# VISIT OF JEAN-MICHEL CORON, Oct. 2014.

Jean-Michel Coron is Professor of Mathematics at the University Pierre et Marie Curie and a senior member of the Institut Universitaire de France. He will give a series of three lectures entitled :

# "Control of systems: The importance of the nonlinearities''

Lecture 1 is a Colloquium on Oct 24 AT 2PM (note the change of time), room 705 intended for a general audience.

Title: "Stabilization of control systems: From Ctesibius's water clock to the regulation of rivers."

Abstract: A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think of the classical experiment of an upturned broomstick on the tip of one's finger.) We present some pioneer devices and works (Ctesibius, Watt, Foucault, Maxwell, Lyapunov...), some recent mathematical results, and an application to the regulation of the rivers La Sambre and La Meuse.

Lectures 2 and 3 will be held Tuesday October 28, at 1:40 PM room 705 and Wednesday October 29 at 1:40 PM room SEC 220

Title: Nonlinearities and the control of systems.

Abstract: A control system is a dynamical system on which one can act thanks to what is called the control. For example, in a car, one can turn the steering wheel, press the accelerator pedal etc. These are the control(s). One of the main problems in control theory is the controllability problem. It is the following one. One starts from a given situation and there is a given target. The controllability problem is to see if, by using some suitable controls depending on time, one can move from the given situation to the prescribed target. We study this problem with a special emphasis on the case where the nonlinearities play a crucial role. We first recall some classical results on this problem for finite dimensional control systems. We explain why the main tool used for this problem in finite dimension, namely the iterated Lie brackets, is difficult to use for many important control systems modeled by partial differential equations. We present methods to avoid the use of iterated Lie brackets. We give applications of these methods to various physical control systems (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg-de Vries equations, Schroedinger equations...). Finally we turn to the stabilization problem and present methods to construct stabilizing feedback laws. Applications are presented to some of the previous physical control systems.