Some Examples of Reachable Sets and Optimal Cost Functions That Fail to be Subanalytic

We give examples of: a) a control system x = f(x) + Bu, with f a vector field whose components are quadratic polynomials, and with the controls taking values in the unit cube, such that the time T reachable sets from the origin are not subanalytic; b) a system x = g(x) + Bu, with the components of g cubic polynomials, such that the system is completely controllable but the optimal time function is not subanalytic; c) a linear system with a compact, convex, semialgebraic control constraint for which the time T reachable sets are not subanalytic.