From
the recent thirty years, scientists will never stop exploring the outer space.
To assist the development of travelling into the universe, I devote myself into
providing theoretical support and future indications for designing the optimal
orbit for satellite to travel in a Three-Body
System. This paper offers the optimal orbit for satellite to change path in the
earth-moon system. Also, it provides the path for the satellite to use the
least fuel to go to the L4 and L5 Lagrange points. These inspiring results are
obtained through several steps: to solve the problems caused by the non-linear
character of Three-Body System, I use Koopman eigenfunction to change the system into a
linear one. Data-driven method is adopted to find the most suitable Koopman
eigenfunction to apply control. The traditional LQR operator for linear system
is used to design the optimal orbit for the satellite.
References
[1]
Kaiser, E., Kutz, J.N. and Brunton, S.L. (2018) Data-Driven Discovery of Koopman Eigen Functions for Control. arXiv preprint arXiv:1707.01146
[2]
Brunton, S.L., Proctor, J.L. and Kutz, J.N. (2016) Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems. Proceedings of the National Academy of Sciences of the USA, 113, 3932-3937.
https://doi.org/10.1073/pnas.1517384113
[3]
Gardner, J.P., Mather, J.C., Clampin, M., Doyon, R., Greenhouse, M.A., Hammel, H.B., Hutchings, J.B., Jakobsen, P., Lilly, S.J., Long, K.S., et al. (2006) The James Webb Space Telescope. Space Science Reviews, 123, 485-606.
https://doi.org/10.1007/s11214-006-8315-7
[4]
Benson, J.M. and Mutel, R. (1979) Multibaseline vlbi Observations of the 1612 mhz oh Masers toward nml cygni and vy canis majoris. The Astrophysical Journal, 233, 119-126. https://doi.org/10.1086/157372
[5]
Brunton, S.L., Brunton, B.W., Proctor, J.L. and Kutz, J.N. (2016) Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control. PLoS ONE, 11, e0150171.
https://doi.org/10.1371/journal.pone.0150171
[6]
Koopman, B.O. (1931) Hamiltonian Systems and Transformation in Hilbert Space. Proceedings of the National Academy of Sciences of the USA, 17, 315-318.
https://doi.org/10.1073/pnas.17.5.315
