# American Institute of Mathematical Sciences

April  2014, 10(2): 477-501. doi: 10.3934/jimo.2014.10.477

## Designing rendezvous missions with mini-moons using geometric optimal control

 1 Department of Mathematics, University of Hawaii at Manoa, Honolulu, United States, United States, United States 2 Department of Physics, University of Helsinki, Helsinki, Finland 3 Institute for Astronomy, University of Hawaii at Manoa, Honolulu, United States 4 Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, Paris, France

Received  March 2013 Revised  August 2013 Published  October 2013

Temporarily-captured Natural Earth Satellites (NES) are very appealing targets for space missions for many reasons. Indeed, NES get captured by the Earth's gravity for some period of time, making for a more cost-effective and time-effective mission compared to a deep-space mission, such as the 7-year Hayabusa mission. Moreover, their small size introduces the possibility of returning with the entire temporarily-captured orbiter (TCO) to Earth. Additionally, NES can be seen as interesting targets when examining figures of their orbits. It requires to expand the current state-of-art of the techniques in geometric optimal control applied to low-thrust orbital transfers. Based on a catalogue of over sixteen-thousand NES, and assuming ionic propulsion for the spacecraft, we compute time minimal rendezvous missions for more than $96%$ of the NES. The time optimal control transfers are calculated using classical indirect methods of optimal control based on the Pontryagin Maximum Principle. Additionally we verify the local optimality of the transfers using second order conditions.
Citation: Monique Chyba, Geoff Patterson, Gautier Picot, Mikael Granvik, Robert Jedicke, Jeremie Vaubaillon. Designing rendezvous missions with mini-moons using geometric optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 477-501. doi: 10.3934/jimo.2014.10.477
##### References:

show all references

##### References:
 [1] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [2] W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 [3] Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405 [4] Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 [5] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [6] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [7] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [8] Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 [9] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [10] A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 [11] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [12] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [13] Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

2019 Impact Factor: 1.366