Publications

2016

• A Hamilton-Jacobi-Bellman approach for the optimal control of an abort landing problem
  
  • Assellaou Mohamed
  • Bokanowski Olivier
  • Desilles Anna
  • Zidani Hasnaa
  , 2016, pp.3630-3635. (10.1109/CDC.2016.7798815)
  DOI : 10.1109/CDC.2016.7798815
• A semi-Lagrangian algorithm in policy space for hybrid optimal control problems
  
  • Ferretti Roberto
  • Sassi Achille
  , 2016. The mathematical framework of hybrid system is a recent and general tool to treat control systems involving control action of heterogeneous nature. In this paper, we construct and test a semi-Lagrangian numerical scheme for solving the Dynamic Programming equation of an infinite horizon optimal control problem for hybrid systems. In order to speed up convergence, we also propose an acceleration technique based on policy iteration. Finally, we validate the approach via some numerical tests in low dimension.
• Transient simulation of an electrical rotating machine achieved through model order reduction
  
  • Montier Laurent
  • Henneron Thomas
  • Clenet Stéphane
  • Goursaud Benjamin
  Advanced Modeling and Simulation in Engineering Sciences, Springer, 2016, 3 (1).
• Kernel Density Estimation applied to the chance-constrained Goddard problem
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  , 2016.
• Robust optimal sizing of a hybrid energy stand-alone system
  
  • Billionnet Alain
  • Costa Marie-Christine
  • Poirion Pierre-Louis
  European Journal of Operational Research, Elsevier, 2016, 254, pp.565 - 575. This paper deals with the optimal design of a stand-alone hybrid system composed of wind turbines, solar photovoltaic panels and batteries. To compensate for a possible lack of energy from these sources, an auxiliary fuel generator guarantees to meet the demand in every case but its use induces important costs. We have chosen a two-stage robust approach to take account of the stochastic behavior of the solar and wind energy production and also of the demand. We seek to determine the optimal system, i.e. the one that generates a minimum total cost when the worst case scenario relating to this system occurs. We use a constraint generation algorithm where each sub-problem (the recourse problem) can be reformulated by a mixed-integer linear program and hence solved by a standard solver. We also propose a polynomial time dynamic programming algorithm for the recourse problem and show that, in some cases, this algorithm is much more efficient than mixed-integer linear programming. Finally, we report computational experiments on instances constructed from real data, that show the efficiency of the proposed approach and we study the addition of constraints linking the uncertainty in consecutive time periods. (10.1016/j.ejor.2016.03.013)
  DOI : 10.1016/j.ejor.2016.03.013
• Fiabilité et optimisation des calculs obtenus par des formulations intégrales en propagation d'ondes
  
  • Bakry Marc
  , 2016. Dans cette thèse, on se propose de participer à la popularisation des méthodes de résolution de problèmes de propagation d'onde basées sur des formulations intégrales en fournissant des indicateurs d'erreur a posteriori utilisable dans le cadre d'algorithmes de raffinement autoadaptatif. Le développement de tels indicateurs est complexe du fait de la non-localité des normes associées aux espaces de Sobolev et des opérateurs entrant en jeu. Des indicateurs de la littérature sont étendus au cas de la propagation d'une onde acoustique. On étend les preuves de convergence quasi-optimale (de la littérature) des algorithmes autoadaptatifs associés dans ce cas. On propose alors une nouvelle approche par rapport à la littérature qui consiste à utiliser une technique de localisation des normes, non pas basée sur des inégalités inverses, mais sur l'utilisation d'un opérateur Λ de localisation bien choisi.On peut alors construire des indicateurs d'erreur a posteriori fiables, efficaces, locaux et asymptotiquement exacts par rapport à la norme de Galerkin de l'erreur. On donne ensuite une méthode pour la construction de tels indicateurs. Les applications numériques sur des géométries 2D et 3D confirment l'exactitude asymptotique ainsi que l'optimalité du guidage de l'algorithme autoadaptatif.On étend ensuite ces indicateurs au cas de la propagation d'une onde électromagnétique. Plus précisément, on s'intéresse au cas de l'EFIE. On propose des généralisations des indicateurs de la littérature. On effectue la preuve de convergence quasi-optimale dans le cas d'un indicateur basé sur une localisation de la norme du résidu. On utilise le principe du Λ pour obtenir le premier indicateur d'erreur fiable, efficace et local pour cette équation. On en propose une seconde forme qui est également, théoriquement asymptotiquement exacte.
• On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2016, 322, pp.224-247. We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\texttt{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{loc}$ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods. (10.1016/j.jcp.2016.06.037)
  DOI : 10.1016/j.jcp.2016.06.037
• Development and use of higher-order asymptotics to solve inverse scattering problems
  
