Some properties of the value function and its level sets for affine control systems with quadratic cost
Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems~: $\ds{\dot{x}=f_0(x)+\sum_{i=1}^m u_if_i(x)}$, with a cost of the form~: $\ds{C(u)=\int_0^T \sum_{i=1}^m u_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.