Geodesic
Logarithmic spirals in optimal control problems with control in a disk
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 75-88

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We study a neighborhood of singular second-order extremals in optimal control problems that are affine in a two-dimensional control in a disk. We study the stabilization problem for a linear system of second-order differential equations for which the origin is a singular second-order extremal. This problem can be considered as a perturbation of an analog of the Fuller problem with two-dimensional control in a disk. We prove that for this class of problems, optimal solutions keep their form of logarithmic spirals that arrive at a singular point in a finite time, while optimal controls make an infinite number of revolutions along the circle. Finally, we present a brief review of problems whose solutions have the form of such logarithmic spirals.
Keywords: two-dimensional control in a disk, singular extremal, blow-up of a singularity, logarithmic spiral, Hamiltonian system, Pontryagin's maximum principle 
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     title = {Logarithmic spirals in optimal control problems with control in a disk},
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M. I. Ronzhina; L. A. Manita. Logarithmic spirals in optimal control problems with control in a disk. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 75-88. http://geodesic.mathdoc.fr/item/INTO_2024_233_a6/