julia> using OptimalControl

julia> using OptimalControl.ExaLinAlg

julia> using MadNLPMumps

julia> t0 = 0
0

julia> tf = 1
1

julia> x0 = [-1, 0]
x2-element Vector{Int64}:
 -1
  0

julia> xf = [0, 0]
2-element Vector{Int64}:
 0
 0

julia> A = [0 1; 0 0]
2×2 Matrix{Int64}:
 0  1
 0  0

julia> b = [0, 1]
2-element Vector{Int64}:
 0
 1

julia> o = @def begin
           t ∈ [t0, tf], time
           x ∈ R², state
           u ∈ R, control
           x(t0) == x0
           x(tf) == xf
           ẋ(t) == A * x(t) + u(t) * b
           0.5∫( u(t)^2 ) → min
       end
solAbstract definition:

    t ∈ [t0, tf], time
    x ∈ R², state
    u ∈ R, control
    x(t0) == x0
    x(tf) == xf
    ẋ(t) == A * x(t) + u(t) * b
    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.,

        ϕ₋ ≤ ϕ(x(0), x(1)) ≤ ϕ₊, 

    where x(t) ∈ R² and u(t) ∈ R.


julia> solve(o, :exa, :madnlp)
▫ This is OptimalControl version v1.1.6 running with: direct, exa, madnlp.

▫ The optimal control problem is solved with CTDirect version v0.17.4.

   ┌─ The NLP is modelled with ExaModels and solved with MadNLP suite.
   │
   ├─ Number of time steps⋅: 250
   └─ Discretisation scheme: midpoint

▫ This is MadNLP version v0.8.12, running with mumps

Number of nonzeros in constraint Jacobian............:     1754
Number of nonzeros in Lagrangian Hessian.............:     1000

Total number of variables............................:      753
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:      504
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 inf_compl lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  5.0000000e-03 1.10e+00 3.96e-16 0.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  6.0000967e+00 3.20e-16 6.08e-04 0.00e+00  -1.0 6.08e+00  -4.0 1.00e+00 1.00e+00h  1
   2  6.0000960e+00 3.36e-12 1.27e-07 0.00e+00  -5.7 3.81e-03  -4.5 1.00e+00 1.00e+00h  1
   3  6.0000960e+00 2.13e-16 3.55e-10 0.00e+00  -8.6 3.20e-05  -5.0 1.00e+00 1.00e+00h  1

Number of Iterations....: 3

                                   (scaled)                 (unscaled)
Objective...............:   6.0000960015360283e+00    6.0000960015360283e+00
Dual infeasibility......:   3.5549475516094553e-10    3.5549475516094553e-10
Constraint violation....:   2.1337098754514727e-16    2.1337098754514727e-16
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   3.5549475516094553e-10    3.5549475516094553e-10

Number of objective function evaluations             = 4
Number of objective gradient evaluations             = 4
Number of constraint evaluations                     = 4
Number of constraint Jacobian evaluations            = 4
Number of Lagrangian Hessian evaluations             = 3
Total wall-clock secs in solver (w/o fun. eval./lin. alg.)  = 24.643
Total wall-clock secs in linear solver                      =  0.026
Total wall-clock secs in NLP function evaluations           =  0.000
Total wall-clock secs                                       = 24.670

EXIT: Optimal Solution Found (tol = 1.0e-08).
• Solver:
  ✓ Successful  : true
  │  Status     : SOLVE_SUCCEEDED
  │  Message    : MadNLP
  │  Iterations : 3
  │  Objective  : 6.000096001536028
  └─ Constraints violation : 2.1337098754514727e-16

• Boundary duals: [0.0, 0.0, 0.0, 0.0]