Moon Landing Problem#
The Moon landing problem is a version of the soft landing problem presented in [Med64]. The problem is simplified to have one degree-of-freedom and normalized such that the Moon’s gravity is unity. The goal is to minimize the amount of fuel consumed or, stated differently, maximize the final mass, while bringing the lander down to the surface for a soft landing.
State and control variables#
This system has three state variables, the altitude (\(h\)), velocity (\(v\)), and mass (\(m\)) of the lander.
This system has one control variable, (\(T\)), the thrust applied to the vehicle.
The dynamics of the system are given by
(70)#\[\begin{align}
\dot{h} &= v \\
\dot{v} &= -1 + \frac{T}{m} \\
\dot{m} &= -\frac{T}{2.349}
\end{align}\]
Problem Definition#
We seek to maximize the final mass of the vehicle while bringing it to a soft landing.
(71)#\[\begin{align}
\mathrm{Minimize} \, J &= m_f
\end{align}\]
The initial conditions are
(72)#\[\begin{align}
h_0 &= 1 \\
v_0 &= -0.783 \\
m_0 &= 1
\end{align}\]
and the terminal constraints are
(73)#\[\begin{align}
h_f &= 0 \\
v_f &= 0
\end{align}\]
Additionally, the thrust is constrained to be positive but remain under 1.227.
(74)#\[\begin{align}
0 \le T \le 1.227
\end{align}\]
Defining the ODE#
The following implements the dynamics of the Moon landing problem described above.
import numpy as np
import openmdao.api as om
class MoonLandingProblemODE(om.ExplicitComponent):
def initialize(self):
self.options.declare('num_nodes', types=int)
def setup(self):
nn = self.options['num_nodes']
# inputs
self.add_input('h', val=np.ones(nn), units=None, desc='Altitude')
self.add_input('v', val=np.ones(nn), units='1/s', desc='Velocity')
self.add_input('m', val=np.ones(nn), units=None, desc='Mass')
self.add_input('T', val=np.ones(nn), units=None, desc='Thrust')
# outputs
self.add_output('h_dot', val=np.ones(nn), units='1/s', desc='Rate of change of Altitude')
self.add_output('v_dot', val=np.ones(nn), units='1/s**2', desc='Rate of change of Velocity')
self.add_output('m_dot', val=np.ones(nn), units='1/s', desc='Rate of change of Mass')
# partials
ar = np.arange(nn)
self.declare_partials(of='h_dot', wrt='v', rows=ar, cols=ar, val=1.0)
self.declare_partials(of='v_dot', wrt='m', rows=ar, cols=ar)
self.declare_partials(of='v_dot', wrt='T', rows=ar, cols=ar)
self.declare_partials(of='m_dot', wrt='T', rows=ar, cols=ar, val=-1/2.349)
self.declare_partials(of='m_dot', wrt='T', rows=ar, cols=ar, val=-1/2.349)
def compute(self, inputs, outputs, discrete_inputs=None, discrete_outputs=None):
v = inputs['v']
m = inputs['m']
T = inputs['T']
outputs['h_dot'] = v
outputs['v_dot'] = -1 + T/m
outputs['m_dot'] = -T/2.349
def compute_partials(self, inputs, partials, discrete_inputs=None):
m = inputs['m']
T = inputs['T']
partials['v_dot', 'T'] = 1/m
partials['v_dot', 'm'] = -T/m**2
Solving the Moon landing problem with Dymos#
The optimal solution to this problem is known to have bang-bang control. That is, the control has a “jump” that render it discontinuous in time. Capturing this behavior accurately requires the use of grid refinement for the Gauss-Lobatto and Radau pseudospectral transcriptions but the Birkhoff pseudospectral transcription can be used to handle this behavior without the use of any grid refinement. The following code shows the use of the Birkhoff pseudospectral transcription to solve the problem.
