Exact Controllability and Observation of Structural Acoustics and Thermoelastic Systems

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0208121AvalosThis project is concerned with studying exact boundary controllability properties of those systems of coupled partial differential equations (PDE's) which govern structural acoustic flow within a chamber. Exact and null boundary controllability problems for two-dimensional systems of thermoelasticity will also be studied. In part, the work will entail a study of the dual problem; namely, the attainment of related observability inequalities for solutions of homogeneous adjoint equations. In line with the intended engineering applications, the focus will be on situations which allow control of the structural acoustic dynamics on as small a (boundary) control region as possible. Moreover, this project is aimed at finding conditions on the geometry and prescribed controls so that, with control implemented on the flexible portion of the acoustic chamber only, one will have exact controllability of the acoustic flow, for arbitrary initial data of finite energy. It is anticipated that key ingredients in the work will include the following: (i) sharp trace regularity for the wave equation in the absence of the so-called Lopatinski condition (intrinsic to the wave equation under Neumann boundary conditions); (ii) microlocal analytical estimates which will allow the absorption of tangential wave traces by time derivatives on the boundary; and (iii) recent results involving Carleman's estimates for the wave equation with controlled Neumann part of the boundary. In addition, the project will focus on problems of linear and (globally) nonlinear exact controllability for thermoelastic systems. In particular, thermoelastic PDE's will be considered which have their associated (non-Lipschitz) nonlinearities in place; e.g., the von Karman bracket and the quasilinearities which appear in the modeling of extensible plates. This work will attempt to use, in an essential way, the now-known analyticity of linearized thermoelastic models and our recent stability work for uncontrolled (but fully nonlinear) thermoelastic systems. Examples of coupled partial differential equations (PDE's), such as those to be investigated, have long existed in the literature. However, recent innovations in smart material technology, and the potential applications of these innovations within the context of control engineering design, have greatly increased the interest in these PDE models. The project is aimed at obtaining information about certain qualitative properties of these equations, which in turn can be used to design effective control laws for the structures/interactions that these equations govern. For example, structural acoustic PDE's are used to model the interaction of an aircraft cabin's interior acoustic field with the surrounding walls of the cabin. For the benefit of the passengers, it is desirable to negate or control pressure disturbances that act directly on the interior acoustic field. These disturbances typically emanate from outside the cabin environment; e.g., vibrations due to aircraft engine and propeller noise, or effects due to weather turbulence. In practice, engineers attempt to control this external noise by placing piezoelectric actuators/sensors on a portion of the cabin wall, these devices to act in such a way so as to remove, or at least lessen, the harmful acoustic pressure effects. However, the efficacy of this technology is profoundly sensitive to the shape of the cabin, as well as to the particular region of the cabin walls where the actuators are placed. The goals of this project include: (i) the precise mathematical characterization of those cabin geometries for which active control design by piezoelectric actuation is indeed possible; and (ii) when such control design is practicable, the construction of a reliable method to prescribe the amount and region of control actuation which will be necessary to maintain a calm acoustic field within the cabin.