topics

My main research topics concern control and analysis of PDE (Partial Differential Equations), either linear or nonlinear - but mainly nonlinear. I am interested in understanding results in control theory that have been found by mathematicians in order to apply them for automatic control problems.

Stability of PDEs with Nonlinear Control

I am interested in the asymptotic behavior of linear PDEs whose control is subject to a nonlinearity (as, for instance, the saturation, which models some amplitude constraints on the actuator). There exist many ways to achieve such an objective. The direct way is to build a Lyapunov function, as it is done for hyperbolic systems. Hierarchical method, using Legendre polynomials, has also been applied to some coupled PDE/ODE systems. One may also use the backstepping method, due to Miroslav Krstic and Andrey Smyshlyaev, which is an extension of a technique used earlier for nonlinear ODEs. However, nowadays there is no general methodology to build Lyapunov functions explicitely, even for linear PDE. I even expect that such a general methodology cannot be obtained. One might use some Lyapunov converse theorem or even follow another strategy, as for instance the compactness-uniqueness one, which has been really powerful in recent decades. I would like also to make a link between Lyapunov functionals and the multiplier technique, that has been popularized by Vilmos Komornik. I think also that there is a strong link between Lyapunov functionals and Carleman inequalities, that are a fundamental tool to prove the controllability of PDEs. My main collaborators for this topic are Vincent Andrieu , Eduardo Cerpa, Yacine Chitour and Christophe Prieur.

LASSERRE hierarchy applied to nonlinear PDEs

I am also interested in the numerical aspect of PDEs. More precisely, I am convinced that the Moment/SOS hierarchy (also known as Lasserre hierarchy), developed by Jean Bernard Lasserre and Didier Henrion in recent decades, might be really powerful for such an analysis. Roughly speaking, this technique proposes to solve numerically infinite-dimensional optimization problem on measures by focusing on their moments - these problems are known as Generalized Moment Problems (GMP). Some non convex and nonlinear problems can be rephrased as instances of the GMP and then be solved globally with the Moment/SOS Hierarchy method. In a recent paper, we have rephrased solution to scalar nonlinear hyperbolic PDEs as an instance of the GMP. This result relies on earlier results due to Ronald J. Di Perna who relaxed the notion of weak solution so that the solution that one looks for is no more a function, but a Borel measure. This leads to a formulation linear in the measure. Therefore, we could apply the Moment/SOS hierarchy. One of the interesting aspect of this numerical is that it does not rely on a time/space discretization, but rather on a truncation of the moments of the measure. This allows us to generate discontinuous numerical solutions, thanks to the Christoffel-Darboux kernel. This result might be an interesting first step in order to treat another problems such as the computation of region of attraction, as it has been done in the case of ODEs, or even the computation of controllers. This is the subject of my current post-doc. My main collaborators for this topic are Didier Henrion, Jean Bernard Lassere, Edouard Pauwels and Tillmann Weisser.