Abstract
First-order necessary conditions for optimality reveal the Hamiltonian nature of optimal control problems. Regardless of the overwhelming awareness of this result, the implications that it entails have not been fully explored. We discuss how the symplectic structure of optimal control constrains the flow of sub-volumes in the phase space. Special emphasis is devoted to dynamics in the neighborhood of optimal trajectories and insight is gained into how errors in the initial states affect terminal conditions. Specifically, we prove that if the optimal trajectory does not satisfy a particular condition, then there exists a set of variations in the initial states yielding a greater error in norm when mapped to the terminal time through the state transition matrix. We relate this result to the sensitivity problem in solving indirect problems for optimal control.
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Notes
For repeatability purposes, the shooting problem is solved by means of the fsolve function of MATLAB using default settings
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Acknowledgements
This work was partially supported by the Belgian National Fund for Scientific Research (FNRS) and by the Air Force Office of Scientific Research (AFOSR).
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Dell’Elce, L., Scheeres, D.J. Sensitivity of Optimal Control Problems Arising from their Hamiltonian Structure. J Astronaut Sci 67, 539–551 (2020). https://doi.org/10.1007/s40295-019-00168-1
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DOI: https://doi.org/10.1007/s40295-019-00168-1