the control at this time and see whether the red cross lands in the long-lived island in the next ICM, at $t=$271.26 days (Fig. \[fig:control\] a).

At this point we should address the answer to the first question. Since we know the system dynamics only at discrete times, at every single ICM, a natural choice is a continuous tangential force acting on the RT (or spacecraft/satellite) between two ICMs in the following form, $$\mathbf{f} =  \left\{ \begin{array}{ll}
         F_y\cdot\mathbf{v_y}/|v| & \mbox{if $\;t \in \left[t_{n},\, t_{n} + \Delta t\right]$}, \\
         F_x\cdot\mathbf{v_x}/|v| & \mbox{if $\;t \in \left[t_{n},\, t_{n} + \Delta t\right]$}, \\
         0   & \mbox{otherwise},\end{array}\right.
\label{eq:force}$$ where $F_x$ and $F_y$ are the magnitude of the applied force in directions $\mathbf{v_x}$ and $\mathbf{v_y}$ (where $\mathbf{v_x}$ is the $x$ component, $\mathbf{v_y}$ is the $y$ component of the velocity vector), $t_{n}$ is the instance when the control begins and $\Delta t$ describes the length of the control.

In general, we switch on the propulsion at $t$ (or equivalently when we reach the criterion $v_x=$0 and $v_y<$0) in order to manoeuvre the spacecraft to a given phase space position, $(x,y,v_x,v_y).$ Therefore, we should have an approximate initial guess for the magnitude of the force (thrust or breaking). 