01171nas a2200121   4500008004100000020001400041245007800055210006900133260001500202520075800217100002700975856004701002       2022                        eng d  a1573-869800aA Gradient Flow Equation for Optimal Control Problems With End-point Cost0 aGradient Flow Equation for Optimal Control Problems With Endpoin  c2022/07/073 aIn 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 Γ-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.1 aScagliotti, Alessandro  uhttps://doi.org/10.1007/s10883-022-09604-2