modules

Lyapunov theory is a landmark in the stability of dynamical systems and differential equations which has profoundly influenced both significant mathematical results and important applications. This theory is built around a study of energy-like Lyapunov functions of a system. Such functions are first motivated via simple examples and then developed into a powerful tool for analysis and control of nonlinear systems. We study stability types of equilibria and basins of attraction. Rate of change of energy is manipulated via feedback control to force equilibria to have the desired qualitative properties. Applications to mechanical, bio-chemical and economic systems will be developed. Feedback design techniques such as recursive back-stepping and adaptation are studied.  Energy-like functions play a key role in the qualitative study of the dynamical behaviour of nonlinear systems, replacing algebraic tools like eigenvalues so important for linear systems. Mechanical systems and electrical circuits have naturally defined energy. Energy can be manipulated via external control and especially feedback control. The module will develop a conceptual framework interwoven with several case studies. The emphasis for the mathematics is the application of the theory, not so much in the development of the theory. Case studies will include examples such as inverted pendula, rotating bodies, bio-reactors, etc. On this module, there is ample opportunity for use of computer software (e.g. Maple and similar packages). You will find out how the need to choose suitable Lyapunov functions or stabilising feedbacks lends itself for developing creative mathematical processes and intuition.  Pre-requisite: MTH2003 Differential Equations, MTH2005 Modelling: Theory & Practice or equivalent

_{Please note that all modules are subject to change, please get in touch if you have any questions about this module.}