A MINIMAL TIME OPTIMAL CONTROL FOR A DRONE LANDING PROBLEM

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. Mathematics Subject Classification. 34H10, 49J15, 49K15. Received January 27, 2021. Accepted September 25, 2021.


Introduction
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]. This problem was solved by Meditch [12] in 1964, by using tools from optimal control theory. Nowadays, this became a classical problem and it is mentioned in most of the textbooks on optimal control theory, see e.g. [8,9]. A more modern model consists in replacing the spacecraft with a hybrid gas-electric drone, see Figure 1. A complete answer to the original question requires a painstaking analysis of several critical curves in the phase space. We recently revisited this problem in [10] and we emphasized a possible misunderstanding and confusion between these critical curves.
It is our purpose to tackle here a different, but related, optimal control problem. 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 problem it acts in both directions (upwards and downwards), which translates into −1 ≤ α(t) ≤ 1; it is precisely this additional freedom for α which distinguishes the two problems. If α(t) = 0, the thrust is switched off and the drone is in free fall, while α(t) > 0 (resp. α(t) < 0) means that the thrust is applied against gravity (resp. in the direction of gravity). As the gas (or fuel) is burnt, the mass m(t) of the drone changes over time with a rate of change inversely proportional to α(t). According to Newton's second law, the motion of the drone is then (the upward direction is the positive direction) m(t)ḧ(t) = −gm(t) + α(t). (1.1) Since the target is to land safely in the least possible time, the minimization problem reads where m 0 represents total initial mass (the sum of the initial mass of fuel plus the mass m s of the empty drone). Too heavy drones cannot land safely because their weight would be larger than the full braking power of the thrust. This leads to the inequality gm s < 1 which says that the free fall of an empty drone can be slowed down. But, clearly, if the drone is empty there is no fuel to burn... Then it is physically more realistic to assume that gm 0 < 1 , (1.5) so that the thrust is able to slow down the drone even if it is full of fuel. As outlined in [10], the failure of (1.5) also has a mathematical consequence since the so-called safe landing curve has a turning point. Therefore, (1.5) is unavoidable in a reasonable physical and mathematical setting. Modern techniques of optimal control theory, introduced in the 1980s (see e.g. [15,16], the textbooks [1,3,7,11,14] and the references therein), are often used in literature to study aerospace problems. This approach is well described in the survey article [17], where the author shows how to combine usual techniques of optimal control theory, as the Pontryagin Minimum Principle (PMP), and results of geometric nonlinear optimal control, to analyze some aerospace models. It is out of reach to provide an exhaustive list of references dealing with aerospace problems, let us just mention a few of them. In [6], the optimal control reentry problem of a spatial shuttle is set in the geometric framework. In [4,5], local geometric results are combined with numerical simulations and conjugate point arguments. More recently, [2] deals with geometric analysis and numerical algorithm, based on indirect methods, and provides high numerical precision for optimal trajectories; see also [18] that contains a refined geometric study of the extremals coming from PMP, which allows to implement efficient numerical methods solving the problem of the guidance of a rocket.
According to [17], in order to determine optimal trajectories the first step is to make explicit the minimization conditions from the PMP, while geometric tools provide a complement to the PMP whenever it gives insufficient information about the optimal control problem. However, in our particular model, the PMP leads to a precise description of the optimal control. 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. In Theorem 2.3 we order these four possible controls, we establish which of them better performs in minimizing the landing time. Then we turn to the characterization of the optimal control. In Theorem 3.1 we give a full response: for any triad of initial data (v 0 , h 0 , m 0 ) we determine explicitly the optimal control.
The last purpose of the present paper is focused on the applications. Theorem 3.2 provides precise "friendly" instructions to the pilot of the drone in order to reach a safe landing in minimum time. These instructions are complemented with the analysis of how the optimal control varies in dependence of the initial mass m 0 (amount of fuel): this is described in Figure 5. 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. We also complement these instructions with some numerics: in Section 4, we quote some numerical results that show how disregarding the precise instructions (hence, performing a wrong manoeuvre), the pilot cannot safely land the drone.
