Extremum conditions for constrained scalar control of two nonsynchronous oscillators in the time-optimal control problem

The time-optimal control problem for two nonsynchronous oscillators accelerated from rest with a constrained scalar control is considered. A feature of this problem is that the phase coordinates of the second oscillator again become equal to zero at the terminal time. For a given number of unknown switching times that determine the optimal bang-bang control, necessary extremum conditions in the form of nonlinear matrix equalities are proposed. An analytical form of the curve corresponding to the class of two switchings in the phase space of the first oscillator is found by analyzing necessary and sufficient extremum conditions. This curve separates the reachable sets of the class of three switchings.