we have shown how to compute, at order one, the correction to apply to control the system to some point $x_f + \delta x_f$: it boils down to solving a least-squares problem. Let us keep in mind that our definitive goal is to control systems that are perturbed all along their trajectory, to a fixed final point $x_f$. In other words, from a perturbed point $x(t) + \delta x(t)$ at some time $t \in [0,t_f)$, we want to absorb the perturbation $\delta x(t)$ and still reach the final point $x_f$. Even if this is a slightly different setting, we show that we can apply the same idea if we look at a *backward problem*.

#### Absorbing a perturbation at time $t$.

Let $(\overline{x}(\cdot),\overline{u}(\cdot))$ be a nominal solution of the control system . We assume that when applying in practice the control $\overline{u} = \overline{\mathcal{T}}$, because of model uncertainties and perturbations, we observe a perturbed trajectory $x_{per}(t) = \overline{x}(t) + \delta x(t)$.

Let $t \in [0,t_f]$. Starting from the perturbed point $\overline{x}(t) + \delta x(t)$, which stands as a new initial point, we want to reach the final point $x_f$ in time $t_f-t$. Hence, we look for a control $\overline{u} + 