Properties of the Time-Optimal Control for Lagrangian Single-Degree-of-Freedom Systems

The number of switchings is an important parameter characterizing the complexity of the bang-bang optimal control. A time-optimal control problem for Lagrangian single-degree-of-freedom (single-DOF) systems is considered. The system describes the dynamics of an inertial object under the action of a bounded control force and an external force given as a function of the coordinate. The terminal set consists of the zero point in the phase plane. We analyze the possible motion time along the optimal trajectories with two switchings and obtain corresponding necessary optimality condition. As a result, we suggest a criterion (tool) for quick checking the absence of such trajectories in the phase plane. In this case, the time-optimal feedback control has the simplest structure: the number of switchings is not greater than one for any initial conditions. The corresponding examples are presented.