Focus Period Linköping 2023

PhD Course – Theory and Applications of Contracting Dynamical Systems



Francesco Bullo, Mechanical Engineering, UC Santa Barbara


Theory and Applications of Contracting Dynamical Systems


Over the last two decades, engineers and mathematicians have made remarkable progress on the application of the Banach contraction principle to dynamical systems over networks. The basic contraction property is now understood for discrete and continuous time systems, with respect to Euclidean and non-Euclidean norms, in closed and open systems, and for single agents and networks of systems. In this set of lectures, I will present this theoretical field and its applications. Topics will include (i) the algebraic properties of the induced norms and logarithmic norms of matrices, (ii) contracting dynamics over finite-dimensional vector spaces endowed with Euclidean and non-Euclidean norms, and (iii) weakly-contracting dynamics and monotone dynamics. Numerous examples will be presented, including Hopfield neural networks, interconnected contracting systems, and gradient and primal/dual flows of convex functions.


The reference text for the course is the recent text “Contraction Theory for Dynamical Systems”, KDP, v1.1, 2023, ISBN 979-8836646806, freely available at:


Where: ISY Linkoping University (Room: Systemet*)
Time: September 13 – 15, 2023
Duration: 4h/day (total 12h)

Wednesday September 13: 10:15-12:00 and 15:15-17:00
Tuesday September 14: 10:15-12:00 and 14.15:16:00
Friday September 15: 10:15-12:00 and 14:15-16:00


It will be possible to take the course for credits. Info on the format of the exam and n. of credits (likely 3 credits) will be added later on.


Send an email to C. Altafini at