harmonic oscillator/permanent.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0dad42c1",
   "metadata": {},
   "source": [
    "\n",
    "## Harmonic oscillator\n",
    "\n",
    "$$\n",
    "    \\left\\{\n",
    "    \\begin{array}{l}\n",
    "        \\displaystyle \\min  T, \\\\[0.5em]\n",
    "        \\dot{x}_1(t) = x_2(t), \\; t\\in [0,T]\\\\[0.5em]\n",
    "        \\dot{x}_2(t) = u(t)-x_1(t), t\\in [0,T]  \\\\[0.5em]\n",
    "        u(t) \\in [-1, 1], \\; t\\in [0,T]\\\\[0.5em]\n",
    "        x(0) = (2,2) , \\quad x(T) = 0_{\\mathrm{R}^2}, \n",
    "    \\end{array}\n",
    "    \\right.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c4f1fcc1",
   "metadata": {},
   "outputs": [],
   "source": [
    "using JuMP  \n",
    "using Ipopt\n",
    "using Plots\n",
    "using Plots.PlotMeasures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b8592f83",
   "metadata": {},
   "outputs": [],
   "source": [
    "mutable struct DirectSolution\n",
    "    t; x1; x2 ; p1; p2 ; u \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "6ffddef8",
   "metadata": {},
   "outputs": [],
   "source": [
    "function HO(x0; solution=[], nsteps=400, display=true)\n",
    "\n",
    "    # Create JuMP model, using Ipopt as the solver\n",
    "    if display\n",
    "        pl = 5\n",
    "    else\n",
    "        pl = 1\n",
    "    end\n",
    "    sys = Model(optimizer_with_attributes(Ipopt.Optimizer, \"print_level\" => pl))\n",
    "    set_optimizer_attribute(sys,\"tol\",1e-8)\n",
    "    set_optimizer_attribute(sys,\"constr_viol_tol\",1e-8)\n",
    "    set_optimizer_attribute(sys,\"max_iter\",1000)\n",
    "    \n",
    "    \n",
    "    N  = nsteps     # Grid size\n",
    "    \n",
    "    \n",
    "    @variables(sys, begin\n",
    "            x1[1:N+1]    \n",
    "            x2[1:N+1]   \n",
    "            -1.0 ≤ u[i=1:N+1] ≤ 1.0\n",
    "            0. ≤ Δt ≤ 1. \n",
    "        end)\n",
    "    \n",
    "    T = Δt*(N+1) ;\n",
    "\n",
    "    # Objective\n",
    "    @objective(sys, Min, T)\n",
    "\n",
    "    # Boundary constraints\n",
    "    @constraints(sys, begin\n",
    "        con_x10, x1[1]   == x0[1] \n",
    "        con_x20, x2[1]   == x0[2] \n",
    "        con_x1f, x1[N+1] == 0.0 \n",
    "        con_x2f, x2[N+1] == 0.0\n",
    "        end)\n",
    "\n",
    "    # Dynamics\n",
    "    @NLexpression(sys, dx1[j = 1:N+1], x2[j])\n",
    "    @NLexpression(sys, dx2[j = 1:N+1], u[j] - x1[j])\n",
    "\n",
    "    # Dynamics with Crank-Nicolson scheme\n",
    "    @NLconstraints(sys, begin\n",
    "        con_dx1[j=1:N], x1[j+1] == x1[j] + 0.5 * Δt * (dx1[j+1] + dx1[j])\n",
    "        con_dx2[j=1:N], x2[j+1] == x2[j] + 0.5 * Δt * (dx2[j+1] + dx2[j])\n",
    "        end)\n",
    "    \n",
    "    \n",
    "    # Solve for the control and state\n",
    "    if display\n",
    "        println(\"Solving...\")\n",
    "    end\n",
    "    status = optimize!(sys)\n",
    "    if display\n",
