Simple integrator: constraint minimisation state and control constraint non autonomous

The constraint minimisation simple integrator problem consists in minimising

\[ \int_{0}^{3} e^{-α*t}*u(t) \, \mathrm{d}t\]

subject to the constraints

\[ \dot x(t) = u(t), u(t) \in [0,3], \\ 1 - x(t) - (t-1)^2 \in [-\infty,0]\]

and the limit conditions

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

You can access the problem in the CTProblems package:

using CTProblems
prob = Problem(:integrator, :state_dime_1, :lagrange, :x_cons, :u_cons, :nonautonomous)

title           = simple integrator - state and control constraints - nonautonomous
model    (Type) = OptimalControlModel{NonAutonomous, Fixed}
solution (Type) = OptimalControlSolution

Then, the model is given by

prob.model

The (non autonomous) optimal control problem is given by:

    t ∈ [t0, tf], time
    x ∈ R, state
    u ∈ R, control
    x(t0) == x0, initial_con
    0 ≤ u(t) ≤ 3, u_con
    -Inf ≤ (1 - x(t)) - (t - 2) ^ 2 ≤ 0, x_con
    ẋ(t) == u(t)
    ∫(exp(-α * t) * u(t)) → min

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

    minimize  J(x, u) = ∫ f⁰(t, x(t), u(t)) dt, over [0, 3]

    subject to

        ẋ(t) = f(t, x(t), u(t)), t in [0, 3] a.e.,

        ηl ≤ η(t, x(t)) ≤ ηu, 
        ϕl ≤ ϕ(x(0), x(3)) ≤ 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)