Neste trabalho consideramos sistemas elípticos de tipo hamiltoniano, envolvendo o operador Laplaciano, com uma parte linear dependendo de dois parâmetros e uma perturbação sublinear. Obtemos a existência de pelo menos duas soluções quando a parte linear está perto da ressonância (este fenômeno é chamado de quase ressonância). Mostramos também a existência de uma terceira solução, quando a quase ressonância é em relação ao primeiro autovalor do operador Laplaciano. No caso ressonante obtemos resultados análogos, adicionando mais uma perturbação sublinear. Os sistemas estão associados a funcionais fortemente indefinidos, e as soluções são obtidas através do Teorema de Ponto de Sela e aproximação de Galerkin. / In this work we consider elliptic systems of hamiltonian type, involving the Laplacian operator, a linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least two solutions when the linear part is near resonance (this phenomenon is called almost-resonance). We also show the existence of a third solution when the almost-resonance is with respect to the first eigenvalue of the Laplacian operator. In the resonant case, we obtain similar results, with an additional sublinear term. These systems are associated with strongly indefinite functionals, and the solutions are obtained by Saddle Point Theorem and Galerkin approximation.

Cette thèse s’intéresse au développement d’un contrôleur sûre et robuste en tant que partie intégrante d’un pancréas artificiel. Plus précisément, nous sommes intéressés à contrôler la partie du traitement usuel qui a pour but d’équilibrer la glycémie du patient. C’est ainsi que le développement d’une commande prédictive nonlinéaire robuste basée sur la résolution d’un problème de point selle a été envisagé. Afin de valider les performances du contrôleur dans une situation réaliste, des simulations numériques en utilisant une plate-forme de tests validée par la FDA sont envisagées.Dans une première partie, nous présentons une extension de la classique commande prédictive nonlinéaire dont le but est d’assurer le contrôle robuste de systèmes décrits par des équations différentielles ordinaires non linéaires dans un cadre échantillonné. Ce contrôleur, qui calcule une action de contrôle adéquate en considérant la solution d’un problème de point selle, est appelé saddle point model predictive controller (SPMPC). En utilisant cette commande, il est prouvé que le système converge en temps fini dans un espace borné et, en supposant une certaine structure dans le problème, qu’il est pratiquement stable entrée-état. Ensuite, nous nous sommes intéressés à la résolution numérique. Pour ce faire, nous proposons une méthode de résolution inspirée de la méthode du Langrangien augmenté et qui fait usage de modèles adjoints.Dans un deuxième temps, nous considérons l’application de ce contrôleur au problème du contrôle artificiel de la glycémie. Après une phase de modélisation, nous avons retenu deux modèles : un modèle simple qui est utilisé pour développer la commande et un modèle complexe qui est utilisé comme un simulateur réaliste de patients. Ce dernier est nécessaire pour valider notre approche de contrôle. Afin de calculer une entrée de commande adéquate, la commande SPMPC a besoin de l’état complet du système. Or, les capteurs ne peuvent fournir qu’une valeur du glucose sanguin. C’est pourquoi le développement d’un observateur est envisagé. Ensuite, des simulations sont réalisées. Les résultats obtenus témoignent de l’intérêt de l’approche retenue. En effet, pour tous les patients, aucune hypoglycémie n’a été observée et le temps passé en état hyperglycémique est suffisamment faible pour ne pas être dommageable. Enfin, l’intérêt d’étendre l’approche de commande SPMPC au problème de contrôle de systèmes décrits par des équations différentielles retardées non linéaires dans un cadre échantillonné est formellement investigué. / This thesis deals with the design of a robust and safe control algorithm to aim at an artificial pancreas. More precisely we will be interested in controlling the stabilizing part of a classical cure. To meet this objective, the design of a robust nonlinear model predictive controller based on the solution of a saddle point optimization problem is considered. Also, to test the controller performances in a realistic case, numerical simulations on a FDA validated testing platform are envisaged.In a first part, we present an extension of the usual nonlinear model predictive controller designed to robustly control, in a sampled-data framework, systems described by nonlinear ordinary differential equations. This controller, which computes the best control input by considering the solution of a constrained saddle point optimization problem, is called saddle point model predictive controller (SPMPC). Using this controller, it is proved that the closed-loop is Ultimately Bounded and, with some assumptions on the problem structure, Input-to State practically Stable. Then, we are interested in numerically solving the corresponding control problem. To do so, we propose an algorithm inspired from the augmented Lagrangian technique and which makes use of adjoint model.In a second part, we consider the application of this controller to the problem of artificial blood glucose control. After a modeling phase, two models are retained. A simple one will be used to design the controller and a complex one will be used to simulate realistic virtual patients. This latter is needed to validate our control approach. In order to compute a good control input, the SPMPC controller needs the full state value. However, the sensors can only provide the value of blood glucose. That is why the design of an adequate observer is envisaged. Then, numerical simulations are performed. The results show the interest of the approach. For all virtual patients, no hypoglycemia event occurs and the time spent in hyperglycemia is too short to induce damageable consequences. Finally, the interest of extending the SPMPC approach to consider the control of time delay systems in a sampled-data framework is numerically explored.

