Logarithmic spirals in optimal control problems with control in a disk

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.