import dymos as dm
import matplotlib.pyplot as plt
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.declare_coloring()
p.driver.options['optimizer'] = 'IPOPT'
p.driver.opt_settings['hessian_approximation'] = 'limited-memory'
p.driver.opt_settings['print_level'] = 0
p.driver.opt_settings['linear_solver'] = 'mumps'
p.driver.declare_coloring()
t = dm.Birkhoff(num_nodes=20)
traj = p.model.add_subsystem('traj', dm.Trajectory())
phase = dm.Phase(ode_class=MoonLandingProblemODE, transcription=t)
phase.set_time_options(fix_initial=True, fix_duration=False)
phase.add_state('h', fix_initial=True, rate_source='h_dot')
phase.add_state('v', fix_initial=True, rate_source='v_dot')
phase.add_state('m', fix_initial=True, lower=1e-3, rate_source='m_dot')
phase.add_control('T', lower=0.0, upper=1.227)
phase.add_boundary_constraint('h', loc='final', equals=0.0)
phase.add_boundary_constraint('v', loc='final', equals=0.0)
phase.add_objective('m', scaler=-1)
phase.set_simulate_options(atol=1.0E-1, rtol=1.0E-2)
traj.add_phase('phase', phase)
p.setup(check=True, force_alloc_complex=True)
phase.set_time_val(initial=0.0, duration=1.0)
phase.set_state_val('h', [1.0, 0.0])
phase.set_state_val('v', [-0.783, 0.0])
phase.set_state_val('m', [1.0, 0.2])
phase.set_control_val('T', [0.0, 1.227])
dm.run_problem(p, simulate=False, simulate_kwargs={'times_per_seg': 100}, make_plots=True)
--- Constraint Report [traj] ---
--- phase ---
[final] 0.0000e+00 == h [None]
[final] 0.0000e+00 == v [1/s]
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
Model viewer data has already been recorded for Driver.
INFO: checking out_of_order
INFO: checking system
INFO: checking solvers
INFO: checking dup_inputs
INFO: checking missing_recorders
WARNING: The Problem has no recorder of any kind attached
INFO: checking unserializable_options
INFO: checking comp_has_no_outputs
INFO: checking auto_ivc_warnings
Full total jacobian for problem 'problem' was computed 3 times, taking 0.09401158600007875 seconds.
Total jacobian shape: (63, 124)
Jacobian shape: (63, 124) (2.60% nonzero)
FWD solves: 4 REV solves: 0
Total colors vs. total size: 4 vs 124 (96.77% improvement)
Sparsity computed using tolerance: 1e-25
Time to compute sparsity: 0.0940 sec
Time to compute coloring: 0.0433 sec
Memory to compute coloring: 0.2500 MB
Coloring created on: 2024-08-13 12:33:27
Optimization Problem -- Optimization using pyOpt_sparse
================================================================================
Objective Function: _objfunc
Solution:
--------------------------------------------------------------------------------
Total Time: 0.7355
User Objective Time : 0.0158
User Sensitivity Time : 0.6412
Interface Time : 0.0387
Opt Solver Time: 0.0399
Calls to Objective Function : 16
Calls to Sens Function : 16
Objectives
Index Name Value
0 traj.phase.boundary_vals.m -3.952837E-01
Variables (c - continuous, i - integer, d - discrete)
Index Name Type Lower Bound Value Upper Bound Status
0 traj.phase.t_duration_0 c -1.000000E+21 1.397227E+00 1.000000E+21
