The objective for this project is to investigate issues of modeling, stability, and control for a new partial differential equation shell model, based on intrinsic geometry, which includes nonlinear effects and thermal effects. In contrast to traditional coordinate-based models, the intrinsic geometric (coordinate-free) model facilitates analysis for stabilization and optimal control of the system by exploiting the special geometry of the shell. This project will explore (i) modeling of a nonlinear and thermoelastic shell based on intrinsic geometry, the equations of which appear analogous to those of the full von Karman system of linear elasticity; (ii) analysis of the resultant wellposedness issues of the uncontrolled (free) model, which is specifically of interest in the context of nonlinear models; (iii) control of the model -- both controllability and stabilizibility issues are of paramount importance; and (iv) numerical analysis as a means to verify the modeling. These goals will be accomplished through the adaptation of mathematical techniques that have been developed for the corresponding `flat' case. These are highly nontrivial adaptations because of the fact that variable coefficients are involved. The adaptations are made possible through the use of the intrinsic geometry to implicitly take advantage of the special form of those coefficients and the known relationships among the geometric objects that appear in the equations.
Modeling and control of shell structures is important in many fields, including design of automobiles, aircraft, and submarines, manufacturing of industrial equipment, and medical research. A thorough understanding of the nature and behavior of the partial differential equations governing shell structures is essential for development of effective control strategies and efficient numerical simulation of shell properties and behavior. This research project aims to advance understanding of these equations, with the long-range goal of improving technology for analyzing industrial problems that require control of shell structures.