All the proofs are postponed to Section 5. Some further remarks and possible future perspectives are collected in Section 6.

Characterization of the optimal control
In this section we determine the candidates optimal controls. In the first result we find the possible forms of admissible controls, namely controls able to steer the solution of (1.3) safely to rest, that is, with the physical constraints that (v(τ ), h(τ )) = (0, 0) for some τ > 0 and h(t) > 0 for t ∈ (0, τ ). Since controls may be discontinuous, here and in the sequel their characterization is intended a.e. in [0, τ ], without further mention.
If α is an optimal control for the minimum time problem (1.2)-(1.3), then there exist 0 ≤ t 1 ≤ t 2 < τ such that (2.1) Theorem 2.1, whose proof is given in Section 5.1, shows that, if it exists, the optimal control can take four possible forms. Let us characterize all of them in the following definition.
As stated in the following theorem, whose proof is postponed to Section 5.2, there exists a hierarchy between eligible controls allowing to deduce which of them is the optimal one. Theorem 2.3. Assume (1.4). If an optimal control α for (1.2)-(1.3) exists, then it is necessarily eligible, see Definition 2.2. Therefore, if no eligible control exists, then the optimal control problem has no solution. Moreover: -if there exists an admissible {1}-control, then it is unique, no other eligible controls exist and, hence, it is optimal; -if there exists an admissible {−1, 1}-control, then it is the unique optimal one; -there exist at most two {−1, 0, 1}-controls, and if a {−1, 0, 1}-control exists, the optimal one has the largest switch time t 1 ; -if there exist no admissible {−1, 1}-control, no admissible {−1, 0, 1}-control, and there exists an admissible {0, 1}-control, then the latter is unique and it is optimal.
The overall analysis will allow to deduce both when eligible controls exist, and their optimality. The existence of a {1}-control means that full braking is the unique possible strategy for a safe landing. If it exists, a {−1, 1}control is the optimal one since it accelerates towards safe landing as much as possible, then it fully brakes. The remaining cases require some free fall time (α = 0) since the fuel is not enough to be used at all time.
All the four cases in Theorem 2.3 may occur, but Theorem 2.3 does not explain which is the optimal control, depending on the initial conditions. Theorem 2.3 is used in the next section, in order to provide exact instructions for pilots of the drone for a safe landing, depending on the position, velocity, and mass at the time where the failure occurs.

Instructions for the pilot
In this section we determine the initial data (v 0 , h 0 , m 0 ) ∈ R × R + × [m s , 1/g) that allow for a safe landing of the drone, namely there exists a control α and τ ∈ [0, ∞) such that the solution of (1.3) satisfies h(τ ) = v(τ ) = 0. In Theorem 3.1 we classify the data in dependence of the optimal control that they require, according to Theorem 2.3, while in Theorem 3.2 we translate the results into instructions for the pilot of the drone, in order to reach a safe landing on the moon surface. We introduce several auxiliary functions whose graphs are the boundaries of the regions in the phase plane (v, h) that contain initial data allowing for a safe landing.
Once the mass of the empty drone m s > 0 is given, consider the function In Section 5.3 we prove that v → Ψ(m s , v) is strictly decreasing and that there exists a unique v ms ∈ (0, v + ms ) satisfying Ψ(m s , v) > 0 in [v − ms , v ms ) and Ψ(m s , v ms ) = 0, see Figure 2. Moreover, these numbers enable us to define another positive and strictly decreasing function see the precise definition in (5.23). We also introduce the equation of the critical parabola the point (V m0 , 0) is where the critical parabola intersects the v-axis for v > 0, see Figure 2. We notice that Next we define, for v ∈ [|v − ms |, v + ms ], the function (3.4) and P 0 (v) > R(m 0 ) − v 2 2g for all v > |v − ms |. In Lemma 5.5 in Section 5.3 we prove that there exists a unique w ms ∈ (v ms , v + ms ) such that P 0 (v) > 0 in [|v − ms |, w ms ) and P 0 (w ms ) = 0. Figure 2 depicts the points and curves defined above, in the two cases v ms ≶ V m0 that may both occur, see Lemma 6.3.
