About two-switching control class in the time-optimal control problem of two non-synchronous oscillators

The time-optimal control problem in a system consisting of two non-synchronous oscillators is considered. The studied formulation has a number of distinctive features, such as, for example, that each of the oscillators is controlled by a common bounded scalar control and the goal is to accelerate the first oscillator from rest to a given position in the shortest time. At the terminal moment the phase coordinates of the second oscillator become zero again. The optimal control is a relay mode, so solutions with different numbers of control switchings are the key. The basic one is the three-switching control class and for a larger number of unknown switching moments the necessary optimality conditions are known. The two-switching control class obtained by degeneracies is of interest, where functional dependences are written out for the values of interval durations. Mathematical modeling was carried out to illustrate the obtained analytical results.