Anne Bouillard

Anne Bouillard, Erwan Faou, Maxime Zavidovique

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.

Anne Bouillard

Network Calculus is a theoretical model that aims at providing upper bounds of worst-case performance (such as delay or buffer occupancy). This is a mathematical framework that handles both network modeling and network analysis. As such it has requirements regarding the space of functions needed for a safe analysis. Namely, the functions need to be non-negative, as they model a quantity of data. This results in some pitfall for the analysis, where hypothesis matter. A recent paper by Hamscher et al. states that allowing functions with negative values can also lead to a valid analysis, in cases that would be untractable with the non-negative assumption results, especially when feedback control is present in the system. In this paper, we show that, on the contrary, a more conventional analysis is possible in all the mentioned cases. The key is a detailed analysis of sub-additive functions. Second, we show that the analysis of complex feedback control systems, presented by Hamscher et al. in a second paper that uses functions with negative values, is unsound and has stability issues. We give a corrected analysis, when possible, with conventional hypotheses.