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Jacobi Fields in Optimal Control: One-dimensional Variations

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Abstract

In this paper which is closely related to the previous paper [3] we specify general theory developed there. We study the structure of Jacobi fields in the case of an analytic system and piecewise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piecewise analytic in this case but may have jump discontinuities. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighbourhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps.

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Acknowledgments

We would like to thank Luca Rizzi for pointing out an error in the initial formulation of the oscillation theorem and Jean-Baptiste Caillau, Francesco Rossi and the anonymous referee for helpful discussions and remarks that helped to improve the article.

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Correspondence to Ivan Beschastnyi.

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Agrachev, A., Beschastnyi, I. Jacobi Fields in Optimal Control: One-dimensional Variations. J Dyn Control Syst (2020) doi:10.1007/s10883-019-09467-0

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Keywords

  • Optimal control
  • Second-order conditions
  • Jacobi equations
  • Singular systems