    "        println()\n",
    "    end\n",
    "\n",
    "    # Display results\n",
    "    if display\n",
    "        if termination_status(sys) == MOI.OPTIMAL\n",
    "            println(\"  Solution is optimal\")\n",
    "        elseif  termination_status(sys) == MOI.LOCALLY_SOLVED\n",
    "            println(\"  (Local) solution found\")\n",
    "        elseif termination_status(sys) == MOI.TIME_LIMIT && has_values(sys)\n",
    "            println(\"  Solution is suboptimal due to a time limit, but a primal solution is available\")\n",
    "        else\n",
    "            error(\"  The model was not solved correctly.\")\n",
    "        end\n",
    "        println(\"  objective value = \", objective_value(sys))\n",
    "        println()\n",
    "    end\n",
    "\n",
    "    # Retrieves values (including duals)\n",
    "    x1 = value.(x1)[:]\n",
    "    x2 = value.(x2)[:]\n",
    "    u  = value.(u)[:]\n",
    "    t  = (0:N) * value.(Δt)\n",
    "    \n",
    "    px10 =   dual(con_x10)\n",
    "    px20 =   dual(con_x20)\n",
    "    px1f = - dual(con_x1f)  # Modification : changement de signe\n",
    "    px2f = - dual(con_x2f)  # Modification : changement de signe\n",
    "\n",
    "    # keep original signs for now\n",
    "    sign = 1\n",
    "    \n",
    "     px1 = [ dual(con_dx1[i]) for i in 1:N ]\n",
    "     px2 = [ dual(con_dx2[i]) for i in 1:N ]\n",
    "\n",
    "     px1 = sign * [px10; (px1[1:N-1]+px1[2:N])/2.; px1f]; # We add the multiplier from the limit conditions\n",
    "     px2 = sign * [px20; (px2[1:N-1]+px2[2:N])/2.; px2f]; # We add the multiplier from the limit conditions\n",
    "    \n",
    "     p1 = - [px10; px1[1:N-1]; px1f]; # We add the multiplier from the limit conditions\n",
    "     p2 = - [px20; px2[1:N-1]; px2f]; # We add the multiplier from the limit conditions\n",
    "\n",
    "    return DirectSolution(t, x1, x2 , p1, p2, u)\n",
    "    \n",
    "    \n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "9831d7d6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving...\n",
      "This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.\n",
      "\n",
      "Number of nonzeros in equality constraint Jacobian...:     4804\n",
      "Number of nonzeros in inequality constraint Jacobian.:        0\n",
      "Number of nonzeros in Lagrangian Hessian.............:     8400\n",
      "\n",
      "Total number of variables............................:     1204\n",
      "                     variables with only lower bounds:        0\n",
      "                variables with lower and upper bounds:      402\n",
      "                     variables with only upper bounds:        0\n",
      "Total number of equality constraints.................:      804\n",
      "Total number of inequality constraints...............:        0\n",
      "        inequality constraints with only lower bounds:        0\n",
      "   inequality constraints with lower and upper bounds:        0\n",
