Thesis of Mohamed yassine Arkhis

Contraction framework for incremental stability with applications to small-gain and control design

In this work, we address the incremental stability of dynamical systems and associated control design using the framework of contraction. This is a stability notion that focuses on the behavior of trajectories relative to one another. In the literature, this notion has been studied using two main approaches. The first is the direct incremental approach, in which the distance between trajectories is analyzed directly. The second is a differential framework, called contraction, in which one analyzes the linearized dynamics. While the direct approach answers the question: how does the distance between the solutions evolve relative to the distance between the initial states?, the contraction framework answers: how does the velocity at which solutions move away from each other evolve relative to the velocity of the movement of the initial conditions? We show that both frameworks are equivalent under suitable regularity assumptions. Furthermore, we use the contraction framework to study the incremental stability of interconnections and to design controllers that enforce this property on a given system.

Jury

M. Denis EFIMOV Directeur de recherche Centre Inria de l'Université de Lille Directeur de thèse, M. Nathan VAN DE WOUW Full professor Eindhoven University of Technology Rapporteur, M. Daniele ASTOLFI Chargé de recherche Université Claude Bernard Lyon 1, LAGEPP Rapporteur, Mme Elena PANTELEY Directrice de recherche CNRS, Laboratory of Signals and Systems Examinatrice, M. Pierre-Alexandre BLIMAN Directeur de recherche INRIA Examinateur, M. Romain POSTOYAN Directeur de recherche Université de Lorraine, CRAN Examinateur.

Thesis of the team VALSE defended on 27/10/2026