  • Cornaggia Rémi
  , 2016. The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions. The first part is dedicated to the localization and size identification of a buried inhomogeneity B in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba) quantifying the misfit between B and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with B. To this end, we produce an expansion of J with respect to the size of Ba , providing a polynomial approximation easier to minimize. This expansion, established up to the sixth order in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space. The main purpose of this second part is to characterize a microstructured two-phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (length and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.
• On the development and use of higher-order asymptotics for solving inverse scattering problems.
  
  • Cornaggia Rémi
  , 2016. The purpose of this work was to develop new methods to address inverse problems in elasticity,taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity Bᵗʳᵘᵉ in a 3Delastic domain. In this goal, we focused on the study of functionals J(Bₐ) quantifying the misfit between Bᵗʳᵘᵉ and a trial homogeneity Bₐ. Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Bₐ (location, size, mechanical properties ...) to find the best agreement with Bᵗʳᵘᵉ. To this end, we produced an expansion of J with respect to the size a of Bₐ, providing a polynomial approximation easier to minimize. This expansion, established up to O(a⁶) in a volume integral equations framework, is justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space ℝ³.The main purpose of this second part is to characterize a microstructured two-phases layered1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigen values of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method tore cover the macroscopic parameters (L and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.
• Randomization method and backward SDEs for optimal control of partially observed path-dependent stochastic systems
  
  • Bandini Elena
  • Cosso Andrea
  • Fuhrman Marco
  • Pham Huyên
  , 2016. We consider a unifying framework for stochastic control problem including the following features: partial observation, path-dependence (both with respect to the state and the control), and without any non-degeneracy condition on the stochastic differential equation (SDE) for the controlled state process, driven by a Wiener process. In this context, we develop a general methodology, refereed to as the randomization method, studied in [23] for classical Markovian control under full observation, and consisting basically in replacing the control by an exogenous process independent of the driving noise of the SDE. Our first main result is to prove the equivalence between the primal control problem and the randomized control problem where optimization is performed over change of equivalent probability measures affecting the characteristics of the exogenous process. The randomized problem turns out to be associated by duality and separation argument to a backward SDE, which leads to the so-called randomized dynamic programming principle and randomized equation in terms of the path-dependent filter, and then characterizes the value function of the primal problem. In particular, classical optimal control problems with partial observation affected by non-degenerate Gaussian noise fall within the scope of our framework, and are treated by means of an associated backward SDE.
• Problèmes d'interface en présence de métamatériaux : modélisation, analyse et simulations
  
