A Partial Differential Equation Approach to Robust Control Design of Smart Materials and Structures: Theoretical, Computational, and Experimental Aspects
Irena Lasiecka
Department of Mathematics
The present Supplement Request originates in the research setting of the regular ARO Grant, DAA H04-96-1-0059, to generate a program of new directions of research and educational activities of the PI's. It may be subtitled, "The need for Riemann geometry tools in the control theory of partial differential equations describing smart material and structures: theoretical and computational aspects."
The present Supplernent Request is divided into three parts. Part A documents how the more realistic models, dictated by smart material technology, presents a researcher with formidable additional difficulties, which appear to call for altogether new mathernatical tools of investigation. Part B documents, in a well-ordered chain of three classes of control problems, that these new tools are derived froin differential geometry, in particular, Riemann geometry, as they pertain to PDE's problems. Part C elaborates on the need to couple the expected new theoretical results with a numerical analysis theory, followed by actual computations, to be performed with the help of graduate students.
A New Research Directions Arising from the Activities of the Regular ARO Grant and Driven by the Utility and Relevance of this Research Toward Design and Control of Smart Structures
Design and control of smart structures. Most of the control and optirnization problems for linear and non-linear partial differential equations (PDE's) proposed in the original ARO Grant DAA H04-96-1-0059 have their roots, and their primary motivation, in a new technology of recognized national priority both military and civilian--usually referred to as 'the design and control of smart structures.'
Three classes of mathematical problems in the original ARO grant leading to the saure new direction of research. For the purpose of the present Supplement Request, we single out three classes of mathematical problems, ordered by implication, where the original ARO grant is driven by the technology of smart materials and structures, which lead to the saine new direction of research. They are:
(1) The study of continuous observability (which is an inverse problem) and, by duality, of exact controllability for the hyperbolic-like PDE's (or systems), with variable coefficients in the principal part, as well as in the energy level ternis. PDE's arising in applications, such as the technology of smart inaterials, are of variable coefficients. The dynamical property of exact controllability beside being attractive and useful as a property in itself serves also as a critical prerequisite in the study of related optimization problems (reaching a target, while minimizing a functional of the control function and of the corresponding PDE response).
(2) Study of control and optimization for shells, such as they arise in elastic theory, which are systems of coupled PDE's with intrinsically variable (even singular) coefficients in the principal part. Shells of various form and shape are ubiquitous in
technological applications, as in 'real life.'
(3) Study of systems of coupled PDE's, such as they arise in the problem of noise rednction in a `chaniber' with curved walls (i.e., shells), subject to an exerior noise field, by means of control actions delivered through smart materials (piezo-ceramic patches) bonded to the elastic wall.
More information at www.math.virginia.edu
Project Sponsored By: U.S. Dod - Army - Aro
Start Date: 6/1/2002
- End Date: 10/31/2006
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