tutorial-basic-example-f.md

[Basic example (functional version)](@id basic-f)

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 $x_2$.

<img src="./assets/chariot.png" style="display: block; margin: 0 auto 20px auto;" width="300px">

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 $x(1) = (0, 0)$.

!!! note "Solution and details"

See the page 
[Double integrator: energy minimisation](https://control-toolbox.org/docs/ctproblems/stable/problems/double_integrator_energy.html#DIE) 
for the analytical solution and details about this problem.

using Plots
using Plots.PlotMeasures
plot(args...; kwargs...) = Plots.plot(args...; kwargs..., leftmargin=25px)

First, we need to import the OptimalControl.jl package:

using OptimalControl

Then, we can define the problem

ocp = Model()                                   # empty optimal control problem

time!(ocp, [ 0, 1 ])                            # time interval
state!(ocp, 2)                                  # dimension of the state
control!(ocp, 1)                                # dimension of the control

constraint!(ocp, :initial, [ -1, 0 ])           # initial condition
constraint!(ocp, :final,   [  0, 0 ])           # final condition

dynamics!(ocp, (x, u) -> [ x[2], u ])           # dynamics of the double integrator

objective!(ocp, :lagrange, (x, u) -> 0.5u^2)    # cost in Lagrange form
nothing # hide

!!! note "Nota bene"

There are two ways to define an optimal control problem:
- using functions like in this example, see also the [`Model` documentation](https://control-toolbox.org/docs/ctbase/stable/api-model.html) for more details.
- using an abstract formulation. You can compare both ways taking a look at the abstract version of this [basic example](@ref basic).

Solve it

sol = solve(ocp)
nothing # hide

and plot the solution

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