Master Class Analysis

Control of non linear systems

Jean-Michel Coron

A control system is a dynamical system on which one can act through what is called a control. For example, in a car, one can turn the steering wheel, press the gas pedal etc. For a satellite, propelors or inertia wheels can be used.

One of the main issues in control theory is the controllability problem, which is stated as follows. We start from a given state, and there is a target we aim at. The controllability problem is to see whether we can go from the given state to the desired target by using appropriate, time dependent controls. We first recall some classical results on this problem for finite dimensional control systems. We explain why the main tool for this finite dimension problem, that is iterated Lie brackets, is difficult to use for numerous important control systems modelled by partial differential equations. We present some methods to avoid using these iterated Lie brackets. We give applications of these methods to various physical control systems modelled by partial differential equations (Euler and Navier-Stokes incompressible fluid equations, Saint-Venant equations, Korteweg-de Vries equations...)

Another important problem in control theory is the stabilization one. We can understand it with the classical broom experiment that one keep in equilibrium on its finger. In principle, if the broom is vertical with zero speed, it should stay vertical (with zero speed). As seen experimentaly, it is not the case in practice: if nothing is done the broom falls down. This is because the equilibrium is unstable. So as to avoid the fall, we move the finger in an appropriate way so as to stabilize this unstable equilibrium. The moving finger is a feedback: it depends on the position (and speed) of the broom. Feedback laws are now used in many industries and even in everyday life (for example, thermostatic valves). We consider the link between controllability and stabilizability, and show how useful are the time periodic feedbacks laws. Some applications will be shown to finite dimension control system and to systems modelled by partial differential equations.

Alberto Calderón's conductivity inverse problem

David Dos Santos Ferreira

In the 1950s, Alberto Calderón, at the time working in Argentina as engeneer for the national oil company Yacimientos Petrolíferos Fiscales, and even before starting his PhD under the advise of Antoni Zygmund, got interested to modeling problem coming from petrol exploration techniques.

The problem is to know whether one can determine the electric conductivity of a body by measuring voltage and current at the outskirts.

Mathematically speaking, this boils down to understand whether one can determine the conductivity in an equation of divergence type on a smooth bounded open set from the Dirichlet-to-Neumann map.
The mini-course will present the problem and techniques for solving it based on the construction of solutions to a partial differential equation by the complex geometric optics method.

Course material: slides, notes.

Derivation of fluid mechanics equation from systems of particles

Isabelle Gallagher

Hilbert sixth problem consist in jusifying rigourously the mechanisms leading from classical molecular dynamics to fluid dynamics.

By using Bolztmann equation as an intermediate step in the description, we will explain is this cours how this is possible in simplified, linear frameworks (in the view, inter alia, to derive the heat equation from systems of particles).

Dynamics of dissipative PDEs

Geneviève Raugel

• Definitions and global attractors of a dissipative system
• Gradient systems and convergence to an equilibrium point
• Elements on invariant manifolds
• Applications: radial focusing Klein-Gordon equation with damping - convergence to equilibrium (two methods - recent results)