      "        inequality constraints with only upper bounds:        0\n",
      "\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "   0  4.0099960e+00 2.50e+00 1.00e+02  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0\n",
      "   1  4.0095990e-02 2.47e+00 5.32e+01  -1.0 2.66e+00    -  3.87e-01 1.11e-02f  1\n",
      "   2  5.2014125e+00 2.44e+00 1.13e+04  -1.0 2.63e+00    -  1.11e-04 1.44e-02h  6\n",
      "   3  6.0432946e+00 2.41e+00 1.09e+04  -1.0 3.59e+00    -  2.68e-01 1.10e-02h  6\n",
      "   4  7.0201860e+00 2.37e+00 1.04e+04  -1.0 4.55e+00    -  2.07e-01 1.50e-02h  5\n",
      "   5  8.3893721e+00 2.31e+00 9.69e+03  -1.0 4.87e+00    -  2.01e-01 2.50e-02h  4\n",
      "   6  9.1960173e+00 2.27e+00 9.40e+03  -1.0 4.50e+00    -  3.07e-01 1.82e-02h  5\n",
      "   7  9.9605913e+00 2.23e+00 9.12e+03  -1.0 4.55e+00    -  2.91e-01 1.94e-02h  5\n",
      "   8  1.1372136e+01 2.14e+00 8.49e+03  -1.0 3.73e+00    -  3.37e-01 4.01e-02h  4\n",
      "   9  1.2791943e+01 2.03e+00 7.89e+03  -1.0 4.02e+00    -  4.01e-01 4.87e-02h  4\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  10  1.4004599e+01 1.94e+00 7.39e+03  -1.0 4.38e+00    -  3.73e-01 4.89e-02h  4\n",
      "  11  1.5124686e+01 1.84e+00 6.96e+03  -1.0 6.37e+00    -  2.58e-01 4.93e-02h  3\n",
      "  12  2.1984268e+01 1.20e+00 1.48e+03  -1.0 5.53e+00    -  3.46e-01 3.48e-01w  1\n",
      "  13  3.1792158e+01 3.28e-01 7.62e+02  -1.0 1.12e+01    -  2.83e-01 7.27e-01w  1\n",
      "  14  2.7610343e+01 7.56e-02 4.57e+02  -1.0 4.52e+00    -  1.00e+00 7.83e-01w  1\n",
      "  15  1.5982134e+01 1.76e+00 6.61e+03  -1.0 5.97e+00    -  3.46e-01 4.35e-02h  3\n",
      "  16  1.7144908e+01 1.64e+00 6.09e+03  -1.0 4.88e+00    -  4.90e-01 6.72e-02h  3\n",
      "  17  1.7978763e+01 1.55e+00 5.71e+03  -1.0 4.87e+00    -  4.01e-01 5.64e-02h  3\n",
      "  18  1.9322797e+01 1.39e+00 5.04e+03  -1.0 3.99e+00    -  5.06e-01 1.03e-01h  3\n",
      "  19  2.0623844e+01 1.21e+00 4.33e+03  -1.0 3.42e+00    -  4.88e-01 1.26e-01h  3\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  20  2.5656358e+01 4.55e-01 1.94e+03  -1.0 5.01e+00    -  4.45e-01 6.25e-01H  1\n",
      "  21  2.4303093e+01 5.26e-03 4.74e+01  -1.0 1.66e+00   0.0 1.00e+00 1.00e+00f  1\n",
      "  22  2.3763296e+01 1.12e-03 1.00e+02  -1.0 9.79e-01    -  1.00e+00 1.00e+00f  1\n",
      "  23  2.2581380e+01 3.55e-03 5.20e+01  -1.0 1.29e+00    -  1.00e+00 1.00e+00f  1\n",
      "  24  2.2611162e+01 1.06e-05 4.58e+00  -1.0 1.92e-01  -0.5 1.00e+00 1.00e+00h  1\n",
      "  25  2.2881535e+01 5.69e-05 3.34e+00  -1.0 1.25e-01  -1.0 1.00e+00 1.00e+00f  1\n",
      "  26  2.2988311e+01 1.88e-05 5.10e-01  -1.0 8.70e-02  -1.4 1.00e+00 1.00e+00h  1\n",
      "  27  2.2464094e+01 2.16e-04 3.19e+01  -1.7 1.63e-01  -1.9 1.00e+00 1.00e+00f  1\n",
      "  28  2.0774476e+01 2.28e-03 4.56e+00  -1.7 5.44e-01  -2.4 1.00e+00 1.00e+00f  1\n",