Dans cette thèse, nous étudions plusieurs problèmes géométriques concernant les liens-selles de surfaces plates définies par des formes différentielles méromorphes ayant des pôles de degré arbitrairement grand. Dans le cas des 1-formes holomorphes, les surfaces sont d'aire finie et ont une infinité de liens-selles. Au contraire, pour les 1-formes méromorphes ainsi que les différentielles d'ordre supérieur (quadratiques et au delà) ayant des pôles dont le degré est suffisamment grand, les surfaces sont d'aire infinie et il est courant que le nombre de liens-selles soit fini. Nous étudions trois problèmes au sujet de telles surfaces. Le premier problème est la caractérisation des strates de différentielles dont les surfaces plates correspondantes ont toujours un nombre fini de liens-selles. Nous sommes parvenus à réduire le problème à un simple critère combinatoire relatif au profil de singularités de la strate. Pour ce premier problème, nous nous plaçons dans le cadre le plus général possible : celui des k-différentielles (de la forme f(z)dz^{k}) ayant au moins un pôle d'ordre supérieur ou égal à k. Le deuxième problème est celui de la caractérisation des groupes de Veech des surfaces plates avec pôles. Dans le cas classique des surfaces de (demi)-translation, il s'agit d'un problème très difficile. Ici, au contraire, la rigidité induite par la présence de pôles permet de donner une réponse complète. Enfin, nous proposons une caractérisation complète des familles de nombres complexes pouvant apparaître comme résidus aux pôles d'une différentielle méromorphe appartenant à une strate donnée. Ainsi, la géométrie plate permet de donner une réciproque au théorème des résidus dans laquelle on contrôle la multiplicité des singularités. Ce dernier résultat est le fruit d'une collaboration avec Quentin Gendron. / In this thesis, we consider several geometric problems about saddle connections of flat surfaces defined by meromorphic differentials with poles of arbitrarily large degree. In the case of holomorphic 1-forms, surfaces are of finite area an have infinitely many saddle connections. On the contrary, if we consider meromorphic 1-forms and differentials of higher orders (quadratic and beyond) with at least one pole whose degree is large enough, flat surface are of infinite area and their number of saddle connections may be finite. We study three problems about such surfaces. The first problem is the characterization of the strata of differentials such that the corresponding flat surfaces always have a finite number of saddle connections. We achieved to reduce the problem to a to single combinatorial criterium depending on the singularity pattern of the stratum. When we deal with this problem, we adopt the most general framework of k-differentials (of the form f(z)dz^{k}) with at least one pole of order at least k. The second problem is characterization of Veech groups of flat surfaces with poles. In the classical case of (half)-translation surfaces, it is a very difficult problem. Here, rigidity induced by the poles makes possible to provide a complete answer. Finally, we provide a complete characterization of the families of complex numbers that can appear as residues at the poles of a meromorphic differential belonging to a gien stratum. Thus, flat geometry provides a reciprocal to the residue theorem in which we control the multiplicities of the singularities. This last result is the product of a collaboration with Quentin Gendron.

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are C² -open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

The topic of this diploma thesis is the planning and realisation of a project documentation for a new building of a family viticulture (to be built between existing terraced houses). The building consists of several types of spaces according to the functional use. It connects the living function (an apartment on the second floor and a small guesthouse for degustation purposes), rooms for the store and services (the wine shop, degustation) and the room for stocking (wine storage - barrels/baskets). The building has three floors, the basement is entirely embedded into the terrain. The building will be located approximately 500 metres SW from the city centre of Lednice (584631), within the district of Břeclav in the South-Moravian region, on the lots 756/1, 756/2 and 755.

The theme of diploma thesis is project of guesthouse and restaurant. These are two new buildings on the flattened land plot no.5767, which is located in Pasohlávky. The guesthouse has two floors and a partial basement. Capacity of the guesthouse is 26 beds, one double room is designed as accessible room. In the basement of the guesthouse there is a wellness area and technical facilities of the building. The restaurant is designed as single-storey building with a capacity 50 seats and kitchen with a capacity 250 meals per shift. The main material of vertical load-bearing structure are sand-lime bricks with a certifies contact insulation system. The roof of the guesthouse is saddle roof, the roof of the restaurant is green roof.

The subject of my diploma thesis is a draft of a mountain holiday resort. The building is three-story, semi-basement and with one above-ground floor and attic. There are employee utility rooms and two social rooms with a bar in the basement. Five apartments with own bathroom and kitchenette are built in the two above-ground floors. The building is designed to be made of ceramic and concrete blocks with additional thermal insulation. The roof of the building is saddle with saddle dormers. The land for this new building is situated on a sloping terrain and is located in the western part of Modrava village. The thesis contains a documentary and drawing part and is processed as documentation for the execution of the construction.

This thesis presents a proposal to design a cultural education centre. The object is located in the suburbs of Velké Meziříčí village. It is situated in a residential area near a primary school and a kindergarten. The object will be connected to the local infrastructure and will also have a car park attached. The building will comprise of two operationally connected parts. The first part will consist of several classrooms intended for afterschool clubs, the other will be designed as a gym. The object is of an L shape with two stories above ground. The object is made of hollow ceramic blocks with concrete-steel ceilings. The roof of the building is of a saddle design and is made of wooden roof. The object is designed with steel roofing and profile roofing tiles. The façade is insulated with mineral fibre and ventilated with a system of load bearing steel grid and fibre cement boards. The object will serve children and the youth in activities during their free time. The building is also designed for the disabled.

Diploma thesis deals with the design documentation of hotel building in the village of Dolní Lomná. The aim of the work was to create a unit that will be used for accommodation and catering of the guests and will not disturb the surroundings. The building was designed as a three-storey building. The hotel is devided to an operational part for staff, catering and accommodation for guests. The capacity of the beds is 32. The hotel is designed as a brick construction system. Roofing is as a saddle roof made of wooden truss above the 3rd floor. The thesis is divided into two parts. The first part contains the textual evaluation of the construction of the hotel, the second part consists of the project documentation itself. The project documentation includes architectural and layout design, drawing up of building drawings, design of a fire safety solution of the building, basic thermal technical assessment.