The Control Theory Group has conducted research across a wide array of topics within classical control theory and non-linear partial differential equations. Most recently, members of the group have worked in the following problem areas.

1. Semilinear Elliptic Equations or Systems. To study the computation, analysis and visualization of semilinear elliptic boundary value problems, which arise naturally from applications in astrophysics, reaction-diffusion, and mathematical biology and ecology. To develop various numerical algorithms and use Mefisto Finite-Element package to compute and visualize solutions on complex domains in both 2D and 3D, together with theoretical analysis to detect how a variation of the domain geometry affect the solution profile. (In collaboration with Professor Wei-Min Ni of the University of Minnesota)
2. Chaotic Vibration. The study of chaos or turbulence in nonlinear PDE is a major research area in nonlinear analysis. To study Acoustic chaos to understand chaotic vibration of sound propagation modelled by the wave equation. For example, when a van der Pol nonlinearity is in effect at one endpoint and energy is injected at the other endpoint of a one-dimensional wave equation, several types of routes/sources to/for chaos, including period doubling, homoclinic orbits and Cantor-like invariant sets have been determined and classified. (in collaboration with Sze-Bi Hsu of Tsing Hua University in Taiwan.)
3. Multiple Solution Problem To develop mathematical theory and numerical methods for finding multiple critical points and instability analysis. Multiple solutions with different performance and instability indices exist in many nonlinear problems in natural and social sciences. Finding multiple solutions in a stable numerical way is interesting and important to both theory and applications. We are using Morse theory and other nonlinear functional analysis tools to establish some new local critical point theory from which local minimax numerical methods can be developed to solve multiple solution problems and to do instability analysis.