Control and Numerical Analysis of Nonlinear Intrinsic Shells

Wayne State University, Detroit MI

Investigators

Abstract

This research is in the general area of control theory for PDE models that involve shell (curved surface) dynamics. While a great deal of research has been done for plate and wave dynamics and other so-called flat geometries, very little is known as far as rigorous control theory for shell models. This particularly refers to problems such as controllability, stabilization and optimal control problems. On the other hand, from the point of view of applications this is precisely the more interesting case -- for example an airplane or helicopter cabin is curved, so the structural acoustic case is naturally modeled with shells. While the interest in studying shells need not be defended on physical/application grounds, the mathematical analysis (in particular in the context of control theory) is poorly understood with many challenges. In fact, methods developed for flat geometries are no longer applicable, and approaches such as microlocal analysis or geometric optics had to be introduced in order to resolve the problem. The geometry of the problem is responsible for these problems, and the idea pursued in this research is that a solid understanding of that geometry is the key to surmounting these obstacles. The goal of this project is to capture this geometry through the correct modeling and exploit it in the proofs. The research builds upon recently developed "intrinsic" models due to Michel Delfour and Jean-Paul Zolesio which are coordinate free. This very fact, together with an appropriately developed calculus rooted in geometry, provides a tool for deriving the appropriate inequalities. In previously funded work, models based on the intrinsic geometry have been developed for linear, nonlinear, and thermoelastic shells. Answers to questions such as stabilization, control, and well-posedness for these models have been established, and the linear model has been numerically verified to be accurate. The objective for the current research is to further investigate control and numerical analysis of the nonlinear intrinsic shell model. As such, the following issues will be addressed: Model verification for the nonlinear shell; error analysis in order to derive the optimal rates of convergence for the finite element approximation to the nonlinear shell model; control theory for shells (i.e., issues of stability and/or stabilizibility of the nonlinear model, long time behavior and attractors, and exact controllability from the boundary); and numerical analysis of control problems. In part, the intellectual merit of this work lies in the construction of a complete theory of the control of shell structures. However, the methods involved are themselves of independent interest as very special inverse-type inequalities are derived in the proofs which have applications to other problems as well. In addition, the numerical libraries developed can be applied to many different types of problem. The issue of the shell is an issue of geometry, and the geometrical and topological aspects of this work are very rich. The broader impact of this work will be seen in the integration of education and research, participation of underrepresented groups in mathematics, technology transfer and cross-fertilization with industry.