Selected Preprints, etc.

• Nonlinear Stuctures. A description of the research project that my PhD student Gareth Hegarty is now working on. This concerns the boundary feedback stabilisation of nonlinear elastic structures such as beams.
• Boundary Feedback Stabilization of a Vibrating String with an Interior Point Mass. We study the boundary feedback stabilization for a one-dimensional wave equation with an interior point mass. We show that if the initial data belong to a certain invariant subspace of the semigroup of operators that generates the solution of the system, then the energy will decay like C / time. This improves a result of Hansen and Zuazua who consider decay of solutions belonging to the domain of a power of the infinitesimal generator of the semigroup. This appears to be an important prototype of a phenomenon that occurs in more complex sytems.
• On the Interaction between a Group of Unitary Operators and a Projection. This is a completely abstract version of a method of boundary control that can be used for systems governed by PDEs and (and mixtures of PDEs and ODEs).
• A Smoothing Property of a Hyperbolic System and Boundary Controllability. We investigate a method with which one can deduce controllability results from smoothing properties. Previous applications of the method were for partial differential equations like the Euler-Bernoulli Beam Equation (Petrowski-Hyperbolic). In this paper we study the method's applicability to a strictly hyperbolic system by considering the boundary controllability of a vibrating Timoshenko beam with physical characteristics that may vary along the length of the beam. Two cases are considered: A beam which is clamped at one end, the other end being controlled by a torque and transverse force; and a beam which is hinged at one end, where a
  control torque is applied, and free at the other end, where a control force is applied.
• Sobolev Spaces. This is an introduction to Sobolev Spaces, written while I was teaching a graduate level PDE course and Montana State University. Feel free to use it - let me know if you have any comments.