Reachable sets, control sets and their computation

We consider nonlinear control affine systems with constraint control range. An important object for the global analysis of these systems are reachable sets, sets of points, which are solutions to an initial value problem for a given initial point, an admissible control function and arbitrary time. Among other things they describe control sets, subsets of the state space where the system is approximately completely controllable. In this work a method for the computation of reachable sets will be developed. It is based on numerical schemes for the solution of initial value problems an uses the invariance of reachable sets by computing approximate invariant sets. The different discretization steps and the most important problems of its implementation will be explained. With some two- and three-dimensional examples the power and the limits of this method for systems with complicated dynamics will be shown.