About the continuity of reachable sets of restricted affine control systems

In this paper we prove that for a restricted affine control system on a connected manifold M, the associated reachable sets up to the time t varies continuously in each independent variable: time, state and the range of the admissible control functions. However, as a global map it is just lower semi-continuous. We show a bilinear control system on the plane where the global map has a discontinuity point. According to the Pontryagin Maximum Principal, in order to synthesizes the optimal control the Hausdorff metric continuity is crucial. We mention some references with concrete applications. Finally, we apply the result to the class of Linear control systems on Lie groups.