      "  29  1.8135672e+01 1.16e-02 3.05e+00  -1.7 3.36e+00  -2.9 7.96e-01 4.75e-01f  2\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  30  1.8337195e+01 2.97e-04 2.08e+01  -1.7 4.98e-01  -2.4 1.00e+00 1.00e+00h  1\n",
      "  31  1.7388649e+01 1.56e-03 1.54e+01  -1.7 2.12e+01    -  1.03e-01 3.57e-02f  3\n",
      "  32  1.5791678e+01 5.35e-03 2.04e+00  -1.7 5.24e+00  -2.9 6.59e-01 2.01e-01f  2\n",
      "  33  1.4669849e+01 2.16e-03 6.23e+01  -1.7 7.57e-01  -2.5 1.00e+00 1.00e+00h  1\n",
      "  34  1.3775311e+01 3.09e-03 3.54e+01  -1.7 2.55e+00    -  5.87e-01 3.39e-01h  2\n",
      "  35  1.3893664e+01 7.37e-05 2.08e+00  -1.7 2.37e-01  -2.1 1.00e+00 1.00e+00h  1\n",
      "  36  1.3610248e+01 1.27e-04 9.06e-01  -1.7 6.20e+00    -  2.58e-01 3.53e-02f  4\n",
      "  37  1.3082047e+01 5.32e-04 9.68e+00  -1.7 4.21e-01  -2.5 1.00e+00 1.00e+00h  1\n",
      "  38  1.2044944e+01 7.68e-03 2.02e+02  -1.7 3.56e+00  -3.0 1.00e+00 1.00e+00F  1\n",
      "  39  1.2028718e+01 1.86e-05 9.86e+00  -1.7 4.83e-01    -  1.00e+00 1.00e+00h  1\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  40  1.1776411e+01 1.91e-04 5.03e+00  -1.7 2.21e-01    -  1.00e+00 1.00e+00h  1\n",
      "  41  1.1765951e+01 9.36e-07 1.28e-01  -1.7 3.40e-02    -  1.00e+00 1.00e+00h  1\n",
      "  42  1.0709049e+01 1.70e-03 8.59e+01  -2.5 7.01e-01    -  7.15e-01 1.00e+00f  1\n",
      "  43  9.6289892e+00 1.52e-03 5.10e+01  -2.5 6.38e-01  -2.6 8.01e-01 1.00e+00h  1\n",
      "  44  8.4327545e+00 4.04e-03 5.14e+01  -2.5 6.71e+00  -3.1 1.24e-01 1.76e-01f  1\n",
      "  45  8.0132373e+00 1.53e-03 3.14e+01  -2.5 1.42e+00    -  5.72e-01 6.91e-01f  1\n",
      "  46  7.7331703e+00 1.48e-03 2.57e+01  -2.5 4.30e+00  -3.5 2.93e-01 1.12e-01h  2\n",
      "  47  7.1933616e+00 1.39e-03 2.65e+00  -2.5 1.54e+00  -3.1 1.00e+00 6.48e-01h  1\n",
      "  48  6.9797683e+00 2.90e-04 3.25e-01  -2.5 5.57e-01  -2.7 1.00e+00 1.00e+00f  1\n",
      "  49  6.8516991e+00 5.70e-05 5.00e-02  -2.5 1.89e-01  -2.3 1.00e+00 1.00e+00h  1\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  50  6.2363544e+00 2.24e-04 1.41e+01  -2.5 7.14e-01  -2.7 1.00e+00 1.00e+00H  1\n",
      "  51  6.1209184e+00 3.10e-04 1.33e+01  -2.5 4.39e+00    -  6.76e-01 1.13e-01h  2\n",
      "  52  5.9842515e+00 2.51e-04 1.52e+00  -2.5 7.88e-01    -  1.00e+00 1.00e+00h  1\n",
      "  53  6.0784033e+00 8.42e-05 6.15e-01  -2.5 3.71e-01    -  1.00e+00 1.00e+00h  1\n",
      "  54  6.1105237e+00 1.32e-05 3.32e-02  -2.5 1.66e-01    -  1.00e+00 1.00e+00h  1\n",
      "  55  6.1100890e+00 5.85e-09 1.04e-05  -2.5 5.71e-03    -  1.00e+00 1.00e+00h  1\n",
      "  56  5.7104834e+00 5.29e-04 2.18e+01  -3.8 5.67e-01    -  6.45e-01 1.00e+00f  1\n",
      "  57  5.3638890e+00 8.24e-04 2.26e+00  -3.8 1.15e+00    -  5.47e-01 8.50e-01h  1\n",
      "  58  5.1966510e+00 5.17e-04 7.99e-01  -3.8 1.61e+00    -  4.85e-01 5.72e-01h  1\n",
      "  59  5.0937798e+00 3.36e-04 2.37e-01  -3.8 1.81e+00    -  5.14e-01 5.14e-01h  1\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  60  5.0267370e+00 2.16e-04 1.11e-02  -3.8 1.70e+00    -  6.87e-01 5.61e-01f  1\n",