In order to simplify notations, we finally introduce the function In Remark 5.6 below we show that Z ∈ C 1 [v − ms , w ms ], which justifies the smooth graph in Figure 2. The main theorem on existence of optimal controls in dependence on the initial data then reads.
, then there exists no admissible control; (ii) if v 0 ∈ [v − ms , 0) and h 0 = Γ m0 (v 0 ), then there exists a unique eligible control which is the optimal one, it is a {1}-control; (iii) if either v 0 ∈ (v − ms , 0] and Γ m0 (v 0 ) < h 0 ≤ Ψ(m s , v 0 ), or v 0 ∈ (0, v ms ] and 0 ≤ h 0 ≤ Ψ(m s , v 0 ), then there exists a unique {−1, 1}-control, the optimal one; there exists at most a further eligible control, a {0, 1}-control; , then there exists a unique eligible control which is the optimal one, it is a {0, 1}-control; ( If v 0 > v − ms in (ii), or h 0 < Ψ(m s , v 0 ) in (iii), then some fuel is left at landing. In all the other cases landing occurs with no fuel left.
Item (i) in Theorem 3.1 defines both a "crash-region" and a "lack-of-fuel-region". If v 0 ∈ [v − ms , 0) and 0 < h 0 < Γ m0 (v 0 ), then the drone is too close to the moon and the thrust is not powerful enough to guarantee a safe landing, the drone will reach the surface of the moon (h = 0) with negative velocity (v < 0) and it will crash.
ms , w ms ] and h 0 > Z(v 0 ), then the drone does not contain enough fuel to allow safe landing. In Figure 3 we depict the regions of the (v, h)-plane with different optimal controls, according to Theorem 3. h + v 2 2g = R(m 0 ). We complement Theorem 3.1 with the a 3D-visualization of the dynamics of (1.3) in correspondence of the optimal control. The first picture in Figure 4 represents the orbit for a {1}-optimal control, with no switch and the m-component strictly decreasing in time, starting from m 0 and stopping at some m(τ ) ∈ [m s , m 0 ). The second picture represents the orbit for a {−1, 1}-optimal control, with one switch and the m-component strictly decreasing in time, starting from m 0 and stopping at some m(τ ) ∈ [m s , m 0 ). The third picture represents the orbit for a {−1, 0, 1}-optimal control, with two switches and the m-component strictly decreasing in the first and third time interval, while it is constant in the second interval; in this case there is total fuel consumption so that m(τ ) = m s . The fourth picture represents the orbit for a {0, 1}-optimal control, with one switch and the m-component being constant in the first interval and strictly decreasing in the second interval; also in this extremal case there is total fuel consumption and m(τ ) = m s .
In general, the residual amount of fuel at landing is m(τ ) − m s . In the next statement, we exclude the trivial case where (v 0 , h 0 ) = (0, 0) for which there is no dynamics and τ = 0.
Moreover, there is no fuel left at landing: m(τ ) = m s .
(iv) If one of the following couples of conditions holds .
(3.10) Figure 5. Variation of safe landing initial data (v 0 , h 0 ) as m 0 increases (from black to gray lines).
(v) System (3.10) admits two triples of solutions In all the other cases, there exists no admissible control and safe landing is impossible either because of the lack of fuel (too large distance or velocity) or of an unavoidable crash (drone initially too close to the moon surface).
Finally, we describe how the (planar) regions of initial data (v 0 , h 0 ) ∈ R × R + allowing for a safe landing change as the mass m 0 − m s of initial fuel varies. The rigorous statement would be quite lengthy and unpleasant. Therefore, we prefer to describe graphically the results in Figure 5, where the gray lines represent the deformation of the black ones when increasing the initial amount of fuel. The full justifications are provided in Section 5.6 by studying, in several steps, the variations of each of the lines depicted in Figure 2.
