Partial Differential Equations (PDEs) are commonly used to model complex systems that arise for example in biology, engineering, chemistry, and elsewhere. The parameters (or coefficients) and the source of PDE models are often unknown and are estimated from available measurements. Despite its importance, solving the estimation problem is mathematically and numerically challenging and especially when the measurements are corrupted by noise, which is often the case. Various methods have been proposed to solve estimation problems in PDEs which can be classified into optimization methods and recursive methods. The optimization methods are usually heavy computationally, especially when the number of unknowns is large. In addition, they are sensitive to the initial guess and stop condition, and they suffer from the lack of robustness to noise. Recursive methods, such as observer-based approaches, are limited by their dependence on some structural properties such as observability and identifiability which might be lost when approximating the PDE numerically. Moreover, most of these methods provide asymptotic estimates which might not be useful for control applications for example. An alternative non-asymptotic approach with less computational burden has been proposed in engineering fields based on the so-called modulating functions. In this dissertation, we propose to mathematically and numerically analyze the modulating functions based approaches. We also propose to extend these approaches to different situations. The contributions of this thesis are as follows. (i) Provide a mathematical analysis of the modulating function-based method (MFBM) which includes: its well-posedness, statistical properties, and estimation errors. (ii) Provide a numerical analysis of the MFBM through some estimation problems, and study the sensitivity of the method to the modulating functions' parameters. (iii) Propose an effective algorithm for selecting the method's design parameters. (iv) Develop a two-dimensional MFBM to estimate space-time dependent unknowns which is illustrated in estimating the source term in the damped wave equation describing the physiological characterization of brain activity. (v) Introduce a moving horizon strategy in the MFBM for on-line estimation and examine its effectiveness on estimating the source term of a first order hyperbolic equation which describes the heat transfer in distributed solar collector systems.

Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functions-based method is used to solve the problem and it is compared to a standard Tikhono-based optimization technique.

Cardiovascular diseases (CVDs) remain the leading cause of death worldwide. Patients at risk of evolving CVDs are assessed by evaluating a risk factor-based score that incorporates different bio-markers ranging from age and sex to arterial stiffness (AS). AS depicts the rigidity of the arterial vessels and leads to an increase in the arterial pulse pressure, affecting the heart and vascular physiology. These facts have encouraged researchers to propose surrogate markers of cardiovascular risks and develop simple and non-invasive models to better understand cardiovascular system operations. This work thus fundamentally capitalizes on developing a novel class of low-dimensional physics-based fractional-order models of systemic arteries and exploring the feasibility of fractional differentiation order to portray the vascular stiffness. Fractional-order modeling is a successful paradigm to integrate multiscale and interconnected mechanisms of the complex arterial system. However, this type of modeling alone often fails to efficiently integrate altered variabilities in vascular physiology from various sources of large datasets, multi-modalities, and levels. In this regard, combining fractional-order-based approaches with machine learning techniques presents a unique opportunity to develop a powerful prediction framework that reveals the correlation between intertwined vascular events. This work is divided into three parts. The first part contributes to developing the fractional-order lumped parametric model of the arterial system. First, we propose fractional-order representations to model and characterize the complex and frequency-dependent apparent arterial compliance. Second, we propose fractional-order arterial Windkessel modeling the aortic input impedance and hemodynamic. Subsequently, the proposed models have been applied and validated using both human in-silico healthy datasets and real vascular aging and hypertension. The second part addresses the non-zero initial value problem for fractional differential equations (FDEs) and proposes an estimation technique for joint estimation of the input, parameters, and fractional differentiation order of non-commensurate FDEs. The performance of the proposed estimation techniques is illustrated on arterial and neurovascular hemodynamic response models. The third part explores the feasibility of using machine learning algorithms to estimate the gold-standard measurement of AS, carotid-to-femoral pulse wave velocity. Different modalities have been investigated to generate informative input features and reduce the dimensionality of the time series pulse waves.

Cette thèse traite de l'identification des paramètres dynamiques des robots, en s'appuyant sur les méthodes d'identification en robotique, qui utilisent le modèle dynamique inverse, ou le modèle de puissance, ou le modèle d'énergie du robot. Ce travail revisite le modèle d'énergie en exploitant le caractère intégral des fonctions modulatrices appliquées au modèle de puissance du robot. En outre, les procédures d'intégration sont analysées dans le domaine fréquentiel, et certains groupes de fonctions modulatrices sont sélectionnés afin d'offrir un bon comportement de filtre passe-bas. Ensuite, l'introduction d'un différentiateur algèbrique récemment développé est proposé, nommé différentiateurs de Jacobi. L'analyse est effectuée dans le domaine temporel, et dans le domaine fréquenciel, ce qui met en évidence la propriété de filtrage passe bande et permet de sélectionner les paramètres des différentiateurs. Puis, ces différentiateurs sont appliqués avec succès à l'identification de robot, ce qui prouve leur bonne performance. Les comparaisons entre les différents modèles d'identification, les différenciateurs, les techniques des moindres carrés sont présentées et des conclusions sont tirées dans le domaine de l'identification de robot. / This thesis discusses the identification issues of the robot dynamic parameters. Starting with the well-known inverse dynamic identification model, power and energy identification models for robots, it extends the identification model from an energy point of view, by integrating modulating functions with robot power model. This new identification model avoids the computation of acceleration data. As well, the integration procedures are analyzed in frequency domain so that certain groups of modulating functions are selected in order to offer a good low-pass filtering property. Then, a recently developed high order algebraic differentiator is proposed and studied, named Jacobi differentiators. The analyses are done in both the time domain and in the frequency domain, which gives a clear clue about the differentiator filtering property and about how to select the differentiator parameters. Comparisons among different identification models, differentiators, least square techniques are presented and conclusions are drawn in the robot identification issues.