July 2019

1er juillet 2019 – 10h, salle Tourmalet

Speaker: Florent Koudohode (stage)
Title: Stabilization of the KdV equation
Keywords: stabilization of PDEs, control of PDEs, phantom tracking method.
Abstract: The study of the Korteweg-de Vries equation, which models waves on shallow water surfaces, has known an increased interesed since decades. In this seminar, we will focus on a particular topic, which is the stabilization of the Korteweg-de Vries equation. We prove the small-time global stabilization of the Korteweg-de Vries equation with three controls.
To achieve this, we split the proof into two steps : the global “approximate stabilization », which consists in using the nonlinear term together with the « Phantom tracking method » to build a time-varying feedback law yielding the state very close to 0, and the “small time local stabilization » obtained thanks to a feedback built with the backstepping method and applied to the linearized version of the Korteweg-de Vries equation.
Slides

Speaker: Ngoc Hoang Mai (stage)
Title: Extracting Information from Moments: Positivity Certification, Polynomial Systems Resolution and Generalized Christoffel-Darboux Kernels
Keywords: Moments, positive polynomials, algebraic equations, Christoffel-Darboux kernels.
Abstract: In this talk, I will present the three main research results obtained during my Master’s internship.
The first result provides a new representation of polynomials which are nonnegative on non-compact semialgebraic sets, together with applications to polynomial optimization. These representations generalize both Reznick’s decomposition of positive definite forms as sums of squares of rational functions with uniform denominators and Putinar’s representation of positive polynomials on compact semialgebraic sets.
The second result is an algorithm to find at least one real solution of a given system of polynomial equations. This algorithm is built upon the difference rate of a function involving the polynomials of the system, the logarithm function and the exponential function, and it does not depend on the total degree of the system. We illustrate the efficiency of the algorithm via numerical benchmarks.
The third result is the design of a new numerical method to extract information from the moments of a given signed atomic measure. The underlying algorithm relies on level sets of perturbed Christoffel functions combined with Newton’s method. We illustrate this method with numerical experiments in low-dimensional spaces and show how to apply this method to extract the solutions of polynomial optimization problems.
Slides