We analyze two optimal problems for a class of nonlinear system on the real projective line Popf1 induced by a class of bilinear control system: the angular system. Two functional costs are considered: time-optimal and quadratic. According to the Pontryagin Maximum Principle, in the time-optimal case we show that if the angle system PS satisfies the controllability property, then there exists a minimal time bang-bang trajectory connecting any two points on P1, the noncontrollable case was discussed in closed form in (SIAM J. Control Optim. 2009; 48(4):26362650). On the other hand, in the quadratic cost, the optimal control is a continuous function (Proyecciones J. Math. 2010; 29(2):145164). A comparison is also established between the structure of the solutions for the two optimal problems: time-optimal and quadratic in the controllable and noncontrollable cases. The extremals are obtained from the adjoint system given by the Pontryagin Maximum Principle onto P1 via radial projection. An example is given. Copyright (C) 2011 John Wiley & Sons, Ltd.