Pontryagin Bang-Bang Control

Time-optimal control of a double integrator

u = +1

ẋ₁ = x₂ (position → velocity)

ẋ₂ = u (acceleration)

|u| ≤ 1 (bounded control)

Initial Position (x₁)

3.0

Initial Velocity (x₂)

2.0

Simulation Speed

1.0x

Display Options

Switching Curve Trajectory Phase Flow Control Regions

0.00

Position x₁

0.00

Velocity x₂

0.00

Time

u = +1

u = -1

Switching Curve

The switching curve is: x₁ = -½ sgn(x₂) x₂²

Bang-bang principle: For time-optimal control, the control only takes extreme values (+1 or -1), never intermediate values.

The trajectory follows parabolas in phase space, switching exactly once when crossing the optimal switching curve.

Click anywhere on the phase plane to start a trajectory from that point. Watch how the optimal controller drives the system to the origin.