a.e.
   $(H_2)$            $\forall i \in \llbracket1,m \rrbracket$, $u_i$ does not chatter.
  --------- -------------------------------------------------------------------------------------

A control is chattering when it switches infinitely many times over a compact time interval (see [@zhu2016; @fuller1963]). Therefore, our method does not apply to those controls. However, when the solution of an optimal control problem chatters, provided that it is possible, one could consider a sub-optimal solution, with only a finite number of switching times.

In the context of optimal control, we will denote the cost under the form $$C(u) = \int^{t_f}_0{f^0(t,x(t),u(t))\,dt}.
\label{cost}$$

We recalled in the introduction the (classical) definitions of the end-point mapping, of a needle-like variation (\[eq\_needle\]) and the expansion of the end-point mapping subject to a needle-like variation (\[eq\_var\]).

Reduced end-point mapping {#reduced_epm}
-------------------------

In this subsection, we give the definition of the reduced end-point mapping, and show a differentiability property.

Let us consider a bang-bang control $u(\cdot)$, and its associated trajectory $x(\cdot)$. For the sake of simplicity, we make the additional assumption that for every switching time $t_j$, one and only one component of the control commutes. Therefore, provided we specify the initial value of each component, the control $u$ is entirely characterized by the switching times of its components and 