## 5 Conclusions  
In this work, motivated by the challenging astrodynamic environment, as quoted by [PERSON] [36] \"among the most extreme found in the Solar System\", we perform a qualitative assessment on how the State-Dependent Riccati Equation (SDRE) suboptimal control formulation would behave in the highly nonlinear and strongly perturbed environment within multi-body NEA systems, to the authors' knowledge a never before undertaken. We made a qualitative assessment on the suitability of the implementation of SDRE in body-fixed hovering and reference-trajectory tracking of a spacecraft in the highly nonlinear environment around Near-Earth Objects (NEOs) [38], a class of small and faint objects of the Solar System with high relevance either in scientific, economic, or planetary defense-related aspects, targets of various rendezvous missions executed/planned by different space agencies [149]. As stated before, we intended to prove the suitability of using the SDRE approach in the highly disturbed environment around Small Solar System Bodies. For real missions, it is necessary to make deeper studies. In our numerical results, the absolute and relative errors are of the order of \(10^{-14}\), as defined in the MATLAB\({}^{\otimes}\) environment we used for the calculations, which means our numerical integrations imparted, for instance, position errors of at most \(10^{-9}\) m (maximum distance \(\sim 10^{5}\) m, see Figure 8). In our results, the great flexibility offered by the state-dependent (SD) factorization in the specification of the control-effort or state-error coefficient matrices, trading off the convergence of the controlled trajectory against fuel consumption, already known in many fields as cancer treatment, satellite or robotic formation dynamics, or in energy harvesting, confirm that this control approach is also highly effective when addressing the hovering of a spacecraft around NEO, as we demonstrated in the cases of body-fixed hovering around the NEAs (16) Psyche and the much smaller main body of the triple system (153591) 2001 SN263, or in the highly stringent case of trajectory tracking in the highly nonlinear environment within the triple NEA system (153591) 2001 SN263, all envisaging the best possible trade-off between time of convergence and fuel consumption, a central aspect in such heavenly missions. This effectiveness can be observed particularly in Figure 7, which depicts control effort and time of convergence of the controlled trajectories around (16) Psyche ranging from a half-time convergence with \(\sim\)20 times lower cost. Figures 14 and 15 show convergence and control cost cases for (153591) 2001 SN263's depicting a trajectory converging 10 times faster but with up to 100 times higher cost. Figure 16: Trajectories for a spacecraft approaching the stable equilibrium point \((x_{x},y_{y})=(0,\,1450.75)\) (\(\ast\)) in the \(Oxy\) effective potential for 2001 SN263-ALPHA. Close view at right. The two curves were obtained with small changes in the coefficient matrices. Dashed black line represents the body surface. As a future extension to this work, we suggest tackling the inertial hovering approach around the Lagrange points of the triplet 2001 SN263 and a sensitivity analysis considering the influence of uncertainties engendered by errors in the physical parameters of the bodies [79; 150]. We also intend to perform an analysis/determination of the basins of attraction for the system, a significant aspect considering closer observation orbits or touch-and-go and descent maneuvers. Conceptualization: A.F.B.d.A.P. and J.M.B. ; writing, original draft preparation: A.M.T. and E.C. ; writing, review and editing: A.M.T., A.F.B.d.A.P., E.C., J.M.B. and V.R. ; supervision: [PERSON] and J.M.B. ; funding acquisition: V.R. All authors have read and agreed to the published version of the manuscript. This paper was supported by the Ministry of Science and Higher Education of the Russian Federation under Agreement No. FSSF-2024-0005. Not applicable. No new data were created or analyzed in this study.