# Basic example

Let us consider a wagon moving along a rail, whom acceleration can be controlled by a force $u$. We denote by $x = (x_1, x_2)$ the state of the wagon, that is its position $x_1$ and its velocity $x_2$.

We assume that the mass is constant and unitary and that there is no friction. The dynamics we consider is given by

\[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = u(t), , \quad u(t) \in \R,\]

which is simply the double integrator system. Les us consider a transfer starting at time $t_0 = 0$ and ending at time $t_f = 1$, for which we want to minimise the transfer energy

\[ \frac{1}{2}\int_{0}^{1} u^2(t) \, \mathrm{d}t\]

starting from the condition $x(0) = (-1, 0)$ and with the goal to reach the target $x(1) = (0, 0)$.

See the page Double integrator: energy minimisation for the analytical solution and details about this problem.

First, we need to import the `OptimalControl.jl`

package:

`using OptimalControl`

Then, we can define the problem

```
@def ocp begin
t ∈ [ 0, 1 ], time
x ∈ R², state
u ∈ R, control
x(0) == [ -1, 0 ]
x(1) == [ 0, 0 ]
ẋ(t) == [ x₂(t), u(t) ]
∫( 0.5u(t)^2 ) → min
end
```

Solve it

`sol = solve(ocp)`

```
Method = (:direct, :adnlp, :ipopt)
This is Ipopt version 3.14.13, running with linear solver MUMPS 5.6.1.
Number of nonzeros in equality constraint Jacobian...: 1205
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 101
Total number of variables............................: 404
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 305
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e-01 1.10e+00 1.92e-14 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -5.0000000e-03 1.81e-01 1.78e-15 -11.0 6.04e+00 - 1.00e+00 1.00e+00h 1
2 6.0023829e+00 8.88e-16 1.78e-15 -11.0 6.01e+00 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 2
(scaled) (unscaled)
Objective...............: 6.0023829460295719e+00 6.0023829460295719e+00
Dual infeasibility......: 1.7763568394002505e-15 1.7763568394002505e-15
Constraint violation....: 8.8817841970012523e-16 8.8817841970012523e-16
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 1.7763568394002505e-15 1.7763568394002505e-15
Number of objective function evaluations = 3
Number of objective gradient evaluations = 3
Number of equality constraint evaluations = 3
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 3
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 2
Total seconds in IPOPT = 0.036
EXIT: Optimal Solution Found.
```

and plot the solution

`plot(sol, size=(600, 450))`