Difference Methods for Differential Inclusions: A Survey

International audienceThe main objective of this survey is to study convergence properties of difference methods applied to differential inclusions. It presents, in a unified way, a number of results scattered in the literature and provides also an introduction to the topic. Convergence proofs for the classical Euler method and for a class of multistep methods are outlined. It is shown how numerical methods for stiff differential equations can be adapted to differential inclusions with additional monotonicity properties. Together with suitable localization procedures, this approach results in higher-order methods. Convergence properties of difference methods with selection strategies are investigated, especially strategies forcing convergence to solutions with additional smoothness properties. The error of the Euler method, represented by the Hausdorff distance between the set of approximate solutions and the set of exact solutions is estimated. First-and second-order approximations to the reachable sets are presented