\delta u$ such that $$E(\overline{x}(t) + \delta x(t),\overline{u}+\delta u,t_f-t) = x_f.$$ Assume for a moment that the perturbation of the control $\delta u$ is small in $L^{\infty}$ norm. Then, at least formally, one can write $$E(\overline{x}(t),\overline{u},t_f-t) + \frac{\partial E}{\partial x_0}(\overline{x}(t),\overline{u},t_f-t)\cdot \delta x(t) + \frac{\partial E}{\partial u}(\overline{x}(t),\overline{u},t_f-t)\cdot \delta u  + o(\|\delta x(t)\|+\|\delta u\|)= x_f.$$ Therefore, at order one, we look for a solution of the (linear) equation $$\frac{\partial E}{\partial x_0}(\overline{x}(t),\overline{u},t_f-t)\cdot \delta x(t) + \frac{\partial E}{\partial u}(\overline{x}(t),\overline{u},t_f-t)\cdot \delta u  = 0.
\label{eq_perturbations}$$

However, we do not want, in this paper, to apply small perturbations in the $L^{\infty}$ norm, as they would not result in bang-bang controls (However, this is similar to what is done while performing a Ricatti procedure to stabilize a system or track a reference trajectory). Nevertheless, reducing the end-point mapping to the switching times enables us to preserve the bang-bang structure: in the formalism previously introduced, we need to solve the nonlinear system of equations $$E(\overline{x}(t) + \delta x(t), \overline{\mathcal{T}} + \delta \mathcal{T}, t_f - t) = x_f.$$

The equation becomes $$\frac{\partial E}{\partial \mathcal{T}}(\overline{x}(t),\overline{\mathcal{T}},t_f-t)\cdot \delta \mathcal{T} = - \frac{\partial E}{\partial x_0}(\overline{x}(t),\overline{\mathcal{T}},t_f-t)\cdot \delta x(t),
\label{eq_perturbations_1}$$ where the expression $\partial E / \partial \mathcal{T}$ is given by Proposition \[prop\_differentiability\_epm\].

#### A 