Ph. D. Project
State observers and control of a class of coupled PDE. Application to Vlasov-Poisson equations
2019/01/28 - 2022/01/28
This research topic focuses on the state estimation and control of a class of Partial Differential Equations (PDEs). The system will be analyzed in 1D and 2D
discrete space form. One of the main motivations concerns the application of these approaches to the Vlasov-Poisson equation. The latter describes the
evolution of the distribution function of charged particles in a fusion plasma.

The spatio-temporal estimation of the distribution function is therefore an important step to analyze, do supervision or control the fusion. Most of the
work in the literature on the Vlasov-Poisson equations concerns the analysis and discretization of these equations, but very few results exist on the control
[1] - [2] - [3] and even less so, not to say at all, about observation. This is a hard point because there are very few mathematical tools in infinite dimension to
make the observation and even less when several PDEs are coupled and non-linear. An alternative solution is to use finite dimensional nonlinear estimators
and controllers on a discretized model of the Vlasov-Poisson equation. This approach can only be effective if the approximated model converges towards
the real solution when the discretization step converges to zero.

Cette stratégie a été utilisée récemment pour résoudre le problème de l'observation et de la commande dans les transferts énergétiques par conduction et
rayonnement (EDP fortement couplées), et a donné des résultats très prometteurs [4][5]. Cependant, nous sommes confrontés à faire l'estimation spatio-
temporelle et la commande d'un système de très grande dimension dont l'état peut atteindre plusieurs milliers de points. Il est utile de souligner que la
simulation de tels systèmes peut durer plusieurs jours, voire même quelques semaines, sur un seul processeur. Les défis à relever lors de ce travail de
recherche peuvent être résumés comme suit :

This strategy has been used recently to solve the problem of state observers and control in conduction and radiation energy transfer (EDP strongly
coupled), and gave very promising results [4] [5]. However, we are faced with the spatio-temporal estimation and the control of a very large scale system
whose state can reach several thousand points. It is worth pointing out that simulation of such systems can take several days or even weeks on a single
processor. The challenges to be addressed in this research can be summarized as follows:

A - Develop a state observer of the approximated system in finite dimension and deduce the stability conditions.

B - Develop control laws in finite dimension and deduce the stability conditions. Extend these results to establish observer-based controllers.

C - The last point concerns the development and implementation of a simulation code. Performances of the obtained state estimator / controller should be
evaluated in terms of computational time, precision and robustness compared to the errors of modeling.


1 - Coron, J.-M., Glass, O., and Wang, Z. Exact boundary controllability for 1-d quasilinear hyperbolic systems with a vanishing characteristic speed. SIAM
Journal on Control and Optimization 48, 5 (2009), 3105-3122.
2 - Glass, O., and Han-Kwan, D. On the controllability of the vlasov-poisson system in the presence of external force fields. Journal of Differential
Equations 252, 10 (2012), 5453-5491.
3 - Glass, O., and Han-Kwan, D. On the controllability of the relativistic vlasov-maxwell system. Journal de Mathématiques Pures et Appliquées 103, 3
(2015), 695-740.
4 - Ghattassi M., Boutayeb, M., « Non linear controller design for a class of parabolic-hyperbolic systems », Journal of Non Linear Systems and
Applications,Vol. 5, pp. 15-20, 2016.
5 - Ghattassi M., Boutayeb M. & Roche J. R.. Reduced order observer of finite dimensional radiative-conductive heat transfer systems. SIAM J. on Cont.
and Optimization, 56 (4), pp.2485-2512, 2018.
Control Identification Diagnosis