A gradient flow equation for optimal control problems with end-point cost

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 that consists in minimizing a weighted sum of an end-point cost and the squared $2$-norm of the control. We study the gradient flow induced by this functional on the Hilbert space of the admissible controls, and we prove a convergence result by means of the Simon-Lojasiewicz inequality. Finally, we prove that, if we let the weight of the end-point cost tend to infinity, a $\Gamma$-convergence result holds, 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.