Abstract
We study a variant of the classical safe landing optimal control problem in aerospace engineering, introduced by Miele (1962), where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. A more modern model consists in replacing the spacecraft by a hybrid gas-electric drone. Assuming that the drone has a failure and that the thrust (representing the control) can act in both vertical directions, the new target is to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the drone.
Highlights
In 1962 Miele raised the question of how to land safely a spacecraft on the moon surface, so as to use the least possible amount of fuel, see Section 4.8 of [13]. This problem was solved by Meditch [12] in 1964, by using tools from optimal control theory
Imagine that the drone has a failure during flight so that the target becomes to land safely in the least possible time: how should a pilot drive the drone on the moon surface in a minimum time? To model this problem, we introduce the variables h(t) = height of the drone at time t, v(t) = h (t) = velocity of the drone, m(t) = mass of the drone, α(t) = thrust at time t
To fully solve the minimum time problem for the safe landing of the drone, by using the Pontryagin Minimum Principle (PMP) in Theorem 2.1 we show that the possible optimal controls are only of four kinds, depending on the switch on/off of the thrust
Summary
In 1962 Miele raised the question of how to land safely a spacecraft on the moon surface (a so-called moon lander), so as to use the least possible amount of fuel, see Section 4.8 of [13]. To fully solve the minimum time problem for the safe landing of the drone, by using the PMP in Theorem 2.1 we show that the possible optimal controls are only of four kinds, depending on the switch on/off of the thrust. Theorem 3.2 provides precise “friendly” instructions to the pilot of the drone in order to reach a safe landing in minimum time These proofs are quite lengthy and involve a large number of basic (but quite delicate) computations throughout the paper; this is the price to pay for having precise answers to these practical aerospace queries.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
