can be represented by a vector: $$\left((u_{10}, \ldots,u_{m0}),\left(t_1,i_1\right), \ldots, \left(t_N,i_N\right), t_f \right) \in \Omega \times {\mathbb{R}}^{2N+1},$$ where $u_{i0} \in \{a_i,b_i\}$ is the initial value for the control $u_i(\cdot)$ ($i \in \llbracket1,m\rrbracket)$, $N$ is the total number of switching times, $t_f$ is the final time, and $i_j$ is the component of the control that switches at time $t_j$. As this representation entirely characterizes the control, we will use indistinctly the notation $u$ and $\left((u_{10}, \ldots,u_{m0}),\left(t_1,i_1\right), \ldots, \left(t_N,i_N\right), t_f \right)$ to speak about the control whose components switch at the times $t_j$. In the literature, $\left(\left(t_1,i_1\right), \ldots, \left(t_N,i_N\right)\right)$ is often called a switching sequence.

Had we wanted to allow simultaneous switching of multiple components, we would need to consider controls represented by: $$\left((u_{10}, \ldots,u_{m0}),\left(t_1,\mathcal{I}_1\right), \ldots, \left(t_N,\mathcal{I}_N\right), t_f \right),$$ where $\mathcal{I}_j \subset \llbracket1,m \rrbracket$ represents the set of components that switch at time $t_j$.

We define the *reduced end-point mapping* by $$E(x_0,(u_{10}, \ldots,u_{m0}),\left(t_1,i_1\right), \ldots, \left(t_N,i_N\right),t_f) = x_u(x_0,t_f),$$ where $u$ is the control represented by $\left((u_{10}, \ldots,u_{m0}),\left(t_1,i_1\right), \ldots, \left(t_N,i_N\right), t_f \right)$, and $x_u(x_0,t_f)$ is the associated state at time $t_f$, starting at $x_0$.

Note that in [@Maurer2005; @Maurer2004], the authors also reduce a bang-bang control to its switching points, in order to formulate an 