Thèse Systèmes à Retards K-Contraction Observateurs et Application aux Neurosciences H/F

Doctorat.Gouv.Fr

Nous recherchons un étudiant brillant et enthousiaste, désireux de développer des outils théoriques de control, avec un goût pour l'interdisciplinarité. Des connaissances solides en mathématiques appliquées, théorie du contrôle et/ou systèmes de dimension infinie sont requises. Un bon niveau d'anglais est également nécessaire.

Établissement : Université Paris-Saclay GS Sciences de l'ingénierie et des systèmes École doctorale : Sciences et Technologies de l'Information et de la Communication Laboratoire de recherche : Laboratoire des Signaux et Systèmes Direction de la thèse : Antoine CHAILLET ORCID 0000000340950468 Début de la thèse : 2026-10-01 Date limite de candidature : 2026-07-01T23:59:59 Ce projet doctoral vise à introduire la notion de k-contraction pour les systèmes non-linéaires à retard et à développer des outils permettant de la garantir en pratique. Cette notion de k-contraction, plus faible que la contraction classique, impose que tout k-volume tende vers zero le long des solutions. Sur cette base, un deuxième objectif est de developper des observateurs innovants, exploitant les retards pour garantir une estimation de l'état en temps fini. Enfin, ces développements théoriques seront utilisés pour une application aux neurosciences, à savoir l'atténuation sélective d'oscillations cérébrales pathologiques. Contraction is a property of dynamical systems that imposes that any pair of solutions converge to one another (but not necessarily to a fixed point) [7,1]. Contractive systems enjoy many powerful properties that can be exploited in control theory. A weaker notion is that of k- contraction, in which the distance between any pair of solutions is not necessarily requested to tend to zero, but rather that any volume of dimension k is requested to shrink along the system's solutions [9]. For instance, 2-contraction imposes that any surface is contracted by the system's flow. A particular feature of 2-contractive systems is that they may have several equilibrium points, but they cannot have limit cycles [6]. The theory of contraction and k-contraction is still at its infancy for time-delay systems. In view of the ubiquity of delay sources in engineering, physics and biology applications (mechanical slack, transport phenomena, non-instantaneous commu- nication,...), a first goal of this PhD thesis is to develop tools to guarantee k-contraction for nonlinear time-delay systems (TDS), using Lyapunov- based conditions. A fundamental challenge lies in the fact that the very notion of k-volumes must be carefully addressed since the state space of a TDS is infinite- dimensional. We will start by studying k-contraction for input-free TDS, and then extend it to systems with inputs to ensure k-contraction modulo the disturbance magnitude, in the spirit of the celebrated input-to-state stability (ISS) property [4].

Based on this theoretical framework, a second objective of this PhD thesis is to derive innovative state observers. Observers can be interpreted as algorithmic sensors: their goal is to estimate hidden state variables by relying only on the measurements available on the system [2]. We propose to purposely add delays in the observer (even when the considered system is delay-free) to obtain better convergence and robustness properties. This approach has already proved efficient in prescribed-time observers design for linear systems [5]. We will extend this methodology to nonlinear systems by using Kazantzis-Kravaris/Luenberger (KKL) observers.