      "  61  4.9827762e+00 1.25e-04 7.09e-02  -3.8 1.29e+00    -  1.00e+00 7.74e-01f  1\n",
      "  62  4.9733791e+00 8.61e-06 5.07e-04  -3.8 3.87e-01    -  1.00e+00 1.00e+00f  1\n",
      "  63  4.8818312e+00 1.73e-04 1.70e-01  -5.7 1.57e+00    -  3.93e-01 4.88e-01f  1\n",
      "  64  4.8467885e+00 1.46e-04 1.34e-01  -5.7 2.79e+00    -  2.62e-01 3.13e-01h  1\n",
      "  65  4.8234776e+00 1.03e-04 1.04e-01  -5.7 2.74e+00    -  2.58e-01 2.93e-01h  1\n",
      "  66  4.8065006e+00 7.28e-05 7.81e-02  -5.7 2.81e+00    -  2.66e-01 2.92e-01h  1\n",
      "  67  4.7940404e+00 5.12e-05 5.75e-02  -5.7 2.60e+00    -  2.48e-01 2.95e-01h  1\n",
      "  68  4.7846990e+00 3.59e-05 4.14e-02  -5.7 2.24e+00    -  2.64e-01 3.10e-01h  1\n",
      "  69  4.7785155e+00 2.64e-05 2.99e-02  -5.7 1.85e+00    -  3.52e-01 2.90e-01h  1\n",
      "iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls\n",
      "  70  4.7735599e+00 1.93e-05 1.82e-02  -5.7 3.44e+00    -  7.63e-01 3.13e-01h  1\n",
      "  71  4.7647585e+00 1.89e-05 4.09e-03  -5.7 1.91e+00    -  2.22e-01 8.39e-01h  1\n",
      "  72  4.7630138e+00 1.78e-06 6.53e-03  -5.7 8.00e-01    -  5.77e-01 1.00e+00f  1\n",
      "  73  4.7629260e+00 7.93e-08 2.42e-03  -5.7 4.96e-01    -  1.00e+00 9.34e-01h  1\n",
      "  74  4.7629285e+00 3.31e-10 3.56e-05  -5.7 1.06e-01    -  1.00e+00 1.00e+00f  1\n",
      "  75  4.7629288e+00 1.09e-12 1.09e-08  -5.7 2.94e-03    -  1.00e+00 1.00e+00h  1\n",
      "  76  4.7605721e+00 7.93e-07 1.75e-04  -8.6 2.94e-01    -  8.22e-01 8.02e-01f  1\n",
      "  77  4.7599909e+00 1.13e-07 6.57e-05  -8.6 1.32e-01    -  9.84e-01 9.84e-01h  1\n",
      "  78  4.7599813e+00 3.56e-11 4.27e-07  -8.6 2.52e-03    -  1.00e+00 1.00e+00f  1\n",
      "  79  4.7599813e+00 5.55e-16 1.61e-10  -8.6 7.88e-05    -  1.00e+00 1.00e+00h  1\n",
      "\n",
      "Number of Iterations....: 79\n",
      "\n",
      "                                   (scaled)                 (unscaled)\n",
      "Objective...............:   1.1870277447107702e+00    4.7599812562901880e+00\n",
      "Dual infeasibility......:   1.6056276234042979e-10    6.4385667698512342e-10\n",
      "Constraint violation....:   5.5511151231257827e-16    5.5511151231257827e-16\n",
      "Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00\n",
      "Complementarity.........:   2.5061976990388848e-09    1.0049852773145927e-08\n",
      "Overall NLP error.......:   2.5061976990388848e-09    1.0049852773145927e-08\n",
      "\n",
      "\n",
      "Number of objective function evaluations             = 179\n",
      "Number of objective gradient evaluations             = 80\n",
      "Number of equality constraint evaluations            = 179\n",
      "Number of inequality constraint evaluations          = 0\n",
      "Number of equality constraint Jacobian evaluations   = 80\n",
      "Number of inequality constraint Jacobian evaluations = 0\n",
