Ph. D. Project
Finite dimensional control of a class of nonlinear partial differential equations.
2019/09/01 - 2022/08/31
Other supervisor(s):
This research subject, which is part of the scientific collaboration between the CRAN and the LIM - Ecole Polytechnique of Tunisia, deals with the control
of a class of nonlinear Partial Differential Equations (PDEs). It addresses both the problem of controlling nonlinear systems of very large dimensions and
the development of reduced order algorithm for a real-time implementation. 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 plasma.

Most of the work in the literature on the Vlasov-Poisson equations concerns the analysis and discretization of these equations [6] to [9], but very few results
exist on the control design. To our knowledge, the few works in this field are described in [1] - [2] - [3]. This is a hard point because there are very few
mathematical tools in infinite dimension and even less when several PDEs are coupled and non-linear. An alternative solution is to use nonlinear finite-
dimensional control approaches on a discretized model of the Vlasov-Poisson equation. This technique can only be effective if the approximated model
converges to the real solution when the discretization step converges to zero.

This strategy has been used recently to solve the problem of observation and control in conduction and radiation energy transfers (strongly coupled PDEs),
and has given very promising results [4] [5]. However, the discrete system obtained is a very large system whose state can reach several thousand points. It
is worth pointing out that the simulation of such systems can take several days or even weeks. The challenges to be addressed in this research work can be
summarized as follows: First, one must develop a stable discretization strategy, ensuring convergence to the real solution when the step converges to zero.
This is a very important point since it ensures that the estimator that will be developed will also converge towards the real solution. Then develop control
strategies of the approximated system in finite dimension and deduce the convergence conditions. It is also a question of evaluating the performance of the
controller obtained in terms of computation time, precision and robustness compared to the errors of modeling. The last point concerns the development and
implementation of reduced-order algorithms to reduce computation time for real-time requirements. The validation of the results will be done through
numerical simulations.

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 Journal
on Control and Optimization, 56 (4), pp.2485-2512, 2018.
6 Shyi‐Shiun Lee , Shih‐Tuen Lee & Jaw‐Yen Yang. Numerical solution of the system of Vlasov‐Poisson equations. Journal of the Chinese Institute
of Engineers, Vol.22,No.3,pp.341-350(1999)
7 Rossmanith JA, Seal DC. A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations. Journal
of Computational Physics , 230 (2011) 6203-6232
8 T. Utsumi, T. Kunugi, J. Koga. A numerical method for solving the one dimensional Vlasov-Poisson equation in phase space. Computer Physics
Communications (108), pp. 159-179, 1998
9 Xiaofeng Cai ; Wei Guo, Jing-Mei Qiu. A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator
splitting. Journal of Computational Physics 354 (2018) 529-551
Control Identification Diagnosis