Synthesis of the time-optimal motion of an oscillator along a broken-line path

A planar problem of the time-optimal displacement of a two-dimensional osci llator into a given point on a broken-line ps,th with the suppression of re lative oscillations in the end of the control process is studied. It is ass umed that the speed of the equilibrium position is controllable within pres cribed bounds and the initial position is arbitrary. The examined system of equations turns out to be nonsmooth. Within the asymptotic approach (the c ontrol acl:ions are relatively small), a program and synthesis of optimal c ontrols, as well as optimal trajectories and the minimal time of motion for arbitrary initial data, are constructed. The structure of synthesis is ana lyzed, and some qualitative effects are discovered. In particular, the opti mal time of motion is found to be a discontinuous function of the state var iables. The results can be of interest in solving the problems of precision control of complex technical systems containing internal moving elements.