Time-optimal synthesis for left-invariant control systems on SO(3)

Consider the control system (Sigma) given by (x) over dot = x( f + ug), where x is an element of SO(3), | u| <= 1, and f, g is an element of so(3) define two perpendicular left-invariant vector fields normalized so that parallel to f parallel to = cos(alpha) and parallel to g parallel to = sin(alpha), alpha is an element of]0, pi/4[. In this paper, we provide an upper bound and a lower bound for N(alpha), the maximum number of switchings for time-optimal trajectories of (Sigma). More precisely, we show that N-S(alpha) <= N(alpha) <= N-S(alpha) + 4, where N-S(alpha) is a suitable integer function of alpha such that N-S(alpha) alpha (-->) over bar )over tilde>0 pi/(4 alpha). The result is obtained by studying the time-optimal synthesis of a projected control problem on RP2, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere S-2. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.