Plenary speakers

Bifurcation analysis of systems with delays: what can it do for you?
Given a nonlinear delay differential equation (DDE) arising in some modelling context, how can one efficiently characterise its behaviour in dependence on model and control parameters? The answer is: by employing advanced tools from bifurcation theory as implemented, for example, in the package DDE-BIFTOOL. The starting point for any such study is the continuation of steady states and periodic orbits in a single parameter while monitoring their stability properties. In this way, their bifurcations of codimension one are detected, which can then be continued as curves in a two-parameter plane, while checking for bifurcation points of codimension two. As we will demonstrate, this dynamical systems approach allows one to assemble the ‘web’ or ‘roadmap’ of bifurcations in a parameter plane of interest — for wide classes of DDEs, including those with state-dependent delays.

Bernd Krauskopf is Professor of Applied Mathematics at The University of Auckland. He received an MSc from RWTH Aachen and then a PhD from the University of Groningen. He held temporary positions at Cornell University and Amsterdam before joining the University of Bristol in 1998, where he worked until joining The University of Auckland in 2011. Professor Krauskopf’s research is in dynamical systems theory and its applications, and he has published over 200 academic journal papers. He made fundamental contributions to theory and led impactful research programmes that introduced new techniques to solve real-world problems, including determining the observable dynamics of prototypical laser systems, analysing and improving aircraft ground manoeuvring, and understanding feedback mechanisms in control and climate modelling. Professor Krauskopf has been PI on large grants, including an Advanced Research Fellowship, the Dodd-Walls Centre of Research Excellence and a Marsden grant, and received substantial industry funding.

Lyapunov stability tests for linear time-delay systems
After a brief review of the Lyapunov approach to the stability of time-delay systems, the stability results based on functionals with prescribed quadratic derivatives are introduced. In analogy with the delay-free linear case, a stability criterion (necessary and sufficient conditions) expressed in terms of the system delay Lyapunov matrix is presented. The interest in extending the results to several classes of delay systems is explained. Recent results that improve the tractability of the criterion are outlined. Finally, the perspectives of this line of research are discussed.

Sabine Mondié (S’96-M’99) received the B.S. degree in industrial engineering from the ITESM, Mexico City, and the M.S. and Ph.D. degrees in electrical engineering from CINVESTAV, Mexico City and IRCyN, Nantes, France, in 1983 and 1996, respectively. She is a professor at the Department of Automatic Control at CINVESTAV, Mexico City, Mexico since 1996, and is currently head of the Department. She has been chair for education, vice-chair of the IFAC Technical Committee 2.2. on “Linear Control Systems” and is a member of the IFAC Council for the 2023–2026 triennium. She has served as Associate Editor for several journals in control, including Systems & Control Letters and European Journal of Control. She is currently Associate Editor of IEEE Transactions on Automatic Control. Her research work is focused on time-delay systems, their stability and robustness properties, as well as delay applications in engineering and biology. She has directed/co-directed over 18 Ph.D. and 25 master theses and authored/co-authored 80 journal and 130 conference papers.

Exploring delayed dynamics and control strategies of human balancing
In an aging society, investigating the biomechanisms underlying human balancing is of significant importance, especially for the purpose of fall prevention. Central to this is the interpretation of control strategies employed by the central nervous system. In this talk, we focus on two control strategies, the proportional-derivative controller and the proportional-derivative-acceleration controller, with which human balancing systems are represented by retarded delay differential equations and neutral delay differential equations, respectively. We analyze nonlinear dynamics of the delayed systems by a proposed symbolic computation scheme. Additionally, we propose an adjoint sensitivity based method for system and controller parameter identification. Our findings reveal that humans tends to employ some optimal control strategies to maintain balance.

Li Zhang received the B.S. degree in Aircraft Design in 2004 and the PhD degree in Dynamics and Control in 2012 from Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China. She is a professor at the college of aerospace engineering and is currently vice director of Center for Dynamics of Intelligent Equipment at NUAA. Her research interests include time delay systems, nonlinear dynamics and human balancing problems.

Fifty years of numerical analysis of delay differential equations
Initial value problems for delay differential equations (DDEs) are commonly used in mathematical modeling, in particular in  physics, engineering, biology, medicine, economics and pharmacometrics. Their numerical integration is often a challenging task due to several reasons, such as the lack of regularity, the possible state dependency of the delay, the possible distributed form of the delay and the necessity of a dense output. These features make an extension of ODE methods tricky. In this talk, starting from the first numerical methods, till most recent methodologies, I will give an overview of several decades of research on the numerical integration of initial value problems for DDEs, focusing - in particular - on stiff and differential algebraic problems.
This talk is inspired by a long-term collaboration with A. Bellen, D. Breda, E. Hairer, S. Maset, L. Torelli, R. Vermiglio and M. Zennaro.

Nicola Guglielmi is full professor in Numerical Analysis at Gran Sasso Science Institute, School of Advanced Studies since 2018 and has been a visiting professor at the Courant Institute, University of Geneva, University of Zurich, McGill University and Georgia Institute of Technology. He is an associate editor of Siam Journal on Numerical Analysis since 2013. His research focuses on numerical methods for differential equations, stability issues, eigenvalue optimization, matrix and control theory. Moreover it addresses implementation issues for stiff and implicit delay equations. With Ernst Hairer he is the co-author of the software RADAR5, which is now distributed commercially by ICON. He was awarded the New Talent Prize (SciCADE, Fraser Island) in 1999 and a young researcher prize (Volterra Centennial, Tempe) in 1996.