Minimal-time control for a double integrator with saturated control ±u_max
The system ẍ = u models a point mass with unit mass. The goal is to bring the system from (x₀, v₀) to the origin (0, 0) in minimal time using bounded control |u| ≤ u_max.
The optimal control switches at most once between +u_max and -u_max. The switching curve consists of two parabolas: the system follows one parabola until reaching the switching curve, then follows the other to the origin.