nonlinear control system $$\label{sys_intro}
\dot x(t) = f(t,x(t),u(t)),$$ from a given $x(0)=x_0$ to the target point $x(t_f)=x_f$, with a scalar control $u$ that can only switch between two values, say $0$ and $1$. The general method, as well as all assumptions, will be written in details in a further section.

Let $E(x_0,t_f,u) = x(t_f)$ be the end-point mapping, where $x(\cdot)$ is the solution of starting at $x(0)=x_0$ and associated with the control $u$. One aims at finding a bang-bang control $u$, defined on $[0,t_f]$ for some final time $t_f>0$, such that $E(x_0,t_f,u) = x_f$.

Many problems impose to implement only bang-bang controls, i.e., controls saturating the constraints but not taking any intermediate value. These are problems where only external actions of the kind on/off can be applied to the system.

Of course, such bang-bang controls can usually be designed by using optimal control theory (see [@lee1967foundations; @pontryagin1987; @Trelat1]). For instance, solving a minimal time control problem, or a minimal $L^1$ norm as in [@Caponigro2015], is in general a good way to design bang-bang control strategies. However, due to their optimality status, such controls often suffer from a lack of robustness with respect to uncertainties, model errors, deviations from the target. 