Mathematical Analysis and Control of Interactive Partial Differential Equations

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The project research will focus on the development of a robust control theory for those interactive partial differential equation (PDE) systems which model certain control engineering and physical phenomena. Such systems comprise a coupling of distinct PDE dynamics, each of these typically evolving on its own domain with boundary, and with the coupling typically being accomplished via the various boundary interfaces. The project investigator shall consider the problem of uniform stabilization for a fluid-structure interaction in three dimensions; this PDE constitutes a coupling of Stokes flow and a system of elasticity. In particular, the investigator will attempt to show that the asymptotic behavior of such fluid-structure PDEs can be uniformly controlled by boundary functions, enacted on the boundary of the structure (or possibly of the fluid, although boundary control via the structure has a direct pharmacological interpretation). Having gained some understanding of the boundary control process for this linearized dynamics, the research will in turn be directed to full-blown, nonlinear fluid-structure dynamics, this involving a coupling of the Navier-Stokes equations with systems of elasticity. In addition, research efforts will be made to understand controllability and optimal control theoretic properties of so-called composite "sandwich" plate models, three dimensional structural acoustic systems, and thermoelastic PDEs which come into contact with a moving obstacle and for which there will be a nonlinear generation of frictional heat.The principal intent of this research is to gain a large and precise understanding of those partial differential equations (PDE's) which fall within the class of so-called hybrid structures; such understanding will eventually have implications in control engineering and design. A hybrid structure--which typically describes some given physical phenomenon--is an equation which comprises two or more distinct PDE dynamics, usually by means of some sophisticated coupling mechanism which is intrinsic to the underlying physical modelling. These coupled systems have attracted, and will probably continue to attract, large interest within the mathematical community, inasmuch as such equations model physical interactions which are observed in the realms of science and engineering: specific examples include those under consideration in this project; namely, fluid-structure interactions, structural acoustic interactions, thin plates under the influence of thermal effects and frictional contact, and the "sandwich" plate models used to describe certain structures in material sciences. Qualitative information gleaned from a mathematical study of such hybrid PDEs can in fact have broader impacts upon other disciplines. For example, part of the project will deal with a three dimensional structural acoustic PDE, as it appears in the modelling of certain engineering design methodologies for the control of external engine noise entering the fuselage of an aircraft. The control design process involves the placement of piezoelectric devices on part of the aircraft wall; a voltage is subsequently applied through these devices to attenuate said noise. The principal investigator is aiming for results for three dimensional structural acoustic systems which will suggest the precise configuration the fuselage should take, so as to attain optimal noise attenuation.