Higher Order Methods for Differential Inclusions

A numerical method for rigorous over-approximation of reachable sets of differential inclusions is given. The method gives high order error for a single time step and a uniform bound on the error over the finite time interval. We view differential inclusion as a system with inputs where input is uncertainty or bounded disturbance present in the system. The approach is based on the approximations of inputs by finetely parametrised functions at each time step. We focus on input-affine systems for which we derive single-step error, and give formulas for obtaining errors of O(h), O(h^2), O(h^3). In addition, we give ingredients necessary for obtaining errors of higher orders. Moreover, we give technique for generalizing the method to differential inclusions with constraints. As an illustration of the theory presented, a rigorous numerical result is given.