This dissertation presents innovative unified approaches to understand and predict the motion between potential wells. The theoretical-computational framework, based on the tube dynamics, will reveal how the dissipative and gyroscopic forces change the phase space structure that governs the escape (or transition) from potential wells. In higher degree of freedom systems, the motion between potential wells is complicated due to the existence of multiple escape routes usually through an index-1 saddle. Thus, this dissertation firstly studies the local behavior around the index-1 saddle to establish the criteria of escape taking into account the dissipative and gyroscopic forces. In the analysis, an idealized ball rolling on a surface is selected as an example to show the linearized dynamics due to its special interests that the gyroscopic force can be easily introduced by rotating the surface. Based on the linearized dynamics, we find that the boundary of the initial conditions of a given energy for the trajectories that transit from one side of a saddle to the other is a cylinder and ellipsoid in the conservative and dissipative systems, respectively. Compared to the linear systems, it is much more challenging or sometimes impossible to get analytical solutions in the nonlinear systems. Based on the analysis of linearized dynamics, the second goal of this study is developing a bisection method to compute the transition boundary in the nonlinear system using the dynamic snap-through buckling of a buckled beam as an example. Based on the Euler-Bernoulli beam theory, a two degree of freedom Hamiltonian system can be generated via a two mode-shape truncation. The transition boundary on the Poincar'e section at the well can be obtained by the bisection method. The numerical results prove the efficiency of the bisection method and show that the amount of trajectories that escape from the potential well will be smaller if the damping of the system is increasing. Finally, we present an alternative idea to compute the transition boundary of the nonlinear system from the perspective of the invariant manifold. For the conservative systems, the transition boundary of a given energy is the invariant manifold of a periodic orbit. The process of obtaining such invariant manifold compromises two parts, including the computation of the periodic orbit by solving a proper boundary-value problem (BVP) and the globalization of the manifold. For the dissipative systems, however, the transition boundary of a given energy becomes the invariant manifold of an index-1 saddle. We present a BVP approach using the small initial sphere in the stable subspace of the linearized system at one end and the energy at the other end as the boundary conditions. By using these algorithms, we obtain the nonlinear transition tube and transition ellipsoid for the conservative and dissipative systems, respectively, which are topologically the same as the linearized dynamics. / Doctor of Philosophy / Transition or escape events are very common in daily life, such as the snap-through of plant leaves and the flipping over of umbrellas on a windy day, the capsize of ships and boats on a rough sea. Some other engineering problems related to escape, such as the collapse of arch bridges subjected to seismic load and moving trucks, and the escape and recapture of the spacecraft, are also widely known. At first glance, these problems seem to be irrelated. However, from the perspective of mechanics, they have the same physical principle which essentially can be considered as the escape from the potential wells. A more specific exemplary representative is a rolling ball on a multi-well surface where the potential energy is from gravity. The purpose of this dissertation is to develop a theoretical-computational framework to understand how a transition event can occur if a certain energy is applied to the system. For a multi-well system, the potential wells are usually connected by saddle points so that the motion between the wells generally occurs around the saddle. Thus, knowing the local behavior around the saddle plays a vital role in understanding the global motion of the nonlinear system. The first topic aims to study the linearized dynamics around the saddle. In this study, an idealized ball rolling on both stationary and rotating surfaces will be used to reveal the dynamics. The effect of the gyroscopic force induced by the rotation of the surface and the energy dissipation will be considered. In the second work, the escape dynamics will be extended to the nonlinear system applied to the snap-through of a buckled beam. Due to the nonlinear behavior existing in the system, it is hard to get the analytical solutions so that numerical algorithms are needed. In this study, a bisection method is developed to search the transition boundary. By using such method, the transition boundary on a specific Poincar'e section is obtained for both the conservative and dissipative systems. Finally, we revisit the escape dynamics in the snap-through buckling from the perspective of the invariant manifold. The treatment for the conservative and dissipative systems is different. In the conservative system, we compute the invariant manifold of a periodic orbit, while in the dissipative system we compute the invariant manifold of a saddle point. The computational process for the conservative system consists of the computation of the periodic orbit and the globalization of the corresponding manifold. In the dissipative system, the invariant manifold can be found by solving a proper boundary-value problem. Based on these algorithms, the nonlinear transition tube and transition ellipsoid in the phase space can be obtained for the conservative and dissipative systems, respectively, which are qualitatively the same as the linearized dynamics.

