événement

# Soutenance publique de thèse de doctorat en Sciences mathématiques - Christian MUGISHO ZAGABE

A Koopman operator approach to stability and stabilization of switched nonlinear systems

**Catégorie :**défense de thèse

**Date :**16/10/2024 17:00 - 16/10/2024 20:00

**Lieu :**PA02

**Orateur(s) :**Christian Mugisho Zagabe

**Organisateur(s) :**Alexandre Mauroy

### Jury

- Prof. André FÜZFA (UNamur), président
- Prof. Alexandre MAUROY (UNamur), secrétaire
- Prof. Joseph WINKIN (UNamur)
- Prof. Raphaël JUNGERS (UCLouvain)
- Dr Milan KORDA (LAAS-CNRS, Toulouse)
- Prof. Igor MEZIC (University of California Santa Barbara)

### Abstract

Switched
systems became more and more interesting since they are conceptually
closed to the description of real complex dynamics in which the state is
not necessarily fixed in time but can abruptly change with the
environment. In this context, not only a finite number of subsystems
(said

*modes)*are given to describe the possible state of the system, but also a*switching*(or*commutation*)*law*is assigned to indicate the active mode at each time.The stability theory of such systems is not intuitive since it is influenced by the commutation law, which plays a capital role.

This dissertation investigates the

*uniform stability*(i.e. stability under any commutation laws) and the*switching stabilization*(design of a stabilizing commutation law) problems of switched nonlinear systems.In
the last decades, these problems have mainly been studied for switched
linear systems and partially solved for the nonlinear case.

The strategy exploited here is based on a successful tool today: the

*Koopman operator*. This is a linear operator acting on an infinite-dimensional space of functions valued on the nonlinear system's state space. Roughly speaking, it allows one to transform a nonlinear finite-dimensional dynamics into a linear infinite-dimensional dynamics, from which one can deduce results for the original nonlinear system. More precisely, we utilize the Koopman operator framework to address switched nonlinear systems' uniform stability and stabilization problems.For
the first problem, by using a Lie-algebraic solvability condition, we
show that individual globally asymptotically stable nonlinear vector
fields which admit a common Koopman finite-dimensional invariant
subspace generate a uniformly globally asymptotically stable switched
nonlinear system. In a broader context, we develop a general framework
for studying the (uniform) stability of (switched) nonlinear systems on
the polydisk or the hypercube. This systematic approach allows us to
construct a common Lyapunov function that guarantee global uniform
asymptotic stability on the polydisk or the hypercube. We then apply
this framework to derive systematic criteria for the global stability of
nonlinear systems defined on the polydisk or the hypercube. Finally,
for the second problem, we utilize the previously developed results to
provide a state-dependent switching stabilization strategy from a
systematic Lyapunov function of a convex combination of nonlinear vector
fields.

**Télecharger :**
vCal