1 traj.phase.controls:T_0 c 0.000000E+00 1.900625E-07 1.227000E+00 l
2 traj.phase.controls:T_1 c 0.000000E+00 1.200754E-08 1.227000E+00 l
3 traj.phase.controls:T_2 c 0.000000E+00 2.042712E-09 1.227000E+00 l
4 traj.phase.controls:T_3 c 0.000000E+00 1.360111E-09 1.227000E+00 l
5 traj.phase.controls:T_4 c 0.000000E+00 4.448197E-09 1.227000E+00 l
6 traj.phase.controls:T_5 c 0.000000E+00 4.256705E-01 1.227000E+00
7 traj.phase.controls:T_6 c 0.000000E+00 1.227000E+00 1.227000E+00 u
8 traj.phase.controls:T_7 c 0.000000E+00 1.227000E+00 1.227000E+00 u
9 traj.phase.controls:T_8 c 0.000000E+00 1.227000E+00 1.227000E+00 u
10 traj.phase.controls:T_9 c 0.000000E+00 1.227000E+00 1.227000E+00 u
11 traj.phase.controls:T_10 c 0.000000E+00 1.227000E+00 1.227000E+00 u
12 traj.phase.controls:T_11 c 0.000000E+00 1.227000E+00 1.227000E+00 u
13 traj.phase.controls:T_12 c 0.000000E+00 1.227000E+00 1.227000E+00 u
14 traj.phase.controls:T_13 c 0.000000E+00 1.227000E+00 1.227000E+00 u
15 traj.phase.controls:T_14 c 0.000000E+00 1.227000E+00 1.227000E+00 u
16 traj.phase.controls:T_15 c 0.000000E+00 1.227000E+00 1.227000E+00 u
17 traj.phase.controls:T_16 c 0.000000E+00 1.227000E+00 1.227000E+00 u
18 traj.phase.controls:T_17 c 0.000000E+00 1.227000E+00 1.227000E+00 u
19 traj.phase.controls:T_18 c 0.000000E+00 1.227000E+00 1.227000E+00 u
20 traj.phase.controls:T_19 c 0.000000E+00 1.227000E+00 1.227000E+00 u
21 traj.phase.states:h_0 c -1.000000E+30 1.000000E+00 1.000000E+30
22 traj.phase.states:h_1 c -1.000000E+30 9.924938E-01 1.000000E+30
23 traj.phase.states:h_2 c -1.000000E+30 9.696467E-01 1.000000E+30
24 traj.phase.states:h_3 c -1.000000E+30 9.305227E-01 1.000000E+30
25 traj.phase.states:h_4 c -1.000000E+30 8.738058E-01 1.000000E+30
26 traj.phase.states:h_5 c -1.000000E+30 7.982641E-01 1.000000E+30
27 traj.phase.states:h_6 c -1.000000E+30 7.064505E-01 1.000000E+30
28 traj.phase.states:h_7 c -1.000000E+30 6.061256E-01 1.000000E+30
29 traj.phase.states:h_8 c -1.000000E+30 5.024707E-01 1.000000E+30
30 traj.phase.states:h_9 c -1.000000E+30 3.999428E-01 1.000000E+30
31 traj.phase.states:h_10 c -1.000000E+30 3.034069E-01 1.000000E+30
32 traj.phase.states:h_11 c -1.000000E+30 2.170119E-01 1.000000E+30
33 traj.phase.states:h_12 c -1.000000E+30 1.443228E-01 1.000000E+30
34 traj.phase.states:h_13 c -1.000000E+30 8.734397E-02 1.000000E+30
35 traj.phase.states:h_14 c -1.000000E+30 4.664210E-02 1.000000E+30
36 traj.phase.states:h_15 c -1.000000E+30 2.082882E-02 1.000000E+30
37 traj.phase.states:h_16 c -1.000000E+30 7.075257E-03 1.000000E+30
38 traj.phase.states:h_17 c -1.000000E+30 1.469604E-03 1.000000E+30
39 traj.phase.states:h_18 c -1.000000E+30 9.514736E-05 1.000000E+30
40 traj.phase.states:h_19 c -1.000000E+30 -3.543057E-17 1.000000E+30
41 traj.phase.state_rates:h_0 c -1.000000E+30 -5.470143E-01 1.000000E+30
42 traj.phase.state_rates:h_1 c -1.000000E+30 -5.536763E-01 1.000000E+30
43 traj.phase.state_rates:h_2 c -1.000000E+30 -5.734247E-01 1.000000E+30
44 traj.phase.state_rates:h_3 c -1.000000E+30 -6.058871E-01 1.000000E+30
