(red cross). $v_{x0}=-0.02527$ in all three cases ($v_{x0}$ is the initial velocity component of the $L_5$ point) while $v_{y0}=-0.24692\;-0.228,\;-0.2194$ for panel (a), (b), and (c), respectively. Different initial $v_{y0}$ velocities correspond to smaller, equal, and larger values than that of the $L_5$ point’s $v_{y0}.$ One can clearly see that, depending on the initial velocity, the relative position of the long-lived island compared to the red cross changes. The smaller the $v_{y0}$, the further the long-lived island from the origin.

We can point out that small changes in $v_{y0}$ results an outward/inward drift of the structures in $x-y$ plane. This fact will be the base of our control method, as we shall see later.

How to Keep a Spacecraft in a Confined Region
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In this section we present our strategy on how to keep a test particle in the system, i.e. preventing the scenario in which $r$ exceeds 1.5. Though the orbit is well-defined by the initial conditions, our method does not allow a strict station keeping, but a rather cheap control of the motion that allows for a bounded orbit around the initial one.

The basic idea of the method is to construct ICMs for appropriate initial velocities similar 