Applied Mathematics

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IVAN KUPKA LEGACY: A Tour Through Controlled Dynamics

By Bernard Bonnard, Monique Chyba, David Holcman and Emmanuel Trélat (Eds.)

ISBN-10: 1-60133-026-X; ISBN-13: 978-1-60133-026-0

This volume is a tribute to Ivan Kupka who passed away at Easter 2023.  It contains a collection of articles written by colleagues working on dynamical systems or control theory. Those colleagues interacted with Ivan Kupka directly or indirectly through his articles.

In his earliest work (1963) Ivan proved independently the Kupka-Smale theorem: “in a compact manifold the set of vector fields with the following properties are generic:  all closed orbits are hyperbolic and heteroclinic orbits are transversal”. At the end of the 70s he made significant contributions to controllability properties of invariant vector fields on semi-simple Lie groups, which are being applied to space mechanics and quantum control decades later. In the mid 80s Ivan proved the ubiquity of Fuller phenomenon which allows to describe the complex behaviors of the extremal trajectories in optimal control and provides an obstruction to the construction of optimal syntheses in the sub-analytic category. In 2001, he co-authored a well-cited book “Deterministic Observation Theory and Applications” which deals with geometric observation, state and parameters estimation. Since the end of the 20th century and for the next two decades he made significant contributions to stochastic systems with life and chemical science applications.

The first chapter of this book is an unpublished handwritten paper from Ivan Kupka which discuss optimality of regular extremals. The remaining articles are original contributions devoted to geometric optimal control with applications to Lotka-Volterra equations, Zermelo navigation problem, Quadcoper dynamics, Quantum control and Control of biological systems. Additional articles deal with Hybrid control and Reversible dynamical systems and some contributions couple geometric control which advanced numerical methods in optimal control. This book is a legacy to Ivan Kupka and his research interests and provides a modern perspective for researchers and graduate students to geometric control and its applications connecting applied mathematics and control engineering.