Family of Logarithmic Spirals in Hamiltonian Systems of Dimension 8 with Control in a Disk

We study a neighbourhood of a singular second-order extremal in optimal controlproblems that are affine in control in a disk. We consider the case in which the Hamiltoniansystem has dimension 8 and is a small (in the sense of the action of the Fuller group) perturbationof the Hamiltonian system of the generalized Fuller problem with control in the disk. For thisclass of problems, we prove the existence of extremals in the form of logarithmic spirals that reachthe singular second-order extremal in a finite time, while the control performs infinitely manyrotations around the circle.