  • Vinoles Valentin
  , 2016. Nous nous intéressons à des problèmes de transmission entre diélectriques et métamatériaux, milieux présentant des propriétés électromagnétiques inhabituelles comme des caractéristiques effectives négatives à certaines fréquences. Par exemple, ces milieux peuvent être construits comme des assemblages périodiques de microstructures résonantes et dans ce cas la théorie de l'homogénéisation permet de justifier mathématiquement ces propriétés effectives. En régime harmonique et dans des géométries à variables séparables, des calculs analytiques peuvent être menés. Ils révèlent dans des cas dits critiques des difficultés mathématiques: les solutions n'ont pas la régularité standard, voire le problème peut être mal posé.La première partie étudie ces problèmes de transmission en régime temporel pour lequel les métamatériaux sont modélisés par des modèles dispersifs (modèle de Drude ou de Lorentz). Les difficultés résident dans le choix d'un schéma de discrétisation mais surtout dans la construction de conditions absorbantes. La méthode retenue ici est celle des Perfectly Matched Layers (PMLs). Comme les PMLs classiques sont instables pour ces modèles du fait de la présence d'ondes inverses, nous proposons une nouvelle classe de PMLs pour lesquelles nous menons une analyse de stabilité. Cette dernière permet de construire des PMLs stables. Elles sont ensuite utilisées pour simuler le comportement en temps long d'un problème de transmission; nous illustrons alors le fait que le principe d'amplitude limite peut être mis en défaut en raison de résonances d'interface.La deuxième partie vise à pallier ces phénomènes d'interface en régime harmonique en revenant sur le processus d'homogénéisation classique, pour un milieu dissipatif. Pour des problèmes de transmission, il est connu que les modèles issus de cette méthode perdent en précision du fait de la présence de couches limites à l'interface. Nous proposons un modèle enrichi au niveau de l'interface. En combinant la méthode d'homogénéisation double-échelle et celle des développements asymptotiques raccordés, nous construisons des conditions de transmission non standards faisant intervenir des opérateurs différentiels le long de l'interface. Le calcul de ces conditions nécessite la résolution de problèmes de cellule et de problèmes non standards posés dans des bandes périodiques infinies. Une analyse d'erreur confirme l'amélioration de la précision du modèle. Des simulations numériques illustrent l'efficacité de ces nouvelles conditions. Enfin, cette démarche est reproduite formellement dans le cas des matériaux à fort contraste se comportant comme des métamatériaux. Nous montrons alors que ces nouvelles conditions permettent de régulariser le problème de transmission dans les cas critiques.
• New model for datasets citation and extraction reproducibility in VAMDC
  
  • Zwölf Carlo Maria
  • Moreau Nicolas
  • Dubernet Marie-Lise
  Journal of Molecular Spectroscopy, Elsevier, 2016, 327, pp.122-137. (10.1016/j.jms.2016.04.009)
  DOI : 10.1016/j.jms.2016.04.009
• HJB approach for a multi-boost launcher trajectory optimization problem *
  
  • Bokanowski O
  • Bourgeois E
  • Desilles Anna
  • Zidani Hasnaa
  , 2016. This work deals with an optimization problem for three-stage space launcher. The mission of the launcher is to put a given payload on the GEO orbit with the minimal propellant consumption. The considered flight sequence performs two boosts. The first one steers the launcher to a given GTO orbit. Then, after a ballistic flight, a second boost is used to perform the orbit transfer maneuver from the GTO to the GEO. We decompose the global optimization problem into two optimal control problems and we apply in this context the Hamilton-Jacobi-Bellman (HJB) approach together with a parameter optimization procedure.
• HJB approach for a multi-boost launcher trajectory optimization problem
  
  • Zidani Hasnaa
  • Bokanowski Olivier
  • Bourgeois Eric
  • Desilles Anna
  , 2016. In this work a trajectory optimization problem for three-stage space launcher is studied. The mission of the launcher is to put a given payload on the GEO orbit with the minimal propellant consumption. The considered flight sequence performs two boosts. The first one steers the launcher to a given GTO orbit. Then, after a ballistic flight, a second boost is used to perform the orbit transfer maneuver from the GTO to the GEO. We decompose the global optimization problem into two optimal control problems and we apply in this context the HJB approach together with a parameter optimization procedure.
• Competition and Coalition for Smart Energy Supply
  
