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Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2)

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Abstract

Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality, is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis.

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Acknowledgments

We are grateful to the anonymous reviewers for valuable comments that improved presentation of the paper.

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Authors and Affiliations

  1. Department of Electronic Engineering, Muhammad Ali Jinnah University, Islamabad, Pakistan

    Yasir Awais Butt & Aamer Iqbal Bhatti

  2. Program Systems Institute, Pereslavl-Zalessky, Russia

    Yuri L. Sachkov

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  1. Yasir Awais Butt
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  2. Yuri L. Sachkov
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  3. Aamer Iqbal Bhatti
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Correspondence to Yasir Awais Butt.

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Butt, Y.A., Sachkov, Y.L. & Bhatti, A.I. Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2). J Dyn Control Syst 23, 155–195 (2017). https://doi.org/10.1007/s10883-016-9337-4

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  • DOI: https://doi.org/10.1007/s10883-016-9337-4

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