external disturbances or model uncertainties, we will develop a method to tune the switching times in order to absorb these perturbations and steer the system to the desired target $x_f \in \mathbb{R}^n$.

Here, in this still informal introduction, we show how to use the additional switching times to make the system reach targets $x_f+\delta x_f$ in a neighborhood of $x_f$. The idea is to solve the nonlinear system of equations $$E(x_0,t_f,\mathcal{T}+\delta\mathcal{T}) = x_f +\delta x_f .$$ Using , we propose to solve, at the first order, $$\label{eq_rec}
\frac{\partial E}{\partial\mathcal{T}}(x_0, t_f, \mathcal{T})\cdot \delta\mathcal{T} = \delta x_f ,$$ which makes $n$ equations with $N+\ell$ degrees of freedom. We assume that $N+\ell$ is (possibly much) larger than $n$ and that the matrix in is surjective. Then one can solve by using the *Moore-Penrose pseudo-inverse* $\left(\frac{\partial E}{\partial\mathcal{T}}\right)^{\dagger}$ of $\frac{\partial E}{\partial\mathcal{T}}$ (see [@Golub], or see [@Beutler1; @Beutler2] for a theory in infinite dimension), which yields the solution of minimal Euclidean norm $$\delta\mathcal{T} = \left(\frac{\partial E}{\partial\mathcal{T}}\right)^{\dagger} \cdot \delta x_f ,$$ and we have $$\label{eq_estimate_simple}
\left\| \delta\mathcal{T} \right\|_2 {\leqslant}\frac{\left\| \delta x_f \right\|_2}{\sigma_{min}} ,$$ where $\sigma_{min}$ is the smallest positive singular value of $\frac{\partial E}{\partial\mathcal{T}}$. This estimate gives a natural measure for robustness, that we will generalize.

The 