This thesis is devoted to creating a systematic way of determining all inequivalent orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero curvature. To achieve this, we represent the problem with Killing tensors and employ the recently developed invariant theory of Killing tensors. Killing tensors on the model spaces of spherical and hyperbolic space enjoy a remarkably simple form; even more striking is the fact that their parameter tensors admit the same symmetries as the Riemann curvature tensor, and thus can be considered algebraic curvature tensors. Using this property to obtain invariants and covariants of Killing tensors, together with the web symmetries of the associated orthogonal coordinate webs, we establish an equivalence criterion for each space. In the case of three-dimensional spherical space, we demonstrate the surprising result that these webs can be distinguished purely by the symmetries of the web. In the case of three-dimensional hyperbolic space, we use a combination of web symmetries, invariants and covariants to achieve an equivalence criterion. To completely solve the equivalence problem in each case, we develop a method for determining the moving frame map for an arbitrary Killing tensor of the space. This is achieved by defining an algebraic Ricci tensor. Solutions to equivalence problems of Killing tensors are particularly useful in the areas of multiseparability and superintegrability. This is evidenced by our analysis of symmetric potentials defined on three-dimensional spherical and hyperbolic space. Using the most general Killing tensor of a symmetry subspace, we derive the most general potential “compatible” with this Killing tensor. As a further example, we introduce the notion of a joint invariant in the vector space of Killing tensors and use them to characterize a well-known superintegrable potential in the plane. xiii

Cette thèse aborde les problèmes de la modélisation et de la commande d'un micro-réseau courant continu (CC) en vue de la gestion énergétique optimale, sous contraintes et incertitudes. Le micro-réseau étudie contient des dispositifs de stockage électrique (batteries ou super-capacités), des sources renouvelables (panneaux photovoltaïques) et des charges (un système d'ascenseur motorise par une machine synchrone a aimant permanent réversible). Ces composants, ainsi que le réseau triphasé, sont relies a un bus commun en courant continu, par des convertisseurs dédies. Le problème de gestion énergétique est formule comme un problème de commande optimale qui prend en compte la dynamique du système, des contraintes sur les variables, des prédictions sur les prix, la consommation ou la production et des profils de référence.Le micro-réseau considère est un système complexe, de par l'hétérogénéité de ses composants, sa nature distribuée, la non-linéarité de certaines dynamiques, son caractère multi-physiques (électromécanique, électrochimique, électromagnétique), ainsi que la présence de contraintes et d'incertitudes. La représentation consistante des puissances échangées et des énergies stockées, dissipées ou fournies au sein de ce système est nécessaire pour assurer son opération optimale et fiable.Le problème pose est abordé via l'usage combine de la formulation hamiltonienne a port, de la platitude et de la commande prédictive économique base sur le modelé. Le formalisme hamiltonien a port permet de décrire les conservations de la puissance et de l'énergie au sein du micro-réseau explicitement et de relier les composants hétérogènes dans un même cadre théorique. Les non linéarités sont gérées par l'introduction de la notion de platitude démentielle et la sélection de sorties plates associées au modèle hamiltonien a ports. Les profils de référence sont génères a l'aide d'une para métrisation des sorties plates de telle sorte que l'énergie dissipée soit minimisée et les contraintes physiques satisfaites. Les systèmes hamiltoniens sur graphes sont ensuite introduits pour permettre la formulation et la résolution du problème de commande prédictive _économique a l'échelle de l'ensemble du micro-réseau CC. Les stratégies de commande proposées sont validées par des résultats de simulation pour un système d'ascenseur multi-sources utilisant des données réelles, identifiées sur base de mesures effectuées sur une machine synchrone. / The goals of this thesis is to propose modelling and control solutions for the optimal energy management of a DC microgrid under constraints. The studied microgrid system includes electrical storage units (e.g., batteries, supercapacitors), renewable sources (e.g., solar panels) and loads (e.g., an electro-mechanical elevator system). These interconnected components are linked to a three phase electrical grid through a DC bus and associated DC/AC converters. The optimal energy management is usually formulated as an optimal control problem which takes into account the system dynamics, cost, constraints and reference profiles.An optimal energy management for the microgrid is challenging with respect to classical control theories. Needless to say, a DC microgrid is a complex system due to its heterogeneity, distributed nature (both spatial and in sampling time), nonlinearity of dynamics, multi-physic characteristics, the presence of constraints and uncertainties. Moreover, the power-preserving structure and the energy conservation of a microgrid are essential for ensuring a reliable operation.This challenges are tackled through the combined use of port-Hamiltonian formulations, differential flatness, and economic Model Predictive Control.The Port-Hamiltonian formalism allows to explicitly describe the power-preserving structure and the energy conservation of the microgrid and to connect different components of different physical natures through the same formalism. The strongly non-linear system is then translated into a flat representation. Taking into account differential flatness properties, reference profiles are generated such that the dissipated energy and various physical constraints are taken into account. Lastly, we minimize the purchasing/selling electricity cost within the microgrid using the economic Model Predictive Control with the Port-Hamiltonian formalism on graphs.The proposed control designs are validated through simulation results.

