Cette thèse traite de l'étude de la convergence de la solution du système (HBF)(Heavy Ball with friction) avec quelques contrôles externes et internes comme le contrôle de frottement lambda. L'objectif est de trouver des conditions permettant à la solution x(t) de converger faiblement ou fortement vers des ponts critiques de la fonction $Phi$. En général, on s'intéresse à l'étude de cette équation (ou système d'équations) selon les comportements du contrôle externe $lambda(.)$. Par exemple : 1- $lambda(.)$ borné, minoré par une constante positive, 2- $lambda(.)$ prend la valeur zéro sur des intervalles disjoints, 3- $lambda(.)$ tend vers zéro quand t tend vers l'infini. De même, on considère les cas où le contrôle interne $Phi$ satisfait à certaines conditions 1- $Phi$ convexe, 2-$Phi$ coercive ou fortement convexe, 3-$nablaPhi (x(t)) $ est remplacé par un opérateur différentiel Ax(t) + epsilon(t)x(t) (exemple A=-Laplacien, ou A=-div(B.gradient)) où B est une matrice carrée / This thesis deals with the study of the convergence of the solution of the (HBF) system (Heavy ball with friction) With some external and internal controls such as the friction control. The purpose is to find conditions allowing the solution x(.) to converge weakly or strongly towards critical points of the fonction Phi. In general, we examine this equation (or a system of equations) depending on the behaviours of the external control lambda(.). The following cases have been considered : 1- lambda, bounded and greater than a positive constant, 2-lambda, vanishes on separate intervals, 3-lambda,tends to zero as t tends to infinity. We emphasize some particular cases for the internal control Phi 1- Phi convex, 2- Phi coercive ou fortement convexe, 3-nablaPhi(x(t)) replaced by a linear differential operator Ax(t) + epsilon(t)x(t) (exemple A=-Laplacien, ou A=-div(B.gradient))

This thesis analyses the semialgebraic sets, that is, a finite union of solu- tions to a finite sequence of polynomial inequalities. We introduce a notion of cylindrical algebraic decomposition as a tool for the construction of a semialge- braic stratification and a triangulation of a semialgebraic set. On this basis, we prove several important and well-known results of real algebraic geometry, such as Hardt's semialgebraic triviality or Sard's theorem. Drawing on Morse theory, we finally give a proof of a Thom-Milnor bound for a sum of Betti numbers of a real algebraic set. 1

[pt] A teoria de Morse relaciona a topologia de um espaço aos elementos críticos de uma função escalar definida nele. Isso vale tanto para a teoria clássica quanto para a versão discreta proposta por Forman em 1995. Essas teorias de Morse permitem caracterizar a topologia do espaço a partir de funções definidas nele, mas também permite estudar funções a partir de construções tipológicas derivadas dela, como por exemplo o complexo de Morse-Smale. Apesar da teoria de Morse discreta se aplicar para complexos celulares gerais de forma inteiramente combinatória, o que torna a teoria particularmente bem adaptada para o computador, as funções usadas na teoria não são amostragens de funções contínuas, mas casamentos especiais no grafo que codifica as adjacências no complexo celular, chamadas de casamentos de Morse. Quando usar essa teoria para estudar um espaço topológico, procura- se casamentos de Morse ótimos, i.e. com o menor número possível de elementos críticos, para obter uma informação topológica do complexo sem redundância. Na primeira parte desta tese, investiga-se a complexidade parametrizada de encontrar esses casamentos de Morse ótimos. Por um lado, prova-se que o problema ERASABILITY, um problema fortemente relacionado à encontrar casamentos de Morse ótimos, é W [P ]-completo. Por outro lado, um algoritmo é proposto para calcular casamentos de Morse ótimos em triangulações de 3-variedades, que é FPT no parâmetro do tree- width de seu grafo dual. Quando usar a teoria de Morse discreta para estudar uma função escalar definida no espaço, procura-se casamentos de Morse que capturam a informação geométrica dessa função. Na segunda parte é proposto uma construção de casamentos de Morse baseada em casamentos estáveis. As garantias teóricas sobre a relação desses casamentos com a geometria são elaboradas a partir de provas surpreendentemente simples que aproveitam da caracterização local do casamento estável. A construção e as suas garantias funcionam em qualquer dimensão. Finalmente, resultados mais fortes são obtidos quando a função for suave discreta, uma noção definida nesta tese. / [en] Morse theory relates the topology of a space to the critical elements of a scalar function defined on it. This applies in both the classical theory and a discrete version of it defined by Forman in 1995. Those Morse theories permit to characterize a topological space from functions defined on it, but also to study functions based on topological constructions it implies, such as the Morse-Smale complex. While discrete Morse theory applies on general cell complexes in an entirely combinatorial manner, which makes it suitable for computation, the functions it considers are not sampling of continuous functions, but special matchings in the graph encoding the cell complex adjacencies, called Morse matchings. When using this theory to study a topological space, one looks for optimal Morse matchings, i.e. one with the smallest number of critical elements, to get highly succinct topological information about the complex. The first part of this thesis investigates the parameterized complexity of finding such optimal Morse matching. On the one hand the Erasability problem, a closely related problem to finding optimal Morse matchings, is proven to be W[P]-complete. On the other hand, an algorithm is proposed for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed parameter tractable in the tree-width of its dual graph. When using discrete Morse theory to study a scalar function defined on the space, one looks for a Morse matching that captures the geometric information of that function. The second part of this thesis introduces a construction of Morse matchings based on stable matchings. The theoretical guarantees about the relation of such matchings to the geometry are established through surprisingly simple proofs that benefits from the local characterization of the stable matching. The construction and its guarantees work in any dimension. Finally stronger results are obtained if the function is discrete smooth on the complex, a notion defined in this thesis.