The effort in this project will be focused on the mathematical analysis and control of partial differential equation (PDE) models which describe certain elastic dynamics which are seen in the natural and man-made world. The elastic evolutionary PDE models under present consideration might also be subjected to influences external to the system; e.g., elastic bodies subjected to damping forces across some boundary interface. In consequence, the governing PDE models we will analyze could conceivably constitute a coupling of PDE dynamics which are quite different in character; e.g., a von Karman dynamical plate PDE coupled to a thermal process would give rise to a coupled PDE system which evinces phenomenological traits of both hyperbolic and parabolic PDE, yet could not be said to be either strictly hyperbolic or parabolic. For such evolution equations, linear and nonlinear, we intend to address the following problems: (1) Exact controllability and general reachability properties of those coupled PDE models which describe the interaction between an elastic structure and a surrounding fluid medium. In line with the physical application, the movements of the elastic body are to be controlled by the indirect means of fluid boundary control. (2) Uniform stability properties of structural acoustic systems. In this situation, acoustic waves, interior to a chamber geometry, are coupled to elastic equations which model the flexural vibrations on a portion of the chamber wall; the elastic component here will manifest some quantifiable measure of damping, from weak viscous to super-strong Kelvin-Voight damping. For these systems, and under appropriate geometrical assumptions, we intend to investigate the possibility that the elastic damping is propagated throughout the entire composite system, to the extent that each component--ostensibly undamped wave as well as structurally damped elastic wall component--decays at some discernible rate. (3) Results concerning the asymptotic behavior of solutions, or flows, of certain nonlinear evolutionary plate PDE systems. In particular, we shall concentrate on those systems which are "non-gradient"; that is, there is no available Lyapunov function on the associated finite energy space which can employed to track the long time behavior of the given, possibly non-dissipative, PDE. It is hoped that our work in this connection will culminate in the stabilization of such nonlinear processes to a global compact attractor.
We believe that the results of this project could give benefit much beyond their intrinsic worth as contributions to the discipline of mathematics. For example, in our aforesaid and intended structural acoustics uniform decay investigation, we anticipate that the stability results will depend critically upon the particular chamber geometry which is in play. As a consequence, we believe our research efforts could give a precise characterization of those structural acoustic geometries which will give rise, in long time, to relatively quiescent interior acoustic fields. Such geometrical situations could then conceivably obviate, or at least lessen, the need for the active engineering control of acoustic noise. Moreover, our intended project work in analyzing the long time behavior of nonlinear evolutionary plate dynamics could have immediate Control Engineering implications: Should we find, for a given nonlinear PDE system, that the corresponding flows converge to a global compact attractor of finite fractal dimension, then conceivably the system could be actively controlled numerically by means of a constructed finite-dimensional feedback. In addition, the research generated by this project will serve as the basis from which we will develop and implement a research program for undergraduates interested in numerical PDEs, and in mathematics generally. In particular, the PIs will run the University of Nebraska-Lincoln "Research Experience for Undergraduates" site in Summer 2013. Within the context of a canonical one dimensional setting, and through the partial agency of a commercial computer algebra software package, we will teach, to our undergraduate participants, aspects of the nonlinear theory developed in the course of our project work. Moreover we will actively involve them in research projects concerning the numerical approximation of the solutions, or flows, which correspond to those one dimensional nonlinear processes which fall under the umbrella of our project work.