Dans cette thèse nous présentons les résultats sur l'emploi des systèmes Hamiltoniens à port et des systèmes de contact commandés pour la modélisation et la commande de systèmes issus de la Thermodynamique Irréversible. Premièrement nous avons défini une classe de pseudo-systèmes Hamiltoniens à port, appelée systèmes Hamiltoniens à port irréversibles, qui permet de représenter simultanément le premier et le second principe de la Thermodynamique et inclut des modèles d'échangeurs thermiques ou de réacteurs chimiques. Ces systèmes ont été relevés sur l'espace des phases thermodynamiques muni d’une forme de contact, définissant ainsi une classe de systèmes de contact commandés, c'est-à-dire des systèmes commandés non-linéaires définis par des champs de contacts stricts. Deuxièmement, nous avons montré que seul un retour d'état constant préserve la forme de contact et avons alors résolu le problème d'assignation d'une forme de contact en boucle fermée. Ceci a mené à la définition de systèmes de contact entrée-sortie et l'analyse de leur équivalence par retour d'état. Troisièmement, nous avons montré que les champs de contact n'étaient en général pas stables en leur zéros et avons alors traité du problème de la stabilisation sur une sous-variété de Legendre en boucle fermée. / This doctoral thesis presents results on the use of port Hamiltonian systems (PHS) and controlled contact systems for modeling and control of irreversible thermodynamic processes. Firstly, Irreversible PHS (IPHS) has been defined as a class of pseudo-port Hamiltonian system that expresses the first and second principle of Thermodynamics and encompasses models of heat exchangers and chemical reactors. These IPHS have been lifted to the complete Thermodynamic Phase Space endowed with a natural contact structure, thereby defining a class of controlled contact systems, i.e. nonlinear control systems defined by strict contact vector fields. Secondly, it has been shown that only a constant control preserves the canonical contact structure, hence a structure preserving feedback necessarily shapes the closed-loop contact form. The conditions for state feedbacks shaping the contact form have been characterized and have lead to the definition of input-output contact systems. Thirdly, it has been shown that strict contact vector fields are in general unstable at their zeros, hence the condition for the the stability in closed-loop has been characterized as stabilization on some closed-loop invariant Legendre submanifolds

La réduction fluide des équations cinétiques est un procédé couramment utilisé en physique des plasmas qui a pour objectif de remplacer la fonction de distribution définie dans l'espace des phases par des grandeurs fluides comme la densité et la pression. Cette réduction diminue la complexité du système initial. En contrepartie, la réduction fluide s'accompagne de la nécessité d'effectuer une fermeture sur les moments d'ordre supérieur. Celle-ci est souvent construite ad hoc en se basant sur des arguments physiques (e.g., quantités conservées, existance d'un théorème H, ...). Dans ce manuscrit, on propose un procédé de réduction qui permet de préserver la structure hamiltonienne du modèle cinétique parent. Ceci est important pour assurer qu'aucune dissipation d'origine non physique est introduite dans le modèle fluide, le munissant ainsi d'une structure hamiltonienne dont l'origine peut être suivie jusqu'à celle de la dynamique microscopique des particules. On utilise cette méthode pour construire des modèles fluides non-adiabatiques pour les trois premiers moments de la fonction de distribution associée à l'équation de Vlasov-Poisson à une dimension, i.e., la densité, la vitesse fluide et la pression. Les résultats sont ensuite étendus pour inclure la dynamique du flux de chaleur en considérant des fermetures construites à partir de l'analyse dimensionnelle. On montre également, pour un nombre arbitraire de champs, la relation existant avec le modèle water-bags. L'extension à des dimensions supérieures est étudiée dans le cadre de l'équation drift-cinétique ainsi que de l'équation de Vlasov-Poisson à trois dimensions. / Fluid reduction of kinetic equations is a ubiquitous procedure in plasma physics which aims to replace the distribution function defined in phase space with more concrete fluid quantities defined solely in configuration space such as the density, the fluid velocity and the pressure. This reduction lowers the complexity of the initial system, leading to a gain of physical insight into the phenomena under investigation as well as a significant decrease of the cost of numerical simulations. On the other hand, in order for the fluid reduction to be complete, one needs to perform a closure on the higher order