Most of my research is in two areas: Boundary Control Theory for systems governed by partial differential equations, and Computational Quantum Chemistry.

Boundary Control Theory: Many mechanical systems may be mathematically modelled by a system of partial differential equations and it is often important to implement the control of such systems. One aspect of this is boundary control theory, for which control functions influence the system through boundary conditions (a simple example is bringing a vibrating rod to rest by applying an appropriate force or torque at one end of the rod). The mathematics behind this is very deep and often leads to completely new results about partial differential equations. Walter Littman (University of Minnesota) and I have developed a very promising mathematical method for boundary control. We have improved it and have applied it to a number of systems. In this project we wish to continue to apply it to other types of systems. We hope that it will become widely used and note that it has already been used by other researchers who state that other methods will not work for the systems that they are working on.

Another interesting aspect of this research is the effect of nonlinearities. Most of the research involving controllability of partial differential equations is for linear systems. This is because mathematical treatments of even the linear problems are difficult. About five years ago, I started to investigate the geometrically nonlinear beam problem (the nonlinearities are due to the geometry, but not the elasticity). I soon found by a Lyapunov method that classical solutions are stabilisable by the same feedback laws that are used for linear systems. However, the existence of classical solutions to the system needed to be proved. The existence of weak solutions was not too difficult. I have been working on this question and related issues with a PhD student, Gareth Hegarty. The research is of fundamental importance in the area of control theory, because it deals with the question of whether or not the many systems in the literature that are simplified by linearisation are actually stabilisable.

Computational Quantum Chemistry involves solving a partial differential equation (Schrödinger's equation) which models the physics of molecules. This is a difficult task, because the number of independent variables that appear in the partial differential equation is three times the number of electrons that are in the molecule (i.e. there are hundreds of independent variables). Most conventional methods for numerically solving partial differential equations have a practical limitation of at most three or four variables. Thus our ability to be able to solve Schrödinger's equation and thus predict the chemistry of molecules is a remarkable achievement. My main collaborator in this research is Peter Gill, of the School of Chemistry, University of Nottingham. Our current research on this subject deals with a mathematical model of the "bubble picture" of a molecule. The actual wave functions of molecules reveal that electrons are smeared across an entire molecule, rather than belonging to individual atoms within a molecule. However, chemists have traditionally explained chemical reactions and other chemical properties in terms of the bubble model, or ball and stick model, that is commonly used in pictures of molecules. This research, which we have already made good progress on, aims at a quantitative description of such models by seeking the most accurate electron density function that is a sum of spherically symmetric functions centred at the centre of atomic nuclei. We have just submitted a new paper on this with Andrew Gilbert, also at the University of Nottingham.