Dans une première partie, nous nous intéressons aux systèmes dynamiques gradients gouvernés par des fonctions non lisses, mais aussi non convexes, satisfaisant l'inégalité de Kurdyka-Lojasiewicz. Après avoir obtenu quelques résultats préliminaires pour la dynamique de la plus grande pente continue, nous étudions un algorithme de descente général. Nous prouvons, sous une hypothèse de compacité, que tout suite générée par ce schéma général converge vers un point critique de la fonction. Nous obtenons aussi de nouveaux résultats sur la vitesse de convergence, tant pour les valeurs que pour les itérés. Ce schéma général couvre en particulier des versions parallélisées de la méthode forward-backward, autorisant une métrique variable et des erreurs relatives. Cela nous permet par exemple de proposer une version non convexe non lisse de l'algorithme Levenberg-Marquardt. Enfin, nous proposons quelques applications de ces algorithmes aux problèmes de faisabilité, et aux problèmes inverses. Dans une seconde partie, cette thèse développe une dynamique de descente associée à des problèmes d'optimisation vectoriels sous contrainte. Pour cela, nous adaptons la dynamique de la plus grande pente usuelle aux fonctions à valeurs dans un espace ordonné par un cône convexe fermé solide. Cette dynamique peut être vue comme l'analogue continu de nombreux algorithmes développés ces dernières années. Nous avons un intérêt particulier pour les problèmes de décision multi-objectifs, pour lesquels cette dynamique de descente fait décroitre toutes les fonctions objectif au cours du temps. Nous prouvons l'existence de trajectoires pour cette dynamique continue, ainsi que leur convergence vers des points faiblement efficients. Finalement, nous explorons une nouvelle dynamique inertielle pour les problèmes multi-objectif, avec l'ambition de développer des méthodes rapides convergeant vers des équilibres de Pareto. / In a first part, we focus on gradient dynamical systems governed by non-smooth but also non-convex functions, satisfying the so-called Kurdyka-Lojasiewicz inequality.After obtaining preliminary results for a continuous steepest descent dynamic, we study a general descent algorithm. We prove, under a compactness assumption, that any sequence generated by this general scheme converges to a critical point of the function.We also obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method, with variable metric and relative errors. As an example, a non-smooth and non-convex version of the Levenberg-Marquardt algorithm is detailed.Applications to non-convex feasibility problems, and to sparse inverse problems are discussed.In a second part, the thesis explores descent dynamics associated to constrained vector optimization problems. For this, we adapt the classic steepest descent dynamic to functions with values in a vector space ordered by a solid closed convex cone. It can be seen as the continuous analogue of various descent algorithms developed in the last years.We have a particular interest for multi-objective decision problems, for which the dynamic make decrease all the objective functions along time.We prove the existence of trajectories for this continuous dynamic, and show their convergence to weak efficient points.Then, we explore an inertial dynamic for multi-objective problems, with the aim to provide fast methods converging to Pareto points.

The master's thesis deals mainly with continuous nonlinear dynamical systems that exhibit chaotic behavior. The main goal is to create algorithms for chaos detection and their subsequent testing on known models. Most of the thesis is devoted to the estimation of the Lyapunov exponents, further it deals with the estimation of the fractal dimension of an attractor and summarizes the 0--1 test. The thesis includes three algorithms created in MATLAB -- an algorithm for estimating the largest Lyapunov exponent and two algorithms for estimating the entire Lyapunov spectra. These algorithms are then tested on five continuous dynamical systems. Especially the error of estimation, speed of these algorithms and properties of Lyapunov exponents in different areas of system behavior are investigated.

Large, software intensive systems are typically developed using a feature oriented development paradigm in which feature specifications are derived from domain requirements and features are implemented to satisfy such specifications. Historically, this approach has been followed in the telecommunications industry. More recently, in the automotive industry, features (for e.g. electronic stability control, collision avoidance etc.) are being developed as part of a software product line and a suitable subset of these features is integrated in an automobile model based on market requirements. Typically, features are designed independently by different engineering teams and are integrated later to create a system. Integrating features that are designed independently is extremely hard because the interactions between features are not understood properly and any incompatibilities may lead to costly redesign. In this thesis, we propose a framework for developing feature based systems such that even if features are incompatible, they can be integrated without redesign. Our view is that a feature based system consists of a base system and multiple features (or controllers), each of which independently advise the base system on how to react to an input so as to conform to their respective specifications. Such a system may reach a point of “conflict” between two or more features when they do not agree on a common action that the base system should perform. Instead of redesigning one or more features for resolving a conflict, we propose the novel notion of “conflicttolerance”, which requires features to be “resilient” or “tolerant” with regard to violations of their advice. Thus, unlike a classical feature, a conflicttolerant feature observes that its advice has been overridden, and takes this fact into account before proceeding to offer advice for subsequent behaviour of the base system. Conflict-tolerant features are composed using a priority order such that whenever a conflict occurs between two features, the base system continues with the advice of the higher priority feature. We guarantee that each feature is “maximally” utilized in that its advice is not taken only when there is a conflict with some higher priority controller. We show how to specify conflict-tolerant features for finite state, timed, and hybrid systems and also provide decision procedures for automated verification of finite state and timed systems. This provides a compositional technique for verifying systems which are composed of conflict-tolerant features. Our framework for developing feature based systems enables conflictresolution without redesign. The scope for reusing conflict tolerant features is significantly higher thus reducing design and verification effort.