Double integrator: time minimisation

The problem consists in minimising the final time $t_f$ for the double integrator system

\[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = u(t), \quad u(t) \in [-1,1],\]

and the limit conditions

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

This problem can be interpretated as a simple model for a wagon with constant mass moving along a line without fricton.

First, we need to import the OptimalControl.jl package to define the optimal control problem and NLPModelsIpopt.jl to solve it. We also need to import the Plots.jl package to plot the solution.

using OptimalControl
using NLPModelsIpopt
using Plots

Optimal control problem

Let us define the problem

ocp = @def begin

    tf ∈ R,          variable
    t ∈ [0, tf],     time
    x = (q, v) ∈ R², state
    u ∈ R,           control

    -1 ≤ u(t) ≤ 1

    q(0)  == -1
    v(0)  == 0
    q(tf) == 0
    v(tf) == 0

    ẋ(t) == [v(t), u(t)]

    tf → min

end

Solve and plot

Solve it

sol = solve(ocp; grid_size=200)

▫ This is OptimalControl version v1.1.1 running with: direct, adnlp, ipopt.

▫ The optimal control problem is solved with CTDirect version v0.16.2.

   ┌─ The NLP is modelled with ADNLPModels and solved with NLPModelsIpopt.
   │
   ├─ Number of time steps⋅: 200
   └─ Discretisation scheme: trapeze

▫ This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.

Number of nonzeros in equality constraint Jacobian...:     2004
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:      402

Total number of variables............................:      604
                     variables with only lower bounds:        0
                variables with lower and upper bounds:      201
                     variables with only upper bounds:        0
Total number of equality constraints.................:      404
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 lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.0000000e-01 1.10e+00 2.51e-04   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  2.2780101e-01 1.09e+00 6.69e+01  -6.0 1.06e+01    -  7.72e-02 1.28e-02h  4
   2  7.5366951e-01 1.03e+00 1.70e+02   0.4 9.79e+00    -  2.37e-01 5.37e-02h  3
   3  1.4473632e+00 8.70e-01 9.88e+01   0.2 5.71e+00    -  1.00e+00 1.53e-01H  1
   4  3.3885017e+00 4.33e-03 1.26e+02  -0.6 1.94e+00    -  1.00e+00 1.00e+00h  1
   5  2.2206258e+00 1.28e-03 7.75e+00  -1.5 1.17e+00   0.0 1.00e+00 1.00e+00h  1
   6  2.3300741e+00 2.17e-05 1.11e+00  -3.2 1.09e-01    -  1.00e+00 1.00e+00h  1
   7  2.0599681e+00 9.30e-04 2.59e-01  -3.6 6.91e-01    -  1.00e+00 1.00e+00f  1
   8  2.0440329e+00 2.77e-05 1.22e-02  -4.5 3.49e-01    -  1.00e+00 1.00e+00h  1
   9  2.0027584e+00 1.30e-04 8.64e-04  -5.2 8.17e-01    -  1.00e+00 9.83e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  2.0006050e+00 1.15e-06 4.22e-06  -5.6 2.33e-01    -  9.80e-01 1.00e+00h  1
  11  2.0000587e+00 9.27e-08 2.16e-07  -7.6 4.22e-02    -  9.87e-01 9.93e-01h  1
  12  2.0000500e+00 5.17e-11 6.45e-10 -11.0 2.26e-03    -  9.98e-01 1.00e+00h  1

Number of Iterations....: 12

                                   (scaled)                 (unscaled)
Objective...............:   2.0000499958370912e+00    2.0000499958370912e+00
Dual infeasibility......:   6.4540800308321433e-10    6.4540800308321433e-10
Constraint violation....:   5.1717741200718592e-11    5.1717741200718592e-11
Variable bound violation:   9.9715664614308253e-09    9.9715664614308253e-09
Complementarity.........:   2.5370603077805507e-10    2.5370603077805507e-10
Overall NLP error.......:   6.4540800308321433e-10    6.4540800308321433e-10


Number of objective function evaluations             = 21
Number of objective gradient evaluations             = 13
Number of equality constraint evaluations            = 21
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 13
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 12
Total seconds in IPOPT                               = 1.836

EXIT: Optimal Solution Found.

and plot the solution

plot(sol)