The inverted pendulum has Lagrangian , and equation where is the angle to the vertical. Alternatively, let be the Cartesian coordinates of the position of the pendulum, subject to the constraint .

The Lagrangian is then

and the mechanical equations are

where we have set an horizontal force as the control .

We want to minimize the objective

Figure 1 shows the states , the control and multiplier .

**Numerical simulations:**

*Discretization: Runge Kutta 4 with 400 steps.*

*We take here* , *and* .

*The final conditions are*

*The initial conditions are*