On the problem whether controllability is finitely determined

Controllability of finite dimensional linear systems can be decided via a finite number of matrix operations, with an a-priori known bound on this number of operations. For both polynomial and general nonlinear analytic systems that are affine in the control it remains an open problem whether controllability can at all be decided by a finite number of differentiations of the data. This is closely related to the question whether the value function of the minimumtime optimal control problem is Hölder continuous. This paper briefly surveys some of the historical background behind the questions, and then presents recent progress, with focus on some unexpected properties of custom-designed polynomial systems. Keywords— Nonlinear control, controllability, value function, Hölder continuity