45 traj.phase.state_rates:h_4 c -1.000000E+30 -6.500537E-01 1.000000E+30
46 traj.phase.state_rates:h_5 c -1.000000E+30 -6.965069E-01 1.000000E+30
47 traj.phase.state_rates:h_6 c -1.000000E+30 -7.033109E-01 1.000000E+30
48 traj.phase.state_rates:h_7 c -1.000000E+30 -6.772406E-01 1.000000E+30
49 traj.phase.state_rates:h_8 c -1.000000E+30 -6.489352E-01 1.000000E+30
50 traj.phase.state_rates:h_9 c -1.000000E+30 -6.080881E-01 1.000000E+30
51 traj.phase.state_rates:h_10 c -1.000000E+30 -5.594978E-01 1.000000E+30
52 traj.phase.state_rates:h_11 c -1.000000E+30 -4.988972E-01 1.000000E+30
53 traj.phase.state_rates:h_12 c -1.000000E+30 -4.298995E-01 1.000000E+30
54 traj.phase.state_rates:h_13 c -1.000000E+30 -3.521394E-01 1.000000E+30
55 traj.phase.state_rates:h_14 c -1.000000E+30 -2.705403E-01 1.000000E+30
56 traj.phase.state_rates:h_15 c -1.000000E+30 -1.889000E-01 1.000000E+30
57 traj.phase.state_rates:h_16 c -1.000000E+30 -1.143464E-01 1.000000E+30
58 traj.phase.state_rates:h_17 c -1.000000E+30 -5.363801E-02 1.000000E+30
59 traj.phase.state_rates:h_18 c -1.000000E+30 -1.387949E-02 1.000000E+30
60 traj.phase.state_rates:h_19 c -1.000000E+30 6.127492E-18 1.000000E+30
61 traj.phase.final_states:h_0 c -1.000000E+30 0.000000E+00 1.000000E+30
62 traj.phase.states:v_0 c -1.000000E+30 -7.830000E-01 1.000000E+30
63 traj.phase.states:v_1 c -1.000000E+30 -7.925359E-01 1.000000E+30
64 traj.phase.states:v_2 c -1.000000E+30 -8.208041E-01 1.000000E+30
65 traj.phase.states:v_3 c -1.000000E+30 -8.672708E-01 1.000000E+30
66 traj.phase.states:v_4 c -1.000000E+30 -9.304912E-01 1.000000E+30
67 traj.phase.states:v_5 c -1.000000E+30 -9.969847E-01 1.000000E+30
68 traj.phase.states:v_6 c -1.000000E+30 -1.006724E+00 1.000000E+30
69 traj.phase.states:v_7 c -1.000000E+30 -9.694068E-01 1.000000E+30
70 traj.phase.states:v_8 c -1.000000E+30 -9.288902E-01 1.000000E+30
71 traj.phase.states:v_9 c -1.000000E+30 -8.704215E-01 1.000000E+30
72 traj.phase.states:v_10 c -1.000000E+30 -8.008690E-01 1.000000E+30
73 traj.phase.states:v_11 c -1.000000E+30 -7.141249E-01 1.000000E+30
74 traj.phase.states:v_12 c -1.000000E+30 -6.153610E-01 1.000000E+30
75 traj.phase.states:v_13 c -1.000000E+30 -5.040547E-01 1.000000E+30
76 traj.phase.states:v_14 c -1.000000E+30 -3.872532E-01 1.000000E+30
77 traj.phase.states:v_15 c -1.000000E+30 -2.703927E-01 1.000000E+30
78 traj.phase.states:v_16 c -1.000000E+30 -1.636762E-01 1.000000E+30
79 traj.phase.states:v_17 c -1.000000E+30 -7.677782E-02 1.000000E+30
80 traj.phase.states:v_18 c -1.000000E+30 -1.986720E-02 1.000000E+30
81 traj.phase.states:v_19 c -1.000000E+30 8.770933E-18 1.000000E+30
82 traj.phase.state_rates:v_0 c -1.000000E+30 -6.986133E-01 1.000000E+30
83 traj.phase.state_rates:v_1 c -1.000000E+30 -6.986134E-01 1.000000E+30
84 traj.phase.state_rates:v_2 c -1.000000E+30 -6.986134E-01 1.000000E+30
85 traj.phase.state_rates:v_3 c -1.000000E+30 -6.986134E-01 1.000000E+30
86 traj.phase.state_rates:v_4 c -1.000000E+30 -6.986134E-01 1.000000E+30
87 traj.phase.state_rates:v_5 c -1.000000E+30 -3.997705E-01 1.000000E+30
