Control problems for systems of strongly coupled partial differential equations with variable coefficients.

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

In recent years, the emerging technology of smart materials and structures has brought to the fore of scientific investigation the pressing need to control, optimize, and stabilize dynamical 'interactive' structures, whose components' behavior is governed by partial differential equations (PDEs). A canonical illustrative example of keen national interest is the noise reduction problem in an acoustic chamber (aircraft's or rotorcraft's cockpit or cabin, etc). This couples the oscillatory behavior of the unwanted acoustic pressure (noise field) within the chamber with the elastic vibrations of a flexible wall of the chamber, possibly reinforced by sandwiched layers, and possibly accounting for thermo-elastic effects. Pairwise sets of piezo-electric patches bonded on the flexible wall, once suitably wired, develop an elastic moment that is meant to dampen out the noise in the acoustic chamber. Mathematically, the acoustic pressure is modeled by a second order scalar hyperbolic equation (wave equation), while the flexible wall is modeled by a plate-like Kirchhoff equation with or without structural damping. In the first case, the plate has a parabolic behavior, in the second a hyperbolic behavior, resulting therefore in either hyperbolic/parabolic coupling, or in hyperbolic/hyperbolic coupling of the overall structure. Two additional key, novel, distinguishing features of the present project are: (i) first, the linear or non-linear PDEs describing the coupled structure have variable coefficients in space, which is always the case when the properties of the medium depend from point to point; (ii) and, moreover, the flexible wall may be curved (rather than flat), and thus modeled by a shell (rather than a plate). Accordingly, differential geometric methods are then proposed in the control theoretic analysis of the overall coupled structure, to cope with these two serious difficulties. The goal is to optimally control - according to a pre-assigned optimality criterion - and asymptotically stabilize the coupled structure. The methodology of this project consists in first establishing a mathematical theory, to be followed next by a numerical analysis thereof, to yield effective and computable algorithms. Recent Federal research has convincingly demonstrated smart materials/structures to be a laboratory reality. This is also confirmed by a Workshop Report for the National Science Foundation entitled: "Rebuilding and enhancing the Nation's infrastructures: a role for intelligent material systems and structures", 1993. These new structural concepts actively damp noise and vibration, suppress flutter at trailing edges of airfoils and enable active twist/camber of both fixed wing and rotorcraft; attenuate or suppress water borne signatures (active acoustic signature control), etc. The transition of novel smart structures into new, truly revolutionary platforms faces many obstacles. The most significant is design optimization: both of the smart structure and its communication and control systems. The present project aims at producing a contribution in this area, based on solid mathematical foundations and analysis.