  • Le Cadre Hélène
  • Pagnoncelli Bernardo
  • Homem-De-Mello Tito
  • Bourdin Nicolas
  , 2016. We model the relation between an aggregator taking forward positions in the electricity market and consumers as a Stackelberg game, formulated as a mathematical bilevel program with private information. At the lower-level, the consumers schedule their shiftable loads depending on the price signal sent by the aggregator. At the upper-level, the aggregator optimizes his price profile so as to reach a targeted profit, which is the maximum value guaranteeing that no consumer will leave the coalition while meeting fairness criterion imposed by the cost sharing mechanism that he has chosen. We provide original algorithms based on linear algebra, efficient enumeration schemes and learning to solve the bilevel program with private information, a framework which is hardly never tackled in the literature. Finally, we illustrate the results on a case study by comparing the consumers' bills and coalition size dynamics as functions of the aggregator's targeted profit for a number of cost sharing mechanisms such as stand-alone, separable and non-separable costs and the Shapley value which coincides here with equitable cost sharing.
• Toward viability and adaptive governance of tropical islands agrosystems
  
  • Angeon Valérie
  • Ozier-Lafontaine Harry
  • Gessner Marion
  • Merlot Bérengère
  • Chia Eduardo
  • Saint-Pierre Patrick
  • Desilles Anna
  • Durand Marie-Hélène
  • Bates Samuel
  , 2016, 52, pp.np. GAIA-TROP project contributes to identify and analyze socio-technical conditions for the implementation of agroecological transition in the French Caribbean. We wonder about adaptive capacities of these areas facing exacerbated global changes. The move from a high-yield farming system to a more viable farming system based on natural resources and social interactions has become one of the major issue and challenge for agronomic research. The FAO (2010) supports the end of the productionist model and claims for “smart agriculture” principles which are environmentally friendly. In this context, agroecological transition may be considered as a privileged pathway. Our approach assumes that this transition comes from the ability of all stakeholders (farmers and institutional stakeholders) to build a common paradigm on farming systems’ viability. This can be initiated through participatory approaches leading each stakeholder to transform their production systems. Our aim is to create a decision support tool dedicated to agriculture in an island context in order to better understand agrosystems. This tool will help decisions to ensure viability in the reality of global changes and other types of uncertainty.
• Open periodic waveguides. Theory and computation.
  
  • Vasilevskaya Elizaveta
  , 2016. The present work deals with propagation of acoustic waves in periodic media. These media have particularly interesting properties since the spectrum associated with the underlying wave operator in such media has a band-gap structure: there exist intervals of frequences for which monochromatic waves do not propagate. Moreover, by introducing linear defects in this kind of media, one can create guided modes inside the bands of forbidden frequences. In this work we show that it is possible to create such guided modes in the case of particular periodic media of grid type: more precisely, the periodic domain in question is R2 minus an infinite set of rectangular obstacles periodically spaced in two orthogonal directions (the distance between two neighbour obstacles being ε), which is locally perturbed by diminishing the distance between two columns of obstacles. The results are extended to the 3D case. This work has a theoretical and a numerical aspect. From the theoretical point of view the analysis is based on the fact that, ε being small, the spectrum of the operator associated with our problem is "close" to the spectrum of a problem posed on a graph which is a geometric limit of the domain as ε tends to 0. However, for the limit graph the spectrum can be computed explicitly. Then, we study the spectrum of the non-limit operator using asymptotic analysis. Theoretical results are illustrated by numerical computations obtained with a numerical method developed for study of periodic media: this method is based on the reduction of the initial (linear) eigenvalue problem posed in an unbounded domain to a non-linear problem posed in a bounded domain (using the exact Dirichlet- to-Neumann operator).
• Exact quadratic convex reformulations of mixed-integer quadratically constrained problems
  
  • Billionnet Alain
  • Elloumi Sourour
  • Lambert Amélie
  Mathematical Programming, Springer Verlag, 2016, 158 (1-2), pp.235-266. We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex. (10.1007/s10107-015-0921-2)
  DOI : 10.1007/s10107-015-0921-2
• Construction and analysis of an adapted spectral finite element method to convective acoustic equations
  
