### LSAM 2017: Control of nonlinear systems

Time: 09:00 đến 11:15 ngày 14/08/2017, 09:00 đến 11:15 ngày 15/08/2017, 09:00 đến 11:15 ngày 18/08/2017,

Venue/Location: C2-714, VIASM

Speaker: Jean-Michel Coron (Université Pierre et Marie Curie, France).

Content:

This is the second lecture series of VIASM lecture series in applied mathematics.

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; for a satellite, thrusters or momentum wheels can be used. 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 first recall some classical results on this problem for finite dimensional control systems. We then 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...).

Another important problem in control theory is the stabilization problem, which is fundamental for numerous physical applications. One can understand it with the classical experiment of an upturned broomstick on the tip of one's finger. In principle, if the broomstick is vertical with a vanishing speed, it should remain at the vertical (with a vanishing speed). As one sees experimentally, this is not the case in practice: If we do nothing the broomstick is going to fall down. This is because the equilibrium is unstable.  In order to avoid the fall down, one moves the finger in a suitable way in order to stabilize this unstable equilibrium.  This motion of the finger is a feedback: It depends on the position (and the speed) of the broomstick. Feedback laws are now used in many industries and even in everyday life (e.g. thermostatic faucets). We start this lecture by giving some historical feedbacks and papers (in particular the Watt regulator and the seminal paper by James Clerk Maxwell). Then we consider the link between controllability and stabilization and show the usefulness of periodic time-varying feedback laws. We also consider the case where only part of the state is measured and present some tools to construct explicit stabilizing feedback laws. Applications are presented to finite dimensional control systems and to systems modeled by means of partial differential equations.

A special emphasize is put on control of systems modeled by hyperbolic systems in one space dimension. These systems appear in various real life applications (navigable rivers and irrigation channels, heat exchangers, plug flow chemical reactors, gas pipe lines...). On these systems we study the two previous problems, namely the controllability and the stabilization problems, when the control is on the boundary. In particular, we show how to construct explicit stabilizing feedback laws. We present specific feedback laws which have been implemented for the regulation of the rivers La Sambre and La Meuse.

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