fluid moments. The choice of the closure usually relies on some ad hoc physical arguments (e.g., conserved quantities, existence of an H-theorem, ...). In this manuscript, we present a reduction procedure that preserves the Hamiltonian structure of the parent kinetic model. This is important in order to ensure that no non-physical dissipation is introduced in the resulting fluid model, providing it with a geometric structure that can be traced back to the microscopic dynamics of the particles. We use this procedure to derive non-adiabatic fluid models for the first three fluid moments of the distribution function of the one dimensional Vlasov-Poisson equation, namely the density, the fluid velocity and the pressure. The results are extended to include the dynamics of the heat-flux by considering a closure based on dimensional analysis. For an arbitrary number of fields, we demonstrate the relationship with the water-bags model. Finally, the extension to higher dimensions is investigated through the drift-kinetic equation and the three dimensional Vlasov-Poisson equation.

Tout phénomène, qu’il soit biologique, géologique ou mécanique peut être décrit à l’aide de lois de la physique en termes d’équations différentielles, algébriques ou intégrales, mettant en relation différentes variables physiques. Les objectifs de la thèse sont de montrer comment les systèmes à paramètres distribués peuvent être modélisés par un modèle bond graph, qui est par nature un modèle à paramètres localisés. Deux approches sont possibles : - utiliser une technique d’approximation qui discrétise le modèle initialement sous forme d’équations aux dérivées partielles (EDP) dans le domaine spatial, en supposant que les phénomènes physiques distribués peuvent être considérés comme homogènes dans certaines parties de l’espace, donc localisés. - déterminer la solution des EDP qui dépend du temps et de l’espace, puis à approximer cette solution avec différents outils numériques. Le premier chapitre rappelle quelques méthodes classiques utilisées pour l’approximation des EDP et les modèles bond graphs correspondants.Dans le deuxième chapitre, l’approche port-Hamiltonienne est présentée et son extension aux systèmes à paramètres distribués est proposée. Dans le troisième chapitre, les principaux modèles utilisés pour la représentation des flux de trafic routier sont rappelés et mis en œuvre en simulation. Ceci conduit à des comparaisons, d’une part entre différentes méthodes de résolution numérique et d’autre part entre différents modèles. Dans le quatrième chapitre, une approche originale propose d’étendre la représentation bond graph issue de la méthodologie Computational Fluid Dynamics au flux de trafic, en utilisant un modèle EDP à deux équations proposé par Jiang / Virtually every phenomenon in nature, whether biological, geological, or mechanical, can be described with the aid of the laws of physics, in terms of algebraic, differential, or integral equations relating various quantities of interest. The objectives of the thesis were to show how distributed parameter systems can be modeled using a bond graph model, which is by its nature itself a lumped parameter model. Two ways are possible :- using an approximation technique to discretize the model in the space domain, assuming that physical distributed phenomena can be considered as homogenous in some parts of space, and thus lumped. Different bond graph models can be obtained depending on the technique used.- determining a solution of the PDE depending on space and time, and thus to approximate this solution by means of different kinds of tools.In chapter 1, some classical methods used for approximation of partial differential equations are recalled and the corresponding bond graph model is designed. For each of them advantages and drawbacks are presented.In the second chapter, the port-Hamiltonian approach for distributed parameter system is presented, and a new result is proposed for telegrapher’s equation solving.In the third chapter, the main models used for traffic flow representation are presented and some of them are implemented in simulation. A comparison is done on one hand on different numerical methods applied on the first class of models (1-eq. model) and on the other hand between 1-equation and 2- equation models.In chapter 4, we have proposed an original approach extending Computational Fluid Dynamics bond graph representation to traffic flow, using Jiang’s model

M.Sc. / In this thesis we study the techniques used to calculate the Hamilton operators related to linear optical quantum computing. We also discuss the basic building blocks of linear optical quantum computing (LOQC) by looking at the logic gates and the physical instruments of which they are made.

