Analysis, simulation and control of nonsmooth dynamical systems

Nonsmooth dynamics concerns the study of the time evolution of physical and mathematical systems that are not smooth in the mathematical sense. In other words, nonsmooth dynamical systems are characterized by a lack of differentiability, either of the mappings that enter in theirs formulations, or of theirs solutions with respect to time. In Mechanics, the main instances of non- smooth dynamical systems are multibody systems with Signorini's unilateral contact, Coulomb's friction and impacts. Others instances of nonsmooth dynamical sys- tems can also be found in electrical circuits with ideal components and in Control Theory, mainly with sliding mode control and variable structure systems. More generally, every time a piecewise, possibly set-valued, modeling of systems is invoked, we end up with a nonsmooth system. This is the case, for instance, for hybrid systems in nonlinear control or for piecewise linear modeling of gene regulatory networks in mathematical biology. This Habilitation sum up my research activities since my PhD in the Bipop project. The contributions concern the development and the analysis of numerical methods for the simulation of nonsmooth dynamical systems. A substantial part of my work is devoted to the time-integration of dynamical systems in Mechanics, in Electronics and in Biology. In Control, the related question of the time-discretization of sliding mode control systems has been also the object of an intense attention. The solution procedure for the discrete frictional contact problem has also been studied and developed. These developments have been carried out with some contributions to the modeling of multiple impact laws, electrical ideal components like switches and various industrial applications ranging from the numerical modeling of circuit breakers for Schneider Electric to the ExoMars Rover of the European Space Agency (ESA). Finally, I have to mention that these developments were done with a strong emphasis on software development, mainly through the Siconos code

Raporteurs:

• Mr Jamal Daafouz (Professeur - Université de Lorraine )
• Mme Samir Adly (Professeur - Université de Limoges )
• Mr Jong-Shi Pang (Professeur - Univ. of South California )

Examinators:

• Mr Bernard Brogliato (Directeur de recherches - INRIA )
• Mr Ernst Hairer (Professeur - Université de Genève )
• Mr Pierre Alart (Professeur - Université de Montpellier )