$(\ldots, b_{i_j}, \ldots)$) is used to show a difference with $u(t_j^+)$ (resp. $u(t_j^-)$) on the $i_j$-th component only.

\[rq\_affine\] In the special (and important in practice) case of a control-affine system $$\dot{x}(t) = f_0(x(t)) + \sum_{k=1}^m{u_k(t) f_k(x(t))},$$ the initial condition on $v_j$ can be written much more easily: $$v_j(t_j) = \left\{
\begin{tabular}{rl}
$ (a_{i_j} - b_{i_j}) f_{i_j}(x(t_j))$ & if $u_{i_j}$ switches from $a_{i_j}$ to $b_{i_j}$. \\
$(b_{i_j} - a_{i_j})  f_{i_j}(x(t_j))$ & if $u_{i_j}$ switches from $b_{i_j}$ to $a_{i_j}$.
\end{tabular}
\right.$$

Absorbing perturbations {#backward_epm}
-----------------------

As explained in the introduction, we present in this paper a closed-loop method to actually steer the system towards a point $x_f$, with bang-bang controls, even in the presence of perturbations.

First, for the sake of simplicity, we will explain how to control the system to some point $x_f + \delta x_f$. We will see that this idea can be adapted for our purpose of controlling a perturbed trajectory, by simply reversing the time.

#### Perturbations on the final point.

We briefly generalize the problem introduced in the introduction. Let $$\overline{u} = \left((u_{10}, \ldots,u_{m0}),\left(\overline{t}_1,i_1\right), \ldots, \left(\overline{t}_N,i_N\right), t_f \right) \in \Omega \times {\mathbb{R}}^{2N+1}$$ be a control such that $x_{\overline{u}}(t_f) = x_f$. That is, using the definition of 