This thesis deals with the topic of min-max formulations of robust model predictive control problems. The sets involved in guaranteeing robust feasibility of the min-max program in the presence of state constraints are of particular interest, and expanding the applicability of well understood solvers of linearly constrained quadratic min-max programs is the main focus. To this end, a generalisation for the set of uncertainty is considered: instead of fixed bounds on the uncertainty, state- and input-dependent bounds are used. To deal with state- and input dependent constraint sets a framework for a particular class of set-valued maps is utilised, namely parametrically convex set-valued maps. Relevant properties and operations are developed to accommodate parametrically convex set-valued maps in the context of robust model predictive control. A quintessential part of this work is the study of fundamental properties of piecewise polyhedral set-valued maps which are parametrically convex, we show that one particular property is that their combinatorial structure is constant. The study of polytopic maps with a rigid combinatorial structure allows the use of an optimisation based approach of robustifying constrained control problems with probabilistic constraints. Auxiliary polytopic constraint sets, used to replace probabilistic constraints by deterministic ones, can be optimised to minimise the conservatism introduced while guaranteeing constraint satisfaction of the original probabilistic constraint. We furthermore study the behaviour of the maximal robust positive invariant set for the case of scaled uncertainty and show that this set is continuously polytopic up to a critical scaling factor, which we can approximate a-priori with an arbitrary degree of accuracy. Relevant theoretical statements are developed, discussed and illustrated with examples.

Dans cette thèse, nous nous intéressons à des inclusions de la forme 0∈ f( x) + F(x), où f est une application univoque et F est une application multivoque à graphe fermé. Ces dernières années, diverses méthodes de résolutions d'inclusions de ce type ont été développées par les chercheurs et, après un bref rappel sur quelques notions d'analyse (univoque et multivoque) nous en présentons quelques unes utilisant l'hypothèse de régularité métrique sur l'application multivoque. Dans la suite de notre travail, plutôt que d'utiliser cette hypothèse de régularité métrique, nous lui préférons des hypothèses directement liées à la solution qui sont la semistabilité et l'hemistabilité. Notons que la semistabilité d'une solution x̅ de l'inclusion 0∈G(x) est en fait équivalente à la sous-régularité métrique forte de l'application multivoque G en x̅ pour 0. Après avoir présenté des méthodes utilisant la semistabilité et l'hemistabilité, nous exposons les nouveaux résultats auxquels nous avons abouti qui consistent essentiellement en des améliorations des méthodes présentées. Ce que nous entendons par améliorations se décline en deux points principaux : soit nous obtenons un meilleur taux de convergence, soit nous utilisons des hypothèses plus faibles qui nous permettent d'obtenir des taux de convergence similaires. / In this thesis, we focus on inclusions in the form of 0∈ f( x) + F(x), where f is a single-valued function and F is a set-valued map with closed graph. In the last few years, various methods to solve such inclusions have been developed; after having recalled some notions in analysis (single-valued and set-valued) we present some of them using metric regularity on the set-valued map. Then, instead of considering this metric regularity assumption, we prefer assumptions which are directly connected to the solution, that are semistability and hemistability. One can note that semistabily of a solution x̅ of the inclusion 0∈G(x) is actually equivalent strong metric subregularity on the set-valued map G at x̅ for 0. After having presented some methods using semistability and hemistability, we show the new results we obtained, most of them being improvement of the presented methods. What we mean by improvement is mainly a better convergence rate on the one hand, and weaker assumptions that lead to similar convergence rate, on the other.

The thesis deals with a general access to topological transversality in uniform spaces. / Die Arbeit behandelt einen allgemeinen Zugang zur Topologischen Transversalität in uniformen Räumen.

The thesis deals with a general access to topological transversality in uniform spaces. / Die Arbeit behandelt einen allgemeinen Zugang zur Topologischen Transversalität in uniformen Räumen.