Abstract
Non-oscillatory systems can be defined as systems whose solutions asymptotically approach the set of equilibria, be it a singleton or a set of higher cardinality. Standard contraction theory is only appropriate to deal with the case of a unique globally asymptotically stable equilibrium, whereas an interesting variant, originally proposed by Prof. J. Muldowney, allows to study them under relaxed technical assumptions. His elegant theory, recently recast under the name of 2-contraction (or k-contraction in the general case), is the focus of our talk. It is based on suitable generalization of the variational equation, which are formulated by taking into account the so called second additive compound of the system’s Jacobian, which allows to study how infinitesimal two-dimensional “area” perturbations propagate along the flow of the system. LMIs or Lyapunov functions can be set-up to test for contraction in this relaxed sense and interesting convergence result be achieved even for systems with multiple equilibria. We provide an introduction to the theory, its fundamental results when it comes to the study of the qualitative asymptotic behaviour of nonlinear systems, as well as applications in the area of chemical reaction networks, chaos control, and small-gain theorems under relaxed conditions.
Biographical Information
David Angeli is currently a Professor in Nonlinear Network Dynamics at the Dept of Electrical and Electronic Engineering of Imperial College London after joining the group in 2008 as a Senior Lecturer. He is also an Associate Professor in the Dept of Information Engineering of University of Florence, Italy. He graduated in Computer Science from Florence University in 1996 and subsequently achieved his PhD in Control Theory from the same University (2000). He has held visiting positions at several institutions, including Rutgers University (New Jersey) and INRIA de Rocquencourt (France). He is a Fellow of the IEEE since 2015 and of the IET since 2018. He has authored more than 100 journal papers in the areas of stability of nonlinear systems, chemical reaction networks, model predictive control and smart grids.