5. Structural Acoustic Control Problems: Semigroup and PDE Models

5.1 Orientation The next three chapters deal with the mathematical optimal control theory of linear structural acoustic problems. Physical motivation for studying this kind of problem comes from a variety of engineering applications that arise, for example, in the context of controlling the pressure in a helicopter's cabin or in reducing unwanted cabin noise generated by some exterior field. There is substantial engineering literature dealing with the practical aspects of structural acoustic problems (see [149, 52, 53, 56] and the references therein). The PDE equations describing acoustic interactions have been known for a long time (e.g., [166, 24]). However, the first mathematical (PDE) model for the underlying control problem was provided in [16, 17, 15], where the structural acoustic problem is formulated in the setting of C0 semigroups with unbounded control operators. References [16, 15, 18] also provide several finite-dimensional numerical approximations. However, until recently, relatively little was known about the mathematical properties of these models. In particular, a PDE theory giving useful qualitative and quantitative information about these problems was lacking. This is hardly surprising, inasmuch as the mathematical control theory of PDE models under the influence of unbounded control operators (which model boundary and point control actions) was developed only recently (see, e.g., the books [129, 28, 140] and references therein). Since the stated structural acoustic problem is described by a coupled system of PDEs composed of a parabolic equation coupled with a hyperbolic equation, it is reasonable to speculate that the already-available control theoretic results for (disparate) hyperbolic and parabolic dynamics should be of use in the course of the analysis. Indeed, the now well-developed theory for single equations [129, 28, 140] has been instrumental, particularly in the last several years, in spurring interest and initiating progress toward the development of a mathematical control theory for interactive structures.