Ces travaux de thèse traitent de l'approximation en dimension finie de système de dimension infinie. La classe considérée est celle des systèmes hamiltoniens à ports. Nous étudions dans un premier temps les systèmes d'équations différentielles ordinaires. Sur la base d'un intégrateur énergétique, nous définissons une classe de dynamiques passives discrètes qui est invariante par interconnexion. Nous obtenons alors des conditions de stabilité (LMI) pour des dynamiques en réseau en présence de retards et d'incertitudes, et proposons une méthode de synthèse énergétique stabilisante. Ces développements ont été validés expérimentalement par la mise en oeuvre d'une commande énergétique sur un convertisseur de puissance (Buck). Nous étudions ensuite le formalisme hamiltonien en dimension infinie. Nous proposons une approximation qui combine une semi-discrétisation et un intégrateur énergétique. La composabilité mixte est étudiée et une méthode de synthèse IDA-PBC a été développée. L'ensemble des résultats obtenus sont illustrés numériquement dans le manuscrit. / This thesis work dealing with finite dimensional approximation of infinite dimension system. The class considered is that of Hamiltonian systems in ports. We study initially ordinary differential equations systems. Based on an energy integrator, we define a class of discrete passive dynamics is invariant interconnection. We obtain the stability conditions (LMI) for dynamic network in the presence of delays and uncertainties, and propose a method of stabilizing energy synthesis. These developments were experimentally validated by the implementation of an energy control a power converter (Buck). We then study the Hamiltonian formalism in infinite dimensions. We offer an approximation that combines a semi-discretization and an energy integrator. The mixed composability is studied and a method of synthesis IDA-PBC was developed. All the obtained results are numerically illustrated in the manuscript.

In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses forming a central configuration. An expression of four masses is established depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically it is proved that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central. The singularities of simultaneous binary collisions in collinear four-body problem is regularized by explicitly constructing new coordinates and time transformation in section 3. The motion in the new coordinates and time scale across simultaneous binary collision is at least C^2. Furthermore, the behavior of the motion closing, across and after the simultaneous binary collision, is also studied. Many different types of periodic solutions involving single binary collisions and simultaneous binary collisions are constructed. In section 4, the linear stability is studied for the Kepler orbits of the rhombus four-body problem. We show that, for given four proper masses, there exists a family of periodic solutions for which each body with the proper mass is at the vertex of a rhombus and travels along an elliptic Kepler orbit. Instead of studying the 8 degrees of freedom Hamilton system for planar four-body problem, we reduce this number by means of some symmetry to derive a two degrees of freedom system which then can be used to determine the linear instability of the periodic solutions. After making a clever change of coordinates, a two dimensional ordinary differential equation system is obtained, which governs the linear instability of the periodic solutions. The system is surprisingly simple and depends only on the length of the sides of the rhombus and the eccentricity e of the Kepler orbit. In section 5, index theory for symplectic paths introduced by Y.Long is applied to study the stability of a periodic solution x for a Hamiltonian system. We establish a necessary and sufficient condition for stability of the periodic solution x in two and four dimension.

