The second lectures of LSAM 2017: “Control of nonlinear systems” by Prof. Jean-Michel Coron (Université Pierre et Marie Curie, France) was held on Monday, August 14, 2017. There were about 40 participants on the first day of the course. The course will be continued in the morning of August 15 & 18, 2017.

Prof. Jean-Michel Coron is a French mathematician. He had previously worked in the field of non-linear functional analysis, where he also obtained significant results. He was awarded numerous prizes, like the Fermat prize in 1993, the Jaffé prize in 1995 by the Académie des Sciences, the Dargelos prize in 2002 and the ICIAM Maxwell prize in 2015.

Thanks to the cooperation and support from Viet Capital Bank and the international prestige of Prof. Ngo Bao Chau in particular, as well as the VIASM’s international position in general, the lecture series of LSAM 2017 has been given by two prominent professors in applied mathematics. The second lectures are about the controllability problem and the stabilization problem.

One of the main problems in control theory is the controllability problem. 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 for finite dimensional control systems and then explain why the main tool used for classical problem in finite dimension is difficult to use for many important control systems modeled by partial differential equations. We present methods to avoid this difficulty.

Another important problem in control theory is the stabilization problem. We start this lecture by giving some historical feedbacks. 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. A special emphasize is put on control of systems modeled by hyperbolic systems in one space dimension, which appear in various real life applications. We study the two previous problems when the control is on the boundary. In particular, we show how to construct explicit stabilizing feedback laws.