## Applied Mathematics

### IVAN KUPKA LEGACY: A Tour Through Controlled Dynamics

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

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The first chapter of this book is an unpublished handwritten paper from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.