Research
Our research lies at the intersection of dynamical systems theory, control theory, and operator theory (Koopman operator). It aims at developing new theoretical approaches and numerical tools that are needed for the global analysis of nonlinear systems and networks. The proposed techniques are built on firm theoretical grounds in a general framework.
Main research interests are summarized in the following (connected) topics.
Koopman operator approach to control theory
We consider an alternative operator-theoretic approach to nonlinear (dissipative) systems, which is based on the so-called Koopman operator. Our main research effort aims to develop and present this Koopman operator approach as a relevant theoretical framework and a valuable alternative tool in the broad context of control theory.
Spectral properties and interplay with classical control-theoretic concepts
The Koopman operator approach leads to a spectral description of nonlinear systems, mirroring the classic spectral analysis of linear systems. In this context, we explore the connections between the spectral properties of the operator and mathematical concepts that are relevant to nonlinear control theory (e.g. global stability, contraction, differential positivity). This approach provides systematic numerical methods for studying nonlinear systems. [PDF]
Lie-algebraic approach, geometric control, and application to switched systems
We explore the connections between the Koopman operator framework and the classical Lie-algebraic approach to nonlinear systems. We focus in particular on the global stability problem for nonlinear switched systems. More generally, we are interested in the connections with geometric control theory.
Other research questions and perspectives
Other recent research questions include, but are not limited to, the use of approximation theory in Koopman operator theory and the development of a proper Koopman-based approach to input-output nonlinear systems.
Data-driven analysis: system identification and network inference
Linear identification of nonlinear systems
We propose to identify the linear Koopman operator in the space of observables. This yields an efficient linear technique for nonlinear identification and parameter estimation. Two dual methods are developed and applied to the specific context of network inference. [PDF]
Codes:
Spectral network identification
Combining operator theory and spectral graph theory, we focus on the paradigm of spectral identification of large networks of interacting agents. Our objective is to reveal topological properties of the networks from sparse measurements of the local dynamics at a few nodes. [PDF]
Codes:
Other data-driven methods
We aim at developing new data-driven methods for the analysis of nonlinear systems. Recent work focuses on the use of a reservoir computer to extract the spectral properties of nonlinear systems from data.
Other
Phase sensitivity analysis, isochrons, and isostables
We develop theoretical tools and methods for sensitivity analysis and phase reduction of nonlinear systems (e.g. neurons). Our main interests focus on the numerical computation of the isochrons (see our forward-integration method here) and their extension to equilibria, leading to the notion of isostables. This framework was applied to the sensitivity analysis of bursting neurons. [PDF] [PDF]
Codes:
computation of isochrons computation of isostables
Networks of coupled oscillators
Using stability theory, we investigate the relationship between the global collective behavior of a network (e.g. a region of the brain) and the local dynamics of the individual oscillators (e.g. a neuron). In the case of pulse-coupled oscillators, we have emphasized the existence of a remarkable dichotomic behavior.