  • Hüppe Andreas
  • Cohen Gary
  • Imperiale Sebastien
  • Kaltenbacher Manfred
  Communications in Computational Physics, Global Science Press, 2016. The paper addresses the construction of a non spurious mixed spectral finite element (FE) method to problems in the field of computational aeroacous-tics. Based on a computational scheme for the conservation equations of linear acoustics, the extension towards convected wave propagation is investigated. In aeroacoustic applications, the mean flow effects can have a significant impact on the generated sound field even for smaller Mach numbers. For those convec-tive terms, the initial spectral FE discretization leads to non-physical, spurious solutions. Therefore, a regularization procedure is proposed and qualitatively investigated by means of discrete eigenvalues analysis of the discrete operator in space. A study of convergence and an application of the proposed scheme to simulate the flow induced sound generation in the process of human phonation underlines stability and validity. (10.4208/cicp.250515.161115a)
  DOI : 10.4208/cicp.250515.161115a
• Seismic Wave Amplification in 3D Alluvial Basins: 3D/1D Amplification Ratios from Fast Multipole BEM Simulations
  
  • Meza Fajardo Kristel Carolina
  • Semblat Jean-François
  • Chaillat Stéphanie
  • Lenti Luca
  Bulletin of the Seismological Society of America, Seismological Society of America, 2016, 106 (3), pp.1267-1281. In this work, we study seismic wave amplification in alluvial basins having 3D standard geometries through the Fast Multipole Boundary Element Method in the frequency domain. We investigate how much 3D amplification differs from the 1D (horizontal layering) case. Considering incident fields of plane harmonic waves, we examine the relationships between the amplification level and the most relevant physical parameters of the problem (impedance contrast, 3D aspect ratio, vertical and oblique incidence of plane waves). The FMBEM results show that the most important parameters for wave amplification are the impedance contrast and the so-called equivalent shape ratio. Using these two parameters, we derive simple rules to compute the fundamental frequency for various 3D basin shapes and the corresponding 3D/1D amplification factor for 5% damping. Effects on amplification due to 3D basin asymmetry are also studied and incorporated in the derived rules. (10.1785/0120150159)
  DOI : 10.1785/0120150159
• Solitons acoustiques en milieu variable
  
  • Mercier Jean-François
  • Lombard Bruno
  , 2016.
• Couplage BEM-rayon pour le contrôle non destructif par ultrasons
  
  • Pesudo Laure
  • Bonnet Marc
  • Collino Francis
  • Demaldent Edouard
  • Imperiale Alexandre
  , 2016.
• Dealing with Uncertainty in the Smart Grid: A Learning Game Approach
  
  • Le Cadre Hélène
  • Bedo Jean-Sébastien
  Computer Networks, Elsevier, 2016. We model the smart grid as a decentralized and hierarchical network, made up of three categories of agents: suppliers, generators and captive consumers organized in microgrids. To optimize their decisions concerning prices and traded power, agents need to forecast the demand of the microgrids and the fluctuating renewable productions. The biases resulting from the decentralized learning could create imbalances between demand and supply leading to penalties for suppliers and for generators. We analytically determine prices that provide generators with a guarantee to avoid such penalties, transferring risk to the suppliers. Additionally, we prove that collaborative learning, through coalitions of suppliers among which information is shared, minimizes the sum of their average risk. Simulations, run for a large sample of parameter combinations, using external and internal regret minimization, show that the convergence of collaborative learning strategies is clearly faster than that resulting from individual learning. Finally, we analyze the suppliers' incentives to organize in a grand coalition versus multiple coalitions, and the tightness of the learning algorithm's theoretical bounds. (10.1016/j.comnet.2016.03.002)
  DOI : 10.1016/j.comnet.2016.03.002
• Représentation probabiliste d'équations HJB pour le contrôle optimal de processus à sauts, EDSR (équations différentielles stochastiques rétrogrades) et calcul stochastique.
  
