Context and objectives of the PhD. 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 [3]. This type of observers relies on embedding the nonlinear system to be observed into a linear filter of the output via an invertible transformation [2], thus mimicking the original idea of D. Luenberger [8]. The method’s robustness will be a crucial point to investigate, as always with finite-time observers.

The last objective of this PhD thesis is to exploit these theoretical developments in the context of selective attenuation of brain oscillations for the treatment of Parkinson’s disease. For this disease, gamma brain oscillations (35-80 Hz) are reported to correlate with tremor severity, whereas slower waves (alpha or delta) do not seem to correlate with symptoms. The goal is thus to disrupt pathological oscillations, while leaving healthy activity unaltered. The key idea here is that, on short time-scales, slow brain waves are quasi-static and hence can be assimilated to equilibria. We will thus derive control strategies to make the system 2-contractive while preserving its original equilibria. By doing so, the closed-loop system will be guaranteed to have no limit cycles, yet to keep low-frequency behavior intact. The control law will first be derived by assuming full-state measurement, which is often not compatible with experimental or clinical constraints. We will then rely on the developed observers to estimate the state of the populations inaccessible to measurements, and exploit them for output-feedback policies.

Environment. The recruited PhD candidate will conduct his/her research in the Laboratoire des Signaux et Syst`emes (L2S) at Gif sur Yvette, France. He/she will be co-supervised by Lucas Brivadis (CNRS researcher) and Antoine Chaillet (pro- fessor at CentraleSup ́elec). This PhD thesis is supported by the French-German ANR project “Robusta”, which will recruit another PhD student and a post-doc on the German side. Regular meetings and visits with the German team (Fabian Wirth at Passau University and Andrii Mironchenko at Bayreuth University) will be planned all along the project.

Sought profile. We are looking for a brilliant and enthusiastic candidate, willing to develop theoretical results in control theory, with a taste for interdisciplinarity. Solid background in applied mathematics, control theory and/or infinite dimen- sional systems theory is required. A good English level is also needed.

References:

[1]  D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control, 47(3):410–421, 2002.

[2]  P. Bernard, V. Andrieu, and D. Astolfi. Observer design for continuous-time dynamical systems. Annual Reviews in Control, 53:224–248, 2022.

[3]  L. Brivadis, V. Andrieu, P. Bernard, and U. Serres. Further remarks on KKL observers. Systems & Control Letters, 172:105429, 2023.

[4]  A. Chaillet, I. Karafyllis, P. Pepe, and Y. Wang. The ISS framework for time-delay systems: a survey. Mathematics of Control, Signals, and Systems, 35(2):237–306, 2023.

[5]  R. Engel and G. Kreisselmeier. A continuous-time observer which converges in finite time. IEEE Transactions on Automatic Control, 47(7):1202–1204, 2002.

[6]  M. Li and J. Muldowney. On R.A. Smith’s autonomous convergence theorem. The Rocky Mountain Journal of Mathematics, pages 365–379, 1995.

[7]  W. Lohmiller and J.J. Slotine. On contraction analysis for nonlinear systems. Automatica, 34(6), 1998.

[8]  D. Luenberger. Observing the state of a linear system. IEEE transactions on military electronics, 8(2):74–80, 2007.

[9]  C. Wu, I. Kanevskiy, and M. Margaliot. k-contraction: Theory and applica- tions. Automatica, 136:110048, 2022.

In this project, we aim to develop the foundations of the robust stability theory for nonlinear time-delay systems with external inputs and outputs. We are going to prove powerful characterisations of input-to-output stability and develop sharp criteria for robust stability in terms of Lyapunov functions. We will apply this machinery for stability analysis of large-scale networks of delay systems, with applications to huge neuronal networks in our brain.

We are seeking a highly motivated, thoroughly prepared, and talented candidate ready to study complex problems at the heart of nonlinear control theory.

Strong knowledge in dynamical systems and/or control theory of infinite-dimensional systems is a prerequisite.  Experience with time-delay systems is a plus. Readiness to undertake long-term research visits to Paris and Passau is pre-assumed.

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