In Figure 6 we also represent the 3D region of admissible initial data it is the projection on the plane h 0 = 0 of the curve (2) and the gray surface joining these two curves corresponds to initial data for which the optimal control is {1}. The curve (3) is symmetric with respect to the plane v 0 = 0 of (2) and the yellow surface joining them corresponds to initial data with {0, 1}-optimal controls: the intersection of this surface with planes such as m 0 = C are parabolas. The curve (4) lies in the plane h 0 = 0 and has equation v 0 = w ms : it is connected to (3) with the blue surface that contains initial data for which the optimal control is {−1, 0, 1}. Such control is also optimal in the planar (white) region between the curve (4) and the curve (5) whose equation is v 0 = v ms . Still in the plane h 0 = 0, but now between the lines (5) and the axis (v 0 = 0), the optimal control is {−1, 1}. Moreover, the lines (5)

Numerical evaluation of a wrong manoeuvre from the pilot
In this section we investigate the consequences of a wrong manoeuvre, namely what can happen if the pilot does not follow the optimal control described in Theorem 3.2. We fix some values for the involved parameters. For gravity, we take the approximation g = 10 m/s 2 . We consider a drone of mass m s = 0.05 kg = 50 g with initial fuel of 30 g, that is, m 0 = 0.08 kg = 80 g. These values fulfill condition (1.5). Also take k = 10 −2 so that We argue directly on the original equation (1.1) that we rewrite as To (4.2) we associate initial data which account for the maximum altitude of the drone: h(0) = h 0 > 0,ḣ(0) = v 0 = 0 and we consider the region where the manoeuvre is more delicate for the pilot, namely when the optimal control is of the kind {−1, 0, 1}. How much time is lost if the pilot does not follow with exactitude the instructions in Theorem 3.2 (iv)? Moreover, can a wrong manoeuvre cause the crash of the drone? If we take v 0 = 0, we fall in the first case of Theorem 3.2 (iv) and we have the constraint Ψ(m s , 0) < h 0 < R(m 0 ) which, with the above choice of the parameters, becomes  We then choose as initial data to be associated with (4.2). Hence, (3.10) becomes The last equation yields t 1 ≈ 0.5334 which, inserted into the second equation, yields τ ≈ 3.32; finally, the first equation gives t 2 ≈ 0.8534. Therefore, the instructions for the pilot are: switch the full thrust downwards for a total time of 0.5334, then turn off the thrust until time 0.8534, finally switch the full thrust upwards until safe landing.
If the pilot follows these instructions, then the drone lands safely at time τ = 3.32. The behavior of the optimal dynamics is displayed in the left picture of Figure 7. Note that the condition τ + t 1 − t 2 = 3 ensures that all the fuel is burnt at time τ . So, by maintaining this constraint and with τ = 3.32, let us see the consequences of a wrong manoeuvre from the pilot, for both too early and too late switch t 1 . By taking t 1 = 0.5, we find t 2 = 0.82 and the mid picture in Figure 7; it turns out that the drone never reaches the ground level h = 0. By taking t 1 = 0.6, we find t 2 = 0.92 and the right picture in Figure 7; it turns out that the drone reaches the ground level h = 0 with negative velocity v < 0 and, hence, crashes.
Let us also analyze the case where the pilot is not able to put the drone in free fall, thereby restricting to {−1, 1}-controls. For a better comprehension, we study the dynamics for t ∈ [0, 4] even if |α(t)| ≡ 1 implies that the fuel ends at time t = 3 and the behavior for t > 3 fails to describe a physical situation. In the left picture of Figure 8, we plot the dynamics for the {−1, 1}-control with switch time at t 1 = 0.8; it turns out that the drone reaches the ground (h = 0) with negative velocity (v < 0) and, hence, crashes. In the middle picture we plot the dynamics for the {−1, 1}-control with switch time at t 1 = 0.67; it turns out that the drone never reaches the ground (h > 0) although the velocity vanishes (v = 0) at some time. These results suggest that for some t 1 ∈ (0.67, 0.8) one might obtain a safe landing. We found that this occurs when t 1 = 0.722, see the right picture in Figure 8 where h(t) = v(t) = 0 for t = 3.185; but safe landing is not reached because the drone remains without fuel for t > 3.

Proof of Theorem 2.1
The proof of Theorem 2.1 consists of three steps, the main ingredient being the Pontryagin Minimum Principle (PMP).
