Meeting ID: 910 7928 6959
The analysis and control of infinite-dimensional dynamical systems has many complications that are not present for finite-dimensional problems. There has been substantial interest in recent years in stabilizing equilibria in delay equations using impulses: short-scale bursts that alter the system state very abruptly and are modeled as occurring in zero time. Nearly all published sufficient conditions for stabilization using impulses are stated in terms of linear matrix inequalities and are derived using Lyapunov functionals. I will refer to the problem of stabilizing a delay equation using impulses as the impulsive stabilization problem.
In this talk I will present an alternative approach to the impulsive stabilization problem that avoids the use of Lyapunov functionals. The key ingredients are Floquet theory and invariant manifold theory. The former provides a faithful description of dynamics of periodic linear systems, while the latter can be used to quantify how perturbations to a system alter its behaviour near an equilibrium. The talk will be divided into a few parts. First I will give a brief introduction to some basics of delay differential equations. I will then overview the impulsive stabilization problem before developing the theoretical tools needed to explain how my new stabilization methodology works. I will conclude with some examples.
Bio: Kevin Church received his M.Sc. in Mathematics from the University of Ottawa in 2014. He then completed a Ph.D. in Applied Mathematics in 2019 under the supervision of Xinzhi Liu and Jun Liu at the University of Waterloo. His dissertation was on invariant manifold theory for impulsive functional differential equations, with applications to mathematical biology and control. He is currently a NSERC Postdoctoral Fellow at McGill University, working in the group of Jean-Philippe Lessard on computer-assisted proofs in dynamical systems.