Print

Examples

An optimal control problem can be described as minimising the cost functional

\[g(t_0, x(t_0), t_f, x(t_f)) + \int_{t_0}^{t_f} f^{0}(t, x(t), u(t))~\mathrm{d}t\]

where the state $x$ and the control $u$ are functions subject, for $t \in [t_0, t_f]$, to the differential constraint

\[ \dot{x}(t) = f(t, x(t), u(t))\]

and other constraints such as

\[\begin{array}{llcll} ~\xi_l &\le& \xi(t, u(t)) &\le& \xi_u, \\ \eta_l &\le& \eta(t, x(t)) &\le& \eta_u, \\ \psi_l &\le& \psi(t, x(t), u(t)) &\le& \psi_u, \\ \phi_l &\le& \phi(t_0, x(t_0), t_f, x(t_f)) &\le& \phi_u. \end{array}\]

Let us define the following optimal control problem.

using OptimalControl

ocp = Model()

state!(ocp, 2, "x", ["r", "v"]) # dimension of the state with the names of the components
control!(ocp, 1)           # dimension of the control
time!(ocp, t0=0, tf=1, name="s")    # initial and final time, with the name of the variable time

constraint!(ocp, :initial, lb=[-1, 0], ub=[-1, 0])
constraint!(ocp, :final  , lb=[ 0, 0], ub=[ 0, 0])
A = [ 0 1
      0 0 ]
B = [ 0
      1 ]
dynamics!(ocp, (x, u) -> A*x + B*u)

objective!(ocp, :lagrange, (x, u) -> 0.5u^2)

Then, we can print the form of this optimal control problem:

ocp

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

    minimize  J(x, u) = ∫ f⁰(x(s), u(s)) ds, over [0, 1]

    subject to

        ẋ(s) = f(x(s), u(s)), s in [0, 1] a.e.,

        ϕl ≤ ϕ(x(0), x(1)) ≤ ϕu, 

    where x(s) = (r(s), v(s)) ∈ R² and u(s) ∈ R.

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

You can also define the optimal control problem in an abstract form:

using OptimalControl

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

Then, you can print this optimal control problem:

ocp

The (autonomous) optimal control problem is given by:

    t ∈ [0, 1], time
    x ∈ R ^ 2, state
    u ∈ R, control
    x(0) == [-1, 0], 1
    x(1) == [0, 0]
    ẋ(t) == A * x(t) + B * u(t)
    ∫(0.5 * u(t) ^ 2) → min

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

    minimize  J(x, u) = ∫ f⁰(x(t), u(t)) dt, over [0, 1]

    subject to

        ẋ(t) = f(x(t), u(t)), t in [0, 1] a.e.,

        ϕl ≤ ϕ(x(0), x(1)) ≤ ϕu, 

    where x(t) ∈ R² and u(t) ∈ R.

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