Good luck on your defense, Zehor !

Ph.D Dissertation Defense, By Zehor Belkhatir, PhD Candidate of Professor Meriem T. Laleg, EMANgroup

Zehor Belkhatir received her Master degree from Ecole Polytechnique d’Alger, Algeria in 2012. Currently, she is pursuing her Ph.D. degree at King Abdullah University of Science and Technology (KAUST), Saudi Arabia. Her current research interests include control and estimation of a class of infinite dimensional systems described by partial differential equations and fractional differential equations.​​​
All EMAN group members wish a good luck to our colleague Zehor in her Phd defense which will be next Thursday 12th Apr 2018 from 04:00 PM - 05:30 PM in Building 5, Level 5, Room 5220​.​​ Her Ph.D defense will be about "Estimation Methods for Infinite-Dimensional Systems Applied to the Hemodynamic Response in the Brain​":

Infinite-Dimensional Systems (IDSs), made possible by recent advances in mathematical and computational tools, can be used to model complex real phenomena. However, due to physical, economical, or stringent non-invasiveness constraints on real systems, underlying model characteristics for mathematical models in general, and IDSs in particular, are often missing or subject to uncertainty. Developing efficient estimation techniques to extract missing pieces of information from available measurements is essential. The human brain is a particular example of an IDS with severe constraints on information collection from controlled experiments and invasive sensors. Investigating the brain’s intriguing modeling potential is in fact the main motivation for this work. Here, we will characterize the hemodynamic behavior of the brain using functional magnetic resonance imaging data. In this regard, we propose efficient estimation methods for two classes of IDSs, namely Partial Differential Equations (PDEs) and Fractional Differential Equations (FDEs). This work is divided into two parts. The first part addresses the joint estimation problem of the state, the parameters, and the input for coupled second-order hyperbolic PDE and infinite-dimensional ordinary differential equation using sampled-in-space measurements. We propose two approaches to solve this estimation problem: an early lumping approach and a late lumping approach. The first strategy consists of designing a Kalman-based algorithm that relies on a reduced finite-dimensional model of the IDS. The second approach proposes an infinite-dimensional adaptive estimator based on the Lyapunov method. We study and discuss the structural identifiability of the unknown variables for both cases. The second part contributes to the development of estimation methods for initialized FDEs. Two major challenges arise in estimating fractional differentiation orders and non- smooth pointwise inputs. First, we extend the integer high-order sliding mode observer to jointly estimate the pseudo-state and the input of commensurate FDEs. Second, we propose a modulating function-based algorithm for the joint estimation of the parameters and the fractional differentiation orders of non-commensurate FDEs for which sufficient conditions ensuring the local convergence of the algorithm have been proposed. Subsequently, we extend the proposed technique to handle the estimation problem of smooth and non-smooth pointwise inputs. The performance of the synthesized estimation techniques is illustrated on a neurovascular- hemodynamic response model. Our formulations are efficiently generic to be applied to a wide set of additional applications.

Best of Luck Zehor !