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Symmetry Properties of States and Costates in Impulsive Control Systems

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Abstract

This paper reveals two symmetry properties of the states and costates of a special type of optimal impulsive control problems and, based on these two properties, proposes a new procedure for obtaining the second-order states and costates approximations. It further shows with an example that this procedure is efficient and can be implemented effectively in a computer algebraic system.

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Notes

  1. The relative error of an approximation is defined as \(\frac{|\sigma- \sigma^{*}|}{|\sigma^{*}|}\). σ and σ ∗ denote the approximation and the true value. |.| denotes the 2-norm.

References

  1. London, H.: Second approximation to the solution of the rendezvous equations. AIAA J. 1(7), 1691–1693 (1963)

    Article  MATH  Google Scholar 

  2. Anthony, N., Sasaki, F.: Rendezvous problem for nearly circular orbits. AIAA J. 3, 1666–1673 (1965)

    Article  Google Scholar 

  3. Richardson, D., Mitchell, J.: A third-order analytical solution for relative motion with a circular reference orbit. J. Astronaut. Sci. 51(1), 1–12 (2003)

    MathSciNet  Google Scholar 

  4. Vaddi, S.S., Vadali, S.R., Alfriend, K.T.: Formation flying: accommodating nonlinearity and eccentricity perturbations. J. Guid. Control Dyn. 26(2), 214–223 (2003)

    Article  Google Scholar 

  5. Sengupta, P., Vadali, S.R., Alfriend, K.T.: Second-order state transition for relative motion near perturbed, elliptic orbits. Celest. Mech. Dyn. Astron. 97(2), 101–129 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Zhang, G., Zhou, D.: A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits. Celest. Mech. Dyn. Astron. 107, 319–336 (2010)

    Article  MATH  Google Scholar 

  7. Lawden, D.: Optimal Trajectories for Space Navigation. Butterworths, Stoneham (1963)

    MATH  Google Scholar 

  8. Lion, P.M., Handelsman, M.: Primer vector on fixed-time impulsive trajectories. AIAA J. 6, 127–132 (1968)

    Article  MATH  Google Scholar 

  9. Hiday-Johnston, L.A., Howell, K.C.: Transfers between libration-point orbits in the elliptic restricted problem. Celest. Mech. Dyn. Astron. 58, 317–337 (1994)

    Article  Google Scholar 

  10. Prussing, J.E.: Primer vector theory and applications. In: Conway, B.A. (ed.) Spacecraft Trajectory Optimization. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  11. Carter, T.: State transition matrices for terminal rendezvous studies: brief survey and new example. J. Guid. Control Dyn. 21(1), 148–155 (1998)

    Article  MATH  Google Scholar 

  12. Carter, T.: Optimal power-limited rendezvous with thrust saturation. J. Guid. Control Dyn. 18(5), 1145–1150 (1995)

    Article  Google Scholar 

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Authors and Affiliations

  1. The MathWorks, Inc., Natick, USA

    Weijun Huang

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  1. Weijun Huang
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Correspondence to Weijun Huang.

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Communicated by Bruce A. Conway.

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Huang, W. Symmetry Properties of States and Costates in Impulsive Control Systems. J Optim Theory Appl 160, 597–607 (2014). https://doi.org/10.1007/s10957-013-0404-7

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  • DOI: https://doi.org/10.1007/s10957-013-0404-7

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