Optimal control of the double integrator — minimum-time rocket car
Setup
Control u(t)—
Time elapsed0.00 s
Position x—
Velocity v—
Switch time—
Double integrator: ẋ=v, v̇=u, |u|≤u_max.
Pontryagin: optimal control is bang-bang: u=±u_max.
Co-states: ṗ₁=0, ṗ₂=−p₁; u*=−sign(p₂)·u_max.
Switching curve: v=−sign(x+v²/2u_max)·√(2u_max|x+v|).
Minimum time: one or two arcs, at most one switch.