  In this paper we consider a control system of the form $\dot x = F(x)u$,
linear in the control variable $u$. Given a fixed starting point, we study a
finite-horizon optimal control problem, where we want to minimize a weighted
sum of an end-point cost and the squared $2$-norm of the control. This
functional induces a gradient flow on the Hilbert space of admissible controls,
and we prove a convergence result by means of the Lojasiewicz-Simon inequality.
Finally, we show that, if we let the weight of the end-point cost tend to
infinity, the resulting family of functionals is $\Gamma$-convergent, and it
turns out that the limiting problem consists in joining the starting point and
a minimizer of the end-point cost with a horizontal length-minimizer path.
