Geodesic
Neighborhood of the Second-Order Singular Regime in Problems with Control in a Disk
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 222-236

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We consider Hamiltonian systems that are affine in a two-dimensional control with values in a disk. In the neighborhood of a second-order singular extremal, we study the structure of optimal synthesis and find a family of solutions in the form of logarithmic spirals that make countably many revolutions around a singular point and reach this point in finite time. 
@article{TM_2021_315_a15,
     author = {M. I. Ronzhina and L. A. Manita and L. V. Lokutsievskiy},
     title = {Neighborhood of the {Second-Order} {Singular} {Regime} in {Problems} with {Control} in a {Disk}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {222--236},
     publisher = {mathdoc},
     volume = {315},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a15/}
}
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M. I. Ronzhina; L. A. Manita; L. V. Lokutsievskiy. Neighborhood of the Second-Order Singular Regime in Problems with Control in a Disk. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 222-236. http://geodesic.mathdoc.fr/item/TM_2021_315_a15/