point (see Figure \[fig\_diff\_epm\], where two variations vectors are displayed, for two switching times $t_1$ and $t_2$) $$E\left(x_0, t_f, \mathcal{T} + \delta\mathcal{T}\right) = x_f + \delta t_1\cdot v_{1}(t_f) + \cdots + \delta t_N \cdot v_{N}(t_f) + \mathrm{o}(\| \delta\mathcal{T} \|) .$$ Therefore the end-point mapping $E$ is differentiable with respect to $\mathcal{T}$, and $$\label{eq_diff_epm_intro}
\frac{\partial E}{\partial\mathcal{T}}(x_0, t_f, \mathcal{T})\cdot \delta\mathcal{T} = \delta t_1\cdot v_{1}(t_f) + \cdots + \delta t_N \cdot v_{N}(t_f) .$$ Notice that compared to , the absolute values disappear. We will prove this result in details further in the paper. In particular, the range of this differential is the first Pontryagin cone $K(t_f)$ (see also [@Silva2010]). Obviously, the more switching times (i.e., degrees of freedom), the more accurate the approximation of the reachable set.

We now add *redundant* switching times $(s_1, \ldots, s_\ell)$ for some $\ell \in \N$ in order to generate more degrees of freedom to solve the control problem $$E\left(x_0,t_f,(t_1, \ldots, t_N, s_1, \ldots, s_\ell)\right) = x_f .$$ We order the times in the increasing order and we still denote by $\mathcal{T}$ the vector of all switching times.

#### Redundancy creates robustness.

We will see further that these redundant switching times contribute to make the trajectory robust to 