 deals with a parabolic-hyperbolic coupling (model (i) above), while Section 7.3 focuses on a hyperbolic-hyperbolic coupling (model (ii) above). Issues of well-posedness and regularity of the overall coupled dynamics are dealt with in Section 7.1 for the parabolic-hyperbolic case, and in Section 7.3 in the hyperbolic-hyperbolic case. Section 7.2 introduces and solves the corresponding optimal control problem for the parabolic-hyperbolic coupling, in the presence of a deterministic disturbance, by means of a corresponding Riccati Equation. The same problem is treated in Section 7.4 for the hyperbolic-hyperbolic system, whose regularity — established in Section 7.3-permits one to fall into established abstract theory. As noted in the opening statement, it should be emphasized at the outset that regularity results as well as properties of the gain operators corresponding to optimal control problems are drastically different for these two cases. While in the case of hyperbolic-parabolic coupling there is a transfer of regularity properties from the parabolic component onto the entire structure, yielding additional regularity of the gain operators, this is not the case for the hyperbolic-hyperbolic coupling. This marked difference in the corresponding theories between these two cases has a critical bearing on the practical implementation of control algorithms and related numerical schemes leading to an effective computation of control laws.