88 traj.phase.state_rates:v_6 c -1.000000E+30 1.930710E-01 1.000000E+30
89 traj.phase.state_rates:v_7 c -1.000000E+30 2.474667E-01 1.000000E+30
90 traj.phase.state_rates:v_8 c -1.000000E+30 3.095485E-01 1.000000E+30
91 traj.phase.state_rates:v_9 c -1.000000E+30 3.864465E-01 1.000000E+30
92 traj.phase.state_rates:v_10 c -1.000000E+30 4.751158E-01 1.000000E+30
93 traj.phase.state_rates:v_11 c -1.000000E+30 5.799643E-01 1.000000E+30
94 traj.phase.state_rates:v_12 c -1.000000E+30 6.980260E-01 1.000000E+30
95 traj.phase.state_rates:v_13 c -1.000000E+30 8.307085E-01 1.000000E+30
96 traj.phase.state_rates:v_14 c -1.000000E+30 9.719486E-01 1.000000E+30
97 traj.phase.state_rates:v_15 c -1.000000E+30 1.116772E+00 1.000000E+30
98 traj.phase.state_rates:v_16 c -1.000000E+30 1.252712E+00 1.000000E+30
99 traj.phase.state_rates:v_17 c -1.000000E+30 1.366703E+00 1.000000E+30
100 traj.phase.state_rates:v_18 c -1.000000E+30 1.442978E+00 1.000000E+30
101 traj.phase.state_rates:v_19 c -1.000000E+30 1.469952E+00 1.000000E+30
102 traj.phase.final_states:v_0 c -1.000000E+30 -1.945852E-29 1.000000E+30
103 traj.phase.states:m_0 c 1.000000E-03 1.000000E+00 1.000000E+30
104 traj.phase.states:m_1 c 1.000000E-03 1.000003E+00 1.000000E+30
105 traj.phase.states:m_2 c 1.000000E-03 9.999805E-01 1.000000E+30
106 traj.phase.states:m_3 c 1.000000E-03 1.000026E+00 1.000000E+30
107 traj.phase.states:m_4 c 1.000000E-03 1.000087E+00 1.000000E+30
108 traj.phase.states:m_5 c 1.000000E-03 9.951016E-01 1.000000E+30
109 traj.phase.states:m_6 c 1.000000E-03 9.613252E-01 1.000000E+30
110 traj.phase.states:m_7 c 1.000000E-03 9.060529E-01 1.000000E+30
111 traj.phase.states:m_8 c 1.000000E-03 8.502589E-01 1.000000E+30
112 traj.phase.states:m_9 c 1.000000E-03 7.900012E-01 1.000000E+30
113 traj.phase.states:m_10 c 1.000000E-03 7.303206E-01 1.000000E+30
114 traj.phase.states:m_11 c 1.000000E-03 6.704314E-01 1.000000E+30
115 traj.phase.states:m_12 c 1.000000E-03 6.137581E-01 1.000000E+30
116 traj.phase.states:m_13 c 1.000000E-03 5.605090E-01 1.000000E+30
117 traj.phase.states:m_14 c 1.000000E-03 5.131199E-01 1.000000E+30
118 traj.phase.states:m_15 c 1.000000E-03 4.721855E-01 1.000000E+30
119 traj.phase.states:m_16 c 1.000000E-03 4.392904E-01 1.000000E+30
120 traj.phase.states:m_17 c 1.000000E-03 4.150448E-01 1.000000E+30
121 traj.phase.states:m_18 c 1.000000E-03 4.002624E-01 1.000000E+30
122 traj.phase.states:m_19 c 1.000000E-03 3.952837E-01 1.000000E+30
123 traj.phase.state_rates:m_0 c -1.000000E+30 -5.651950E-08 1.000000E+30
124 traj.phase.state_rates:m_1 c -1.000000E+30 -3.570203E-09 1.000000E+30
125 traj.phase.state_rates:m_2 c -1.000000E+30 -6.070162E-10 1.000000E+30
126 traj.phase.state_rates:m_3 c -1.000000E+30 -4.040443E-10 1.000000E+30
127 traj.phase.state_rates:m_4 c -1.000000E+30 -1.322227E-09 1.000000E+30
128 traj.phase.state_rates:m_5 c -1.000000E+30 -1.265982E-01 1.000000E+30
129 traj.phase.state_rates:m_6 c -1.000000E+30 -3.649207E-01 1.000000E+30
130 traj.phase.state_rates:m_7 c -1.000000E+30 -3.649207E-01 1.000000E+30