Dans l'optique de produire de l'énergie à travers les réactions de fusion, nous sommes amenés à étudier des phénomènes physiques qui ont lieux dans les tokamaks. Les instabilités qui existent dans les tokamaks peuvent fortement dégrader le confinement et ont un impacte sur le fonctionnement de futurs réactions à fusion. Des mesures révèlent un fort transport radial. Même si ce transport radial est en partie est une conséquence des collisions, l'instabilité d'interchange est la source dominante à ce transport puisque le type de plasmas nous intéressant sont faiblement collisionnels. Dans la limite non collisionnelle, la description hamiltonienne permet de décrire le système dynamique des particules du plasmas dans un champ électromagnétique. Nous donnons de l'importance à cette description afin de pouvoir accéder aux outils hamiltoniens.Nous travaillons sur la modélisation et le contrôle hamiltonien du transport radial. Après avoir écrit le modèle hamiltonien des particules d'un plasma magnétisé, nous introduisons les réductions de ce modèle lagrangien en modèles eulériens réduits afin de s'adapter à certains calculs numériques et théoriques. Ces réductions donnent lieux aux équations fluides hamiltonien. Cependant, nous montrons que ces réductions peuvent faire perdre la propriété hamiltonienne. En particulier pour obtenir un modèle ayant la température des ions (puisqu'elle n'est pas négligeable au centre du plasma), nous montrons la procédure conservant la propriété hamiltonienne à partir du modèle sans température des ions.Quant à l'étude du transport radial, nous appliquons une des propriétés hamiltoniennes (le contrôle) afin de créer une barrière de transport par des perturbations du système. Nous étudions de manière idéale l'effet du contrôle à travers la dynamique lagrangienne des traceurs appelés particules test. Nous faisons particulièrement des efforts dans la prise en compte des contraintes numériques et expérimentales. Nous montrons notamment la robustesse du contrôle lors de l'application des perturbations par des sondes de Langmuir.Finalement, nous étudions l'application du contrôle dans un modèle eulérien décrivant la rétroaction du plasmas (à travers la densité et le potentiel électrique) lorsque nous appliquons les perturbations. Cette étape permet de prendre en compte le couplage du système plasma-perturbations. En utilisant un code fluide permettant de décrire le plasma de bord lors de perturbations générées par des sondes de Langmuir. Nous développons un algorithme permettant de calculer le contrôle en tout temps en fonction du potentiel électrique. Nous montrons alors que la valeur moyenne du potentiel électrique joue un rôle important pour l'application du contrôle dans un modèle fluide. / In order to produce energy through fusion reactions, we are led to study of physical phenomena that occur in tokamaks. The instabilities that exist in tokamaks can significantly degrade the confinement and have an impact on the operation of future fusion reactors. Measurements reveal a strong radial transport. Although this is partly a consequence of collisions, the interchange instability is the dominant source to transport since the type of plasmas that interest us are weakly collisional. Within non collisional limit, the Hamiltonian description used to describe the dynamical system of charged particles in an electromagnetic field. We give importance to this description in order to access the Hamiltonian tools.We are working on modeling and control Hamiltonian of radial transport. After writing the Hamiltonian model of particles in a magnetized plasma, we introduced some reductions from Lagrangian models to Eulerian reduced models in order to accommodate some theoretical and numerical calculations. These places give the Hamiltonian fluid equations. However, we show that these reductions may lose the Hamiltonian property. In particular for a model with the ion temperature (not neglected at the center of the plasma), we show the procedure preserving the Hamiltonian property from the model without ion temperature.As for the study of radial transport, we apply one of the Hamiltonian properties (the control) to create a transport barrier by perturbations of the system. We are looking ideally the effect of control through the Lagrangian dynamics of tracers called test particles. We make particular efforts in the consideration of numerical and experimental constraints. We show the robustness of control when applying perturbations by Langmuir probes.Finally, we study the application of control in an Eulerian model describing the feedback of plasma (through the density and the electric potential) when we apply the perturbations. This step allows to take into account the coupling of the system plasma-perturbations. We use a numerical code to describe the plasma at the edge during perturbations generated by Langmuir probes. We develop an algorithm to calculate the control at all times depending on the electric potential. Finally we show that the average value of electric potential plays an important role in the implementation of control in a fluid model.