Bifurcation in a time-optimal problem for a second-order non-linear system

A time-optimal problem for a second-order non-linear system with one degree of freedom is considered. The system describes the dynamics of an inertial object under the action of a control force of limited modulus which appears linearly and a perturbing force which is periodic in coordinate. The terminal set represents points on the abscissa of the phase plane, and the distance between two neighbouring points is equal to the period of the perturbing force. An estimate is obtained for the amplitude of the control for which the control has the simplest structure: the number of switchings is not greater than one.