      "Number of Lagrangian Hessian evaluations             = 79\n",
      "Total seconds in IPOPT                               = 0.563\n",
      "\n",
      "EXIT: Optimal Solution Found.\n",
      "\n",
      "  (Local) solution found\n",
      "  objective value = 4.759981256290188\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Resolution\n",
    "x0  = [2.5; 1.0]\n",
    "\n",
    "sol = HO(x0);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "b5dcbb28",
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plots\n",
    "t  = sol.t\n",
    "x1 = sol.x1\n",
    "x2 = sol.x2\n",
    "u  = sol.u\n",
    "p1 = sol.p1\n",
    "p2 = sol.p2\n",
    "\n",
    "\n",
    "x1_plot   = plot(t, x1,  xlabel = \"t\", ylabel = \"x1\",  legend = false)\n",
    "p1_plot   = plot(t, p1,  xlabel = \"t\", ylabel = \"p1\",  legend = false)\n",
    "x2_plot   = plot(t, x2,  xlabel = \"t\", ylabel = \"x2\",  legend = false)\n",
    "p2_plot   = plot(t, p2,  xlabel = \"t\", ylabel = \"p2\",  legend = false)\n",
    "\n",
    "u_plot    = plot(t, u,   xlabel = \"t\", ylabel = \"u\",   legend = false)\n",
    "x1x2_plot = plot(x1, x2, xlabel = \"x1\", ylabel = \"x2\",  legend = false)\n",
    "\n",
    "plot( x1x2_plot, u_plot, layout = (1,2), size=(800,500), left_margin=5mm)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "49851969",
   "metadata": {},
   "outputs": [
    {
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       "<polyline clip-path=\"url(#clip402)\" style=\"stroke:#ff0000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" stroke-dasharray=\"16, 10\" points=\"-1838.35,554.64 4448.31,554.64 \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create the first plot\n",
    "plot(x1,x2 , xlabel = \"x1\", ylabel = \"x2\", legend = false)\n",
    "#y = [-5, 5]\n",
    "# add the second plot to the first plot\n",
    "#plot!(ones(2)*3, y, line=:stem, xlabel=\"x\", ylabel=\"y\", title=\"interface\", xlim=(-10,10))\n",
    "hline!([0], line=:dash, color=:red)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4d4278d3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute the differences between adjacent elements\n",
    "d = diff(u)\n",
    "\n",
    "# Find the indices where the differences are non-zero\n",
    "switchings = findall(abs.(d) .> 0.7)\n",
    "\n",
    "# Print the result\n",
    "switching_times = t[switchings] ; \n",
    "\n",
    "\n",
    "T  = t[end];\n",
    "\n",
    "t1  = switching_times[1]\n",
    "t2  = switching_times[2] ; "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ed0b3da",
   "metadata": {},
   "source": [
    "# Indirect method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3619f676",
   "metadata": {},
   "outputs": [],
   "source": [
    "using NLsolve\n",
    "using Plots\n",
    "using Plots.PlotMeasures\n",
    "include(\"flow.jl\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "76198fed",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Dynamics\n",
    "function F0(x)\n",
