We consider the problem of landing a spacecraft on the Moon, assuming that aerodynamic and gravitational forces of bodies other than the Moon are negligible, and lateral motion can be ignored. Accordingly, the descent trajectory is vertical, and the thrust vector is tangent to the trajectory.

Because the spacecraft is near the Moon, we assume that the lunar acceleration of gravity has the constant value , that the relative velocity of the exhaust gases with respect to the spacecraft is constant, and that the mass rate is constrained by , where is constant and gives the maximum rate of change of the mass due to burning the fuel.

### Mathematical Approach to a Soft Landing

The problem is to make a soft landing on the surface of the Moon with the minimum amount of fuel.

Here is a sketch of the system immediately preceding the landing.

Following [1, pp. 247-248] and [2], we introduce the following notation and assumptions:

- is time
- is the mass of the spacecraft, which varies as fuel is burned
- is the rate of change of mass, constrained by
- , the gravitational constant near the Moon
- is a constant, the relative velocity of the exhaust gases with respect to the spacecraft
- , the thrust
- is the the height, with
- , the velocity
- , the control function

Recalling our assumptions, aerodynamic forces and gravitational forces of bodies other than the Moon are negligible and lateral motion is ignored. Thus the descent trajectory is vertical and the thrust vector is perpendicular to the ground.

We also suppose that , where is the mass of the spacecraft without fuel and is the initial mass of fuel; , since as we expect that the spacecraft will return to Earth, it needs some fuel for takeoff.

### Equations of Motion

By Newton’s second law ([3, p. 128] and [2]),

(1) |

which can be written as a system of equations

(2) |

(3) |

(4) |

where is a constant. The third equation states that the loss of mass per second (the fuel burned by the jet per second) is proportional to the thrust of the jet.

### The Optimal Control Problem

Our goal is to minimize the fuel consumption, so the cost functional is

(5) |

where is the first time for which

Thus the horizon is , where remains to be determined.

In vector form, if , then

and the problem can be written

(6) |

and finally,

(7) |

From (3) we have that

and by integration over the interval , we get that

It follows that if and only if

Solving for ,

Now we substitute this into (5) to get

This result was published in [2].

**Theorem 1**

**Corollary 1**

From here it follows that instead of (7) we can consider the following cost functional

(8) |

and thus equation (6) with (7) becomes (6) with (8). This is a Mayer optimal control problem (see Chapter 4 of [4]).

### Necessary Conditions for the Mayer Problem

To avoid a lengthy discussion, we state a short version of theorem 4.2.i in [4]. Let the Mayer problem be expressed as

(9) |

(10) |

A pair , , is said to be *admissible* or *feasible* provided is absolutely continuous [5], is measurable, and and satisfy (10). Let be the class of admissible pairs . The goal is to find the minimum of the cost functional over , that is, to find an element such that for all . We introduce the variables , called *multipliers*, and an auxiliary function , called the *Hamiltonian*, defined on by

We define

More assumptions are necessary:

- There exists an element such that for all .
- is closed in .
- The set , is closed in .
- .
- Notation:
- The graph of the optimal trajectory belongs to the interior of .
- does not depend on time and is a closed set.
- The end point of the optimal trajectory is a point of , where possesses a tangent variety (of some dimension , ), whose vectors are denoted by

or by

**Theorem 2**

The Hamiltonian and the equations for the multipliers to (6) and (7) are

so that , , , where and are constants.

For , the minimum of is attained with , and then

This corresponds to *free fall* for the spacecraft.

For , the minimum of is attained with , and then

Thus we find that the control function takes only extreme values and .

If on an interval we have that , and hence

then for , we have

In this case, describes an arc of a parabola of equation

with

If on an interval we have , and hence

then for , we find

**Theorem 3**

### Program for Soft Landing on the Moon

`MoonLanding` is a *Mathematica* program for a soft landing on the Moon. Here `h0` is the initial height, `v0` is the initial velocity, `mass` is the mass of the lander without fuel, `fuel` is the initial fuel, `g` is acceleration due to gravity, `k` is the relative velocity of the exhaust gases, and ` is the rate of change of the mass by burning.`

The correctness of the results drastically depends on the initial values of the variables `z` and `g` that we use in solving the nonlinear system of equations in the program.

This `Manipulate` lets you vary the parameters in real time.

### Acknowledgments

The author expresses his gratitude to Horia F. Pop from Babes-Bolyai University, Faculty of Mathematics and Computer Science in Cluj-Napoca, Romania, for valuable discussions.

### References

[1] | D. E. Kirk, Optimal Control Theory, Englewood Cliffs, NJ: Prentice-Hall, Inc., 1970. |

[2] | J. Meditch, “On the Problem of Optimal Thrust Programming for a Lunar Soft Landing,” IEEE Transactions on Automatic Control, 9(4), 1964 pp. 477-484. doi:10.1109/TAC.1964.1105758. |

[3] | G. Leitmann (ed.), Optimization Techniques: With Applications to Aerospace Systems, Mathematics in Science and Engineering, Vol. 5, New York: Academic Press, 1962. |

[4] | L. Cesari, Optimization—Theory and Applications, Problems with Ordinary Differential Equations, Applications of Mathematics, Vol. 17, New York: Springer, 1983. |

[5] | M. Mureşan, A Concrete Approach to Classical Analysis, New York: Springer, 2009. |

Marian Mureşan, “Soft Landing on the Moon with Mathematica,” The Mathematica Journal, 2012. dx.doi.org/doi:10.3888/tmj.14-13. |

### About the Author

Marian Mureşan is affiliated with Babes-Bolyai University, Faculty of Mathematics and Computer Science, in Cluj-Napoca, Romania. He is interested in analysis, nonsmooth analysis, calculus of variations, and optimal control.

**Marian ****Mureşan**

*Babes-Bolyai University
Faculty of Mathematics and Computer Science
1, M. Kogalniceanu str., 400084, Cluj-Napoca
Romania
*

*mmarianus24@yahoo.com*

*mmarian@math.ubbcluj.ro*