is a matrix of size $n \times (N - j + 1)$.

In this context, the problem of adjusting the system back towards $x_f$ writes: at time $t$, find $(t_j, \ldots, t_N)$ such that $$\tilde{E}(t,{t}_1, \ldots, {t}_N) = x_{per}(t).
\label{prob_backward}$$

We see that reversing the time, we place ourselves in the setting previously described of aiming at a perturbed final point. Therefore, we have the following proposition.

At order one in $\delta x$, the solution of minimal norm of the problem is given by $\overline{\mathcal{T}} + \delta \mathcal{T}$, with $$\delta \mathcal{T} = d\tilde{E}(t,\overline{\mathcal{T}})^{\dagger}\cdot \delta x(t),
\label{eq_recalage}$$ where $d\tilde{E}(t,\overline{\mathcal{T}})^{\dagger}$ denotes the pseudo-inverse of $d\tilde{E}(t,\overline{\mathcal{T}})$. Moreover, we have the estimate $$\left\| \delta \mathcal{T} \right\|_2 {\leqslant}\frac{1}{\sigma_{min}(t)} \left\| \delta x(t) \right\|_2,
\label{estimate_deltat}$$ where $\sigma_{min}(t)$ is the smallest positive singular value of $d\tilde{E}(t,\overline{\mathcal{T}})$.

The scheme of the proof has already been exposed previously in the paper. However, we write it extensively here. Let $\delta \mathcal{T} = \mathcal{T} - \overline{\mathcal{T}}$. The problem writes $$\tilde{E}(t,\overline{\mathcal{T}} + \delta \mathcal{T}) = x_{per}(t).$$ According to Proposition \[prop\_differentiability\_epm\], the backward end-point mapping is differentiable (and we also know how to compute its derivative), so $$\begin{aligned}
\tilde{E}(t,\overline{\mathcal{T}} + \delta \mathcal{T})    &= \tilde{E}(t,\overline{\mathcal{T}}) + d\tilde{E}(t,\overline{\mathcal{T}})\cdot \delta \mathcal{T} + o(\| \delta \mathcal{T} \|) \\
                            