# Prize Spotlight: Jean-Michel Coron

The 2017 Reid Prize honors Coron for fundamental contributions with lasting impact to both the analysis and control of nonlinear ordinary and partial differential equations, and in particular the Coron return method for feedback stabilization of nonlinear systems using time-varying controls, as well as applications of control theory to practical problems. He received the award at the 2017 SIAM Annual Meeting (AN17), held July 10-14, 2017 in Pittsburgh, Pennsylvania. He delivered the Reid Prize lecture, “Feedback Stabilization of Control Systems,” on July 13, 2017.

Coron is currently a full professor at Université Pierre et Marie Curie and a member of the French Academy of Sciences. He received his Thèse d’Etat (PhD) in mathematical sciences from Université Pierre et Marie Curie in 1982. His research in control theory is on the stabilization of nonlinear control systems and the control of systems modeled by means of partial differential equations.

**Q:** *Why are you excited about winning the prize?*

**A:** The W. T. and Idalia Reid Prize is one of the most prestigious prizes in control theory. This is confirmed by the outstanding level of the previous laureates. Hence I am strongly honored to receive this award. It is for me a very important recognition of my work in control theory.

**Q:** *Could you tell us a bit about the research that won you the prize and what it means to the public?*

**A:** Control theory is something which is very natural and therefore very easy to explain. One has a 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 controls.

There are at least two fundamental problems in control theory. First is the controllability problem. 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 desired target. Second is the stabilization problem, which is also very important for numerous 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, 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. One can see the usefulness of this information (position and speed) by trying to do the experiment with closed eyes: it is much more difficult (and dangerous).

I worked on these two problems. On the first, controllability problem, I proposed methods to study the controllability of systems modeled by means of partial differential equations in the case where the nonlinearities play a crucial role. Using these methods I proved new controllability results for various partial differential equations including the Euler equations and the Navier Stokes equations of incompressible fluids, the shallow water equations, the Korteweg-de Vries equations and the Schrödinger equation. On the second, stabilization problem, my main results concern the possibility to stabilize many controllable systems by means of time-varying feedback laws (even if these systems cannot be stabilized by means of continuous feedback laws which do not depend on time), as well as methods to construct explicit stabilizing feedback laws. One of these methods has been applied to the regulation of rivers.

**Q:** *What does SIAM mean to you?*

**A:** SIAM is doing outstanding work to promote the interactions between the mathematics community and other scientific and technological communities. In particular the *SIAM Journal on Control and Optimization*, among the best journals in control theory, bridges the gap between mathematics and control theory from the engineering point of view. The SIAM Conferences on Control and Its Applications are also key to fulfilling this aim.