Moreover, when the Pontryagin maximum principle yields bang-bang controls, such controls have in general a minimal number of switchings: in dimension $3$ for instance, it is proved in [@Krener; @Kupka; @Schattler] (see also [@BonnardChyba; @Bonnard2005; @Trelat2012] for more details on this issue) that, locally, minimal time trajectories of single-input control-affine systems have generically two switchings. Taking into account the free final time, this makes three degrees of freedom, which is the minimal number to generically make the trajectory reach a target point in ${\mathbb{R}}^3$, i.e., to solve three (nonlinear) equations.

In these conditions, a natural idea is to add redundancy to such bang-bang strategies, by enforcing the control to switch more times than necessary. These additional switching times are introduced by *needle-like variations*, as in the classical proof of the Pontryagin maximum principle (see [@lee1967foundations; @pontryagin1987]).

We recall that a needle-like variation $\pi_1=(t_1,\delta t_1,u_1)$ of a given control $u$ is the perturbation $u_{\pi_1}$ of the control $u$ given by $$\label{eq_needle}
u_{\pi_1}(t) = \left\{
\begin{array}{rcl}
u_1  & \textrm{if} & t\in [t_1,t_1+\delta t_1], \\
 u(t) & \textrm{otherwise,}&
\end{array}\right.$$ where $t_1\in [0,t_f]$ is the time at which the spike variation is introduced, $\delta t_1$ is a real number of small absolute value that 