Step 1: use of the PMP.
By the PMP we know that, if an optimal solution (v, h, m, α) exists, then and the transversality condition p(τ ) ∈ N C = R × R × (−∞, 0] (the normal cone at the endpoint). This means that We consider the (autonomous) Hamiltonian which vanishes along an optimal trajectory, namely an optimal control α satisfies and fulfills the following minimality condition: Since for any C 1 , C 2 ∈ R we have min −1≤a≤1 the optimal control is then characterized by the (geometric and analytic) rule Step 2: we prove that The explicit representation (5.3) leaves some incertitude on the value of α on the switch lines (the lines corresponding to equalities in (5.3)) where we merely know that Since this indeterminacy of α may create troubles, our purpose here is to show (5.4). From the adjoint equation (5.1) we deduce that for some γ 1 , γ 2 , γ 3 ∈ R. Notice that, if p 2 (t) > 0 or kp 3 (t) < p2(t) m(t) ≤ 0 (hence α ∈ [−1, 0]), then d dt . . (5.7) By (5.5) we see that either p 2 (t) vanishes for at most one t or p 2 (t) ≡ 0. In the first case, the switch line p 2 = 0 is crossed at most once. The second case is ruled out if we show that γ 1 = 0. In fact, in view of (5.6)-(5.7), also (5.4) follows if we prove that either γ 1 = 0 or α(t) ≡ 1 . (5.8) In order to prove (5.8), we assume that γ 1 = 0 and we show that, necessarily, α(t) ≡ 1. If γ 1 = 0, then p 2 (t) ≡ γ 2 and we analyze the three possibilities for γ 2 . If γ 2 > 0, then p2(t) m(t) > 0 for all t and, hence α(t) ∈ [−1, 0] in [0, τ ], contradicting the safe landing. Indeed, α(t) ≤ 0 in a left neighborhood of τ , would meanv(t) ≤ −g < 0 with the impossibility of reaching v(τ ) = 0 from v(t) < 0.
Step 3: characterization of one-way switches, taking into account the physical constraints. We observe that safe landing tells us that either there are no switches (α(t) ≡ 1) or that γ 1 > 0.
Otherwisev(t) =ḧ(t) ≤ −g < 0 in a left neighborhood of τ , in contradiction with v(τ ) = h(τ ) = 0 coming from v(t) < 0 and h(t) > 0. In turn, by (5.1) and (5.6)-(5.7), the functions governing the switches have a given monotonicity. This is better seen in the "one-way switches pattern" displayed in the figure on the right.
Therefore, the only possibilities are discrete values of α, ordered as follows: This shows that α has to be as in (2.1) for some 0 ≤ t 1 ≤ t 2 < τ .

Proof of Theorem 2.3
The proof of Theorem 2.3 is obtained by combining a number of preliminary results. The first two statements establish a hierarchy of times between the eligible controls determined by Theorem 2.1.
Reasoning as above we obtain a contradiction in both cases, ending the proof.
Proof. The proof follows arguing by contradiction and by mimicking the proof of Proposition 5.1.
Assume that the initial conditions (v 0 , h 0 , m 0 ) in (1.4) are fixed. Let us make explicit the dynamics of (1.3) in the three different regimes for α established by Theorem 2.1.
When α(t) = −1 in (0, t 1 ), the thrust is oriented downwards and (1.3) readṡ ,ṁ(t) = −k, so that, after integration, When α(t) = 0 in (t 1 , t 2 ), the thrust is switched off and the free fall is described bẏ In this case we have conservation of energy, namely When α(t) = 1 in (t 2 , τ ), the thrust is oriented upwards and (1.3) becomeṡ ,ṁ(t) = −k, and, since h(τ ) = v(τ ) = 0, (5.11) Theorem 2.1 shows that eligible controls are the only candidates to be the optimal controls. But not all the categories of eligible controls exist for any initial data (v 0 , h 0 , m 0 ).