  • Bandini Elena
  , 2016. Dans le présent document on aborde trois divers thèmes liés au contrôle et au calcul stochastiques, qui s'appuient sur la notion d'équation différentielle stochastique rétrograde (EDSR) dirigée par une mesure aléatoire. Les trois premiers chapitres de la thèse traitent des problèmes de contrôle optimal pour différentes catégories de processus markoviens non-diffusifs, à horizon fini ou infini. Dans chaque cas, la fonction valeur, qui est l'unique solution d'une équation intégro-différentielle de Hamilton-Jacobi-Bellman (HJB), est représentée comme l'unique solution d'une EDSR appropriée. Dans le premier chapitre, nous contrôlons une classe de processus semi-markoviens à horizon fini; le deuxième chapitre est consacré au contrôle optimal de processus markoviens de saut pur, tandis qu'au troisième chapitre, nous examinons le cas de processus markoviens déterministes par morceaux (PDMPs) à horizon infini. Dans les deuxième et troisième chapitres les équations d'HJB associées au contrôle optimal sont complètement non-linéaires. Cette situation survient lorsque les lois des processus contrôlés ne sont pas absolument continues par rapport à la loi d'un processus donné. Etant donné ce caractère complètement non-linéaire, ces équations ne peuvent pas être représentées par des EDSRs classiques. Dans ce cadre, nous avons obtenu des formules de Feynman-Kac non-linéaires en généralisant la méthode de la randomisation du contrôle introduite par Kharroubi et Pham (2015) pour les diffusions. Ces techniques nous permettent de relier la fonction valeur du problème de contrôle à une EDSR dirigée par une mesure aléatoire, dont une composante de la solution subit une contrainte de signe. En plus, on démontre que la fonction valeur du problème de contrôle originel non dominé coïncide avec la fonction valeur d'un problème de contrôle dominé auxiliaire, exprimé en termes de changements de mesures équivalentes de probabilité. Dans le quatrième chapitre, nous étudions une équation différentielle stochastique rétrograde à horizon fini, dirigée par une mesure aléatoire à valeurs entières μ sur ℝ+ x E, où E est un espace lusinien, avec compensateur de la forme v(dt dx) = dAt φt(dx). Le générateur de cette équation satisfait une condition de Lipschitz uniforme par rapport aux inconnues. Dans la littérature, l'existence et unicité pour des EDSRs dans ce cadre ont été établies seulement lorsque A est continu ou déterministe. Nous fournissons un théorème d'existence et d'unicité même lorsque A est un processus prévisible, non décroissant, continu à droite. Ce résultat s’applique par exemple, au cas du contrôle lié aux PDMPs. En effet, quand μ est la mesure de saut d'un PDMP sur un domaine borné, A est prévisible et discontinu. Enfin, dans les deux derniers chapitres de la thèse nous traitons le calcul stochastique pour des processus discontinus généraux. Dans le cinquième chapitre, nous développons le calcul stochastique via régularisations des processus à sauts qui ne sont pas nécessairement des semimartingales. En particulier nous poursuivons l'étude des processus dénommés de Dirichlet faibles, dans le cadre discontinu. Un tel processus X est la somme d'une martingale locale et d'un processus adapté A tel que [N, A] = 0, pour toute martingale locale continue N. Pour une fonction u: [0, T] x ℝ → ℝ de classe C⁰′¹ (ou parfois moins), on exprime un développement de u(t, Xt), dans l'esprit d'une généralisation du lemme d'Itô, lequel vaut lorsque u est de classe C¹′². Le calcul est appliqué dans le sixième chapitre à la théorie des EDSRs dirigées par des mesures aléatoires. Dans de nombreuses situations, lorsque le processus sous-jacent X est une semimartingale spéciale, ou plus généralement, un processus de Dirichlet spécial faible, nous identifions les solutions des EDSRs considérées via le processus X et la solution u d’une EDP intégro-différentielle associée.
• Le théorème de Noether
  
  • Perez Jérôme
  Quadrature, EDP Sciences, 2016 (100), pp.12 pages. We present the genesis and some applications of the famous Emmy Noether’s invariants theorem proposed a century ago in 1916. The genesis emphasises the historical context of this theorem, and the proposed applications show how this little piece of mind can be used to recover large parts of physics.