Optimal trajectories for angular systems on the projective line

We analyze two optimal problems for a class of nonlinear system on the real projective line Popf ¹ 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 ℙ satisfies the controllability property, then there exists a minimal time bang-bang trajectory connecting any two points on ℙ ¹, the noncontrollable case was discussed in closed form in (SIAM J. Control Optim. 2009; 48(4):2636-2650). On the other hand, in the quadratic cost, the optimal control is a continuous function (Proyecciones J. Math. 2010; 29(2):145-164). 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 ℙ ¹ via radial projection. An example is given.