optimization problem in finite-dimension.

In the following, when writing this reduced end-point mapping, we may consider that the initial point $x_0$ is fixed, as well as the way the components of the control switch (i.e., we consider that the N-tuple $(i_1, \ldots, i_N)$ is fixed), the initial values $u_{i0}$ and the final time $t_f$. In this context, we may forget them in the notations, and denote the reduced end-point mapping by $$E(t_1, \ldots, t_N) = x_u(t_f).$$

A remarkable fact is that the reduced end-point-mapping is differentiable. Compared to the expansion with respect to a needle-like variation, the sign of $\delta t$ does not matter. For the sake of completeness, we give the proof in appendix.

\[prop\_differentiability\_epm\] The reduced end-point mapping is differentiable, and $$dE(t_1 , \ldots, t_N) = 
\begin{pmatrix}
v_1(t_f) & \cdots & v_N(t_f)
\end{pmatrix}
\in \mathcal{M}_{n,N}({\mathbb{R}}),$$ where $v_j(\cdot)$ ($j \in \llbracket1,N \rrbracket$) is the solution of the Cauchy problem, defined for $t {\geqslant}t_j$: $$\begin{aligned}
\dot{v}_j(t) &= \frac{\partial f}{\partial x}(t,x(t),u(t))v_j(t) \\
v_j(t_j) &= \left\{
\begin{tabular}{rl}
$f(t_j,x(t_j),(\ldots, a_{i_j}, \ldots)) - f(t_j,x(t_j),u(t_j^+))$ & if $u_{i_j}$ switches from $a_{i_j}$ to $b_{i_j}$. \\
$f(t_j,x(t_j),(\ldots, b_{i_j}, \ldots)) - f(t_j,x(t_j),u(t_j^+))$ &  if $u_{i_j}$ switches from $b_{i_j}$ to $a_{i_j}$.
\end{tabular}
\right.\end{aligned}$$ The notation $(\ldots, a_{i_j}, \ldots)$ (resp. 