Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$begin{gathered} min int {h(x,u) dt,} hfill dot x = F(x) + uG(x), hfill left| u right| leqslant 1, x in Omega subset mathbb{R}^2 , hfill x(0) = x_0 , x(1) = x_1 . hfill end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.