Investigation of Stability and Approximation Issues via Renorming for Distributed Parameter Systems

University Of North Carolina Greensboro, Greensboro NC

Investigators

Abstract

The project is concerned with issues related to exponential and asymptotic stability of solutions to linear and nonlinear distributed parameter systems. Particular attention is focused on the investigation of how stability properties are preserved by finite dimensional semidiscrete approximation schemes and how stability properties are affected by disturbances or by system parameters. The PI studies most issues by making use of an appropriate renorming of the underlying state space. The idea of the renorming method is that while every mathematical model is associated with a natural norm (which measures an important quantity in the model, such as energy), it is often the case that a judicious choice of a new norm can be especially useful for insight into a specific issue such as exponential stability. Frequently a new norm also allows construction of improved Galerkin approximation schemes. For linear models, the PI considers applications including hybrid systems (i.e. coupled dynamics, such as models of thermoelasticity) and systems of delay equations, as well as questions such as the implications of renorming in feedback control problems. For nonlinear models, the renorming is an issue for investigation at the stages of linearization and approximation. Today's scientists and engineers are using increasingly complex and sophisticated mathematical models, and they often require quite accurate answers to difficult and delicate questions. A particularly important issue for many models, and the main focus of this project, is exponential stability. Typical of questions related to exponential stability would be - do vibrations of a mechanical structure (robot arm, smart material actuator, satellite antenna, digital reading device, ...) dissipate and at what rate; does the flow of a fluid (around a wing or rudder, in a mixing process, ...) stay smooth or become turbulent; do computer simulations preserve the stability behavior of the original model; etc. These and other issues will be investigated using a mathematical method known as renorming, which also often leads to the construction of improved computer simulation methods (approximation schemes). Undergraduate students will be actively involved in some lines of research, giving them an exposure to topical and scientifically relevant mathematical models, training in the use of the latest scientific computing algorithms and software, and an appreciation of applied mathematics.