stands for the duration of the variation, and $u_1 \in [0,1]$ is some arbitrary element of the set of values of controls. When $\delta t_1<0$, one replaces the interval $[t_1,t_1+\delta t_1]$ with $[t_1+\delta t_1,t_1]$ in . It is well known that, if $\vert\delta t_1\vert$ is small enough, the control $u_{\pi_1}$ is admissible (that is, the associated trajectory solution of is well-defined on $[0,t_f]$) and generates a trajectory $x_{\pi_1}(\cdot)$, which can be viewed as a perturbation of the nominal trajectory $x(\cdot)$ associated with the control $u$, and which steers the control system to the final point $$\label{eq_var}
E(x_0,t_f,u_{\pi_1}) = E(x_0,t_f,u) + \vert \delta t_1 \vert\, v_{\pi_1}(t_f) + o(\delta t_1),$$ where the so-called variation vector $v_{\pi_1}(\cdot)$ is the solution of some Cauchy problem related to a linearized system along $x(\cdot)$ (see [@lee1967foundations; @pontryagin1987; @Silva2010] and Proposition \[prop\_differentiability\_epm\]). Recall that the *first Pontryagin cone* $K(t_f)$ is the smallest closed convex cone containing all variation vectors $v_{\pi_1}(t_f)$; it serves as a local convex estimate of the set of reachable points at time $t_f$ (with initial point $x_0$).

![Changing the switching times induces a displacement at the final time.[]{data-label="fig_diff_epm"}](diff_epm_ter.pdf)

Assume that the nominal control $u$, which steers the system from $x_0$ to the target point 