Simple integrator: state energy minimisation nonsmooth turnpike

The state energy minimisation simple integrator problem consists in minimising

\[ \int_{0}^{2} x(t)^2 \, \mathrm{d}t\]

subject to the constraints

\[ \dot x(t) = u(t), u(t) \in [0,\gamma], \\\]

and the limit conditions

\[ x(0) = 1, \quad x(2) = 0.5.\]

You can access the problem in the CTProblems package:

using CTProblems
prob = Problem(:turnpike, :integrator, :state_energy, :x_dim_1, :u_dim_1, :lagrange, :u_cons, :singular_arc)

title           = simple nonsmooth turnpike - state energy min - affine system in u
model    (Type) = OptimalControlModel{Autonomous, Fixed}
solution (Type) = OptimalControlSolution

Then, the model is given by

prob.model

The (autonomous) optimal control problem is given by:

    t ∈ [t0, tf], time
    x ∈ R, state
    u ∈ R, control
    x(t0) == x0, initial_con
    x(tf) == xf, final_con
    -1 ≤ u(t) ≤ 1, u_con
    ẋ(t) == u(t)
    ∫(x(t) ^ 2) → min

The (autonomous) optimal control problem is of the form:

    minimize  J(x, u) = ∫ f⁰(x(t), u(t)) dt, over [0, 2]

    subject to

        ẋ(t) = f(x(t), u(t)), t in [0, 2] a.e.,

        ηl ≤ η(x(t)) ≤ ηu, 
        ϕl ≤ ϕ(x(0), x(2)) ≤ ϕu, 

    where x(t) ∈ R and u(t) ∈ R.

Declarations (* required):
╭────────┬────────┬──────────┬──────────┬───────────┬────────────┬─────────────╮
│ times* │ state* │ control* │ variable │ dynamics* │ objective* │ constraints │
├────────┼────────┼──────────┼──────────┼───────────┼────────────┼─────────────┤
│   V    │   V    │    V     │    X     │     V     │     V      │      V      │
╰────────┴────────┴──────────┴──────────┴───────────┴────────────┴─────────────╯

You can plot the solution.

using Plots
plot(prob.solution)