By equating the two above expressions for v(t) and h(t) we find By taking a linear combination of these two relationships we infer that which shows that τ → t(τ ) is strictly increasing by recalling (1.5). Assume by contradiction that there exist two admissible {−1, 1}-controls α 1 = α 2 , with switch and final times (t 1 , τ 1 ) and (t 2 , τ 2 ), respectively. By the just proved monotonicity, if t 1 < t 2 then τ 1 < τ 2 and two cases may occur.
We are now in position to give the proof of Theorem 2.3. In [10] we defined the switching curve for {0, 1}controls and we showed that, for given initial data m 0 > m s and (v 0 , h 0 ) ∈ R × R + , there exists at most one admissible {0, 1}-control, whose switch position is uniquely determined by the switch time. Hence, collecting these results and the statements from Propositions 5.1-5.3 (see also Prop. 5.13), we obtain the proof of the theorem.

Some auxiliary functions and their properties
We analyze the functions introduced in Section 3 and prove several technical results needed in the proofs of Theorems 3.1 and 3.2.
The first result defines two implicit functions, v = v m and v = v m , relating the velocity v and the mass m, that are needed in the sequel. In order to clarify the geometric context, we believe that the picture on the right may be helpful for the reader. It depicts the functions defined in the statement and in the proof.
The statement is as follows:  Proof. By differentiating with respect to m we infer that which proves that m → v + m is strictly decreasing. Since v + m0 = 0, positivity of v + m is also proved. Similarly, by differentiating and by (1.5), we have (5.12) In particular, v → ∂Ψ ∂v (m, v) is strictly decreasing and v → Ψ(m, v) is strictly concave. Next, notice that m 0 m e g(m0−m)−kv + m = 1 so that, by differentiating with respect to m and by using (1.5), we infer that In particular, since v + m > 0, this shows that m → Ψ(m, 0) is strictly decreasing and, hence, Recall that, by the Lagrange Theorem, for any m ∈ [m s , m 0 ) there exists ξ m ∈ (m, m 0 ) such that This inequality will repeatedly be used throughout the paper. Here, we use (5.15) to obtain Hence, there exists a unique v m ∈ (0, v + m ) such that Ψ(m, v m ) = 0. Using again (5.12)  The behavior of the function P 0 , defined in Section 3, is analyzed in the following lemma.
Lemma 5.5. The function P 0 is strictly decreasing over [|v − ms |, v + ms ] and there exists a unique w ms ∈ (|v − ms |, v + ms ) such that P 0 (w ms ) = 0. Moreover, w ms > v ms .
To prove the last statement of the lemma, we claim that To this end, we notice that, by (5.12), Since kv < kv + ms = g(m 0 − m s ) + log m0 ms < g(m 0 + m s ) + log m0 ms = 2gα, we infer that Q iv (v) > 0; hence, the map v → Q (v) is strictly increasing in [|v − ms |, v + ms ). Furthermore, by recalling (1.5) and the calculus inequality 0 < x 2 ye x−y < 1, valid whenever 0 < y < x < 1, we infer that Therefore, two cases may occur: either In both cases, we get (5.18). The inequality (5.18) implies that P 0 (v ms ) = Q(v ms ) > 0, hence, since P 0 (w ms ) = 0 and P 0 is strictly decreasing by (5.17), the claimed inequality w ms > v ms follows.
Following (3.2), we put At this stage we fix (v 0 , h 0 ) ∈ R × R + and we define, for t ∈ 0, m0−ms k , the function that measures the distance of a trajectory (v(t), h(t)) solving (5.9) (for a control α = −1) from the critical parabola corresponding to an initial mass of fuel m 0 − kt. The next lemma states the main properties of D: recall that v − ms is defined in (3.1), R(m 0 ) and V m0 are defined in (3.2), w ms is characterized by Lemma 5.5.
Lemma 5.7. Let (v 0 , h 0 ) ∈ R × R + and let D be as in (5.19) and P 0 as in (3.4). . ( Proof. Clearly, t → D(t) is smooth and we first prove the following three properties: (5.20) The first property (5.20) 1 follows by replacing t = 0 in (5.19). By using (5.15), we infer that which proves (5.20) 2 . Then we rewrite (5.19) as Assuming