$x_f$, is bang-bang and switches $N$ times between the extreme values $0$ and $1$ over the time interval $[0,t_f]$. We denote by $\mathcal{T} = (t_1, \ldots, t_N)$ the vector consisting of its switching times $0<t_1<\cdots<t_N<t_f$. Then the control $u$ can equivalently be represented by the vector $\mathcal{T}$, provided one makes precise the value of $u(t)$ for $t\in(0,t_1)$. One can also add new switching times: for instance if $u(t)=0$ for $t\in(0,t_1)$, given any $s_1\in(0,t_1)$, the needle-like variation $\pi_1=(s_1,\delta s_1,1)$ (with $\vert\delta s_1\vert$ small enough) is a bang-bang control having two new switching times at $s_1$ and $s_1+\delta s_1$.

In what follows, we designate a bang-bang control either by $u$ or by the set $\mathcal{T}=(t_1, \ldots, t_N)$ of its switching times. This is with a slight abuse because we should also specify the value of $u$ along the first bang arc. But we will be more precise, rigorous and general in a further section. The end-point mapping is then reduced to the switching times, and one has $E(x_0, t_f, \mathcal{T}) = x_f$. A variation $\delta\mathcal{T} = (\delta t_1, \ldots, \delta t_N)$ of the switching times generates $N$ variation vectors $(v_{1}(t_f), \ldots, v_{N}(t_f))$, and the corresponding bang-bang trajectory reaches at time $t_f$ the 