131 traj.phase.state_rates:m_8 c -1.000000E+30 -3.649207E-01 1.000000E+30
132 traj.phase.state_rates:m_9 c -1.000000E+30 -3.649207E-01 1.000000E+30
133 traj.phase.state_rates:m_10 c -1.000000E+30 -3.649207E-01 1.000000E+30
134 traj.phase.state_rates:m_11 c -1.000000E+30 -3.649207E-01 1.000000E+30
135 traj.phase.state_rates:m_12 c -1.000000E+30 -3.649207E-01 1.000000E+30
136 traj.phase.state_rates:m_13 c -1.000000E+30 -3.649207E-01 1.000000E+30
137 traj.phase.state_rates:m_14 c -1.000000E+30 -3.649207E-01 1.000000E+30
138 traj.phase.state_rates:m_15 c -1.000000E+30 -3.649207E-01 1.000000E+30
139 traj.phase.state_rates:m_16 c -1.000000E+30 -3.649207E-01 1.000000E+30
140 traj.phase.state_rates:m_17 c -1.000000E+30 -3.649207E-01 1.000000E+30
141 traj.phase.state_rates:m_18 c -1.000000E+30 -3.649207E-01 1.000000E+30
142 traj.phase.state_rates:m_19 c -1.000000E+30 -3.649207E-01 1.000000E+30
143 traj.phase.final_states:m_0 c -1.000000E+30 3.952837E-01 1.000000E+30
Constraints (i - inequality, e - equality)
Index Name Type Lower Value Upper Status Lagrange Multiplier (N/A)
0 traj.phases.phase->final_boundary_constraint->h e 0.000000E+00 0.000000E+00 0.000000E+00 9.00000E+100
1 traj.phases.phase->final_boundary_constraint->v e 0.000000E+00 -1.945852E-29 0.000000E+00 9.00000E+100
2 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
3 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
4 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
5 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
6 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
7 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
8 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
9 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
10 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
11 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
12 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
13 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
14 traj.phase.state_defects:h e 1.000000E+00 1.000000E+00 1.000000E+00 9.00000E+100
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124 traj.phase.state_rate_defects:m e 0.000000E+00 -1.067257E-12 0.000000E+00 9.00000E+100
Exit Status
Inform Description
0 Solve Succeeded
--------------------------------------------------------------------------------
Problem: problem
Driver: pyOptSparseDriver
success : True
iterations : 18
runtime : 9.1454E-01 s
model_evals : 18
model_time : 1.2056E-02 s
deriv_evals : 17
deriv_time : 3.8814E-01 s
exit_status : SUCCESS
from IPython.display import IFrame
IFrame(src=str(p.get_reports_dir() / 'traj_results_report.html'), width=800, height=1200)
Notes on the solution#
We can see that the collocation solution accurately captures the jump in the thrust. The oscillatory behavior observed is a result of interpolation performed post solution rather than a property of the solution itself.
References#
[Med64]
J. Meditch. On the problem of optimal thrust programming for a lunar soft landing. IEEE Transactions on Automatic Control, 9(4):477–484, 1964. doi:10.1109/TAC.1964.1105758.