    "    return [ x[2], -x[1] ]\n",
    "end\n",
    "\n",
    "function F1(x)\n",
    "    return [ 0.0, 1.0 ]\n",
    "end\n",
    "\n",
    "# Hamiltonians: permanent region\n",
    "H0(x, p) = p' * F0(x)\n",
    "H1(x, p) = p' * F1(x)\n",
    "\n",
    "H(x, p, u) = H0(x, p) + u*H1(x,p) # pseudo-Hamiltonian\n",
    "\n",
    "up(x, p) =  1.0\n",
    "um(x, p) = -1.0\n",
    "\n",
    "Hp(x, p) = H(x, p, up(x, p))\n",
    "Hm(x, p) = H(x, p, um(x, p))\n",
    "\n",
    "# Flows\n",
    "fp  = Flow(Hp)\n",
    "fm  = Flow(Hm);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c38a0e63",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Shooting function\n",
    "function SHOOT(p0, tt1, tt2, TT) \n",
    "        \n",
    "    x1, p1 = fm(t0, x0 , p0, tt1)\n",
    "    x2, p2 = fp(tt1, x1 , p1, tt2)\n",
    "    x3, p3 = fm(tt2, x2 , p2, TT)\n",
    "\n",
    "    s = zeros(eltype(p0), 5)\n",
    "    \n",
    "    s[1:2] = x3 - [ 0.0 , 0.0 ]   # target\n",
    "    \n",
    "    s[3] = H1(x1, p1)             # switching\n",
    "    s[4] = H1(x2, p2)             # switching\n",
    "    s[5] = Hp(x0, p0) - 1.0       # free final time\n",
    "    \n",
    "\n",
    "    return s\n",
    "\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6bd14cf2",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solve\n",
    "t0 = 0.0\n",
    "x0 = [2.0; 2.0]\n",
    "S(ξ) = SHOOT(ξ[1:2], ξ[3], ξ[4], ξ[5]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c5eabe04",
   "metadata": {},
   "outputs": [],
   "source": [
    "ξ_guess = [p1[1] , p2[1], t1, t2, T]; # initial guess"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "41bdda45",
   "metadata": {},
   "outputs": [],
   "source": [
    "S(ξ_guess);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "d3d80ceb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iter     f(x) inf-norm    Step 2-norm \n",
      "------   --------------   --------------\n",
      "     0     1.714934e+00              NaN\n",
      "     1     2.073014e-02     2.755113e+00\n",
      "     2     2.345183e-04     4.622837e-02\n",
      "     3     3.358192e-08     2.176971e-04\n",
      "     4     1.704527e-15     6.097729e-08\n",
      "Results of Nonlinear Solver Algorithm\n",
      " * Algorithm: Trust-region with dogleg and autoscaling\n",
      " * Starting Point: [-0.785230838611897, -0.8555280965841662, 0.8282264006597898, 3.9654476152802056, 5.0195539433926655]\n",
      " * Zero: [1.1045786867955638, 1.2091573735911278, 0.8305664776430566, 3.972159131232845, 5.019356682429274]\n",
      " * Inf-norm of residuals: 0.000000\n",
      " * Iterations: 4\n",
      " * Convergence: true\n",
      "   * |x - x'| < 1.0e-08: false\n",
      "   * |f(x)| < 1.0e-08: true\n",
      " * Function Calls (f): 5\n",
      " * Jacobian Calls (df/dx): 5\n"
     ]
    }
   ],
   "source": [
    "# Solve\n",
    "\n",
    "indirect_sol = nlsolve(S, ξ_guess; xtol=1e-8, method=:trust_region, show_trace=true)\n",
    "println(indirect_sol)\n",
