Interactions between dynamical systems, evolution equations and control – ISDEEC

Evolution equations, or more generally dynamical systems, form one of the predominant classes of models in the sciences, ranging from physics and biology to social sciences. The study of their qualitative behaviour for short or large times has always been one of the main purposes in mathematics and is a very challenging and exciting question.

In this project, we plan to study the qualitative dynamics of partial differential equations, such as parabolic equations, Hamiltonian or hyperbolic
PDEs, non-linear dispersive PDEs, PDEs on graphs... and also of some classes of finite-dimensional differential equations which admit a special structure, such as Hamiltonian equations, geodesic flows, coupled cell networks... This study may take different forms:
- classification of the typical global dynamics and their complexity,
- study of particular trajectories (homoclinic orbits, travelling waves, blow-up solutions), description of bifurcation phenomena,
- control of the dynamics through an external force or a modification of an internal parameter,
- description of the dynamics from partial observations (boundary observation, observations at a finite sequence of times, observation at a partial set of cells in a network...).

Such a work program requires expertise in various mathematical domains: dynamical systems theory, PDE techniques, control theory, geometry, functional analysis... The current trend in mathematics is to high specialisation and the interactions between experts from these different fields have to be enhanced (especially in France). The main purpose of this project is to create and extend such interactions.

We do not only aim at deepening our understanding of the dynamics of evolution equations by developing interactions between specialists from
various domains, having their own points of view, their own problems in mind and using different notions and tools, but we also hope that, in these various fields, new challenging questions will emerge from these interactions.

The main thematics of the project include
Task 1: Dynamical systems and PDEs
Task 2: Control theory and generic dynamics
Task 3: Global dynamics and control theory
Task 4: Hamiltonian dynamics in infinite dimension