    "\n",
    "# Retrieves solution\n",
    "if indirect_sol.f_converged || indirect_sol.x_converged\n",
    "    pp0 = indirect_sol.zero[1:2]\n",
    "    tt1 = indirect_sol.zero[3]\n",
    "    tt2 = indirect_sol.zero[4]\n",
    "    TT  = indirect_sol.zero[5]\n",
    "\n",
    "else\n",
    "    error(\"Not converged\")\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e6fb7cda",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00234007698326677"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tt1 - t1 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "3949d9f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.006711515952639413"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tt2 - t2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "deb071cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.00019726096339134358"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TT - T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "8e12e16e",
   "metadata": {},
   "outputs": [],
   "source": [
    "ode_sol = fm((t0, tt1), x0, [p1[1] ; p2[1]], saveat=0.1) ;\n",
    "ttt1 = ode_sol.t ;\n",
    "xx1 = [ ode_sol[1:2, j] for j in 1:size(ttt1, 1) ] ;\n",
    "pp1 = [ ode_sol[3:4, j] for j in 1:size(ttt1, 1) ] ;\n",
    "uu1 = um.(xx1, pp1)  ;\n",
    "\n",
    "ode_sol = fp((tt1, tt2), xx1[end], pp1[end], saveat=0.1) ;\n",
    "ttt2 = ode_sol.t ;\n",
    "xx2 = [ ode_sol[1:2, j] for j in 1:size(ttt2, 1) ] ;\n",
    "pp2 = [ ode_sol[3:4, j] for j in 1:size(ttt2, 1) ] ;\n",
    "uu2 = up.(xx2, pp2)  ;\n",
    "\n",
    "ode_sol = fm((tt2, TT), xx2[end], pp2[end], saveat=0.1) ;\n",
    "ttt3 = ode_sol.t ;\n",
    "xx3 = [ ode_sol[1:2, j] for j in 1:size(ttt3, 1) ] ;\n",
    "pp3 = [ ode_sol[3:4, j] for j in 1:size(ttt3, 1) ] ;\n",
    "uu3 = um.(xx3, pp3)  ;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "07296967",
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = [ ttt1 ; ttt2 ; ttt3 ]\n",
    "x = [ xx1 ; xx2 ; xx3 ]\n",
    "p = [ pp1 ; pp2 ; pp3 ]\n",
    "u = [ uu1 ; uu2 ; uu3 ]\n",
    "\n",
    "m = length(t)\n",
    "\n",
    "x1 = [ x[i][1] for i=1:m ]\n",
    "x2 = [ x[i][2] for i=1:m ]\n",
    "p1 = [ p[i][1] for i=1:m ]\n",
    "p2 = [ p[i][2] for i=1:m ];\n",
    "\n",
    "x1_plot   = plot(t,  x1, xlabel = \"t\", ylabel = \"x1\",  legend = false)\n",
    "x2_plot   = plot(t,  x2, xlabel = \"t\", ylabel = \"x2\",  legend = false)\n",
    "p1_plot   = plot(t,  p1, xlabel = \"t\", ylabel = \"p1\", legend = false)\n",
    "p2_plot   = plot(t,  p2, xlabel = \"t\", ylabel = \"p2\", legend = false)\n",
    "u_plot    = plot(t,   u, xlabel = \"t\", ylabel = \"u\",   legend = false, size=(800,400)) #, linetype=:steppre)\n",
    "x1x2_plot = plot(x1, x2, xlabel = \"x1\", ylabel = \"x2\",  legend = false)\n",
    "\n",
    "plot(x1_plot, x2_plot, x1x2_plot, u_plot, p1_plot ,p2_plot , layout = (3,3), size=(800,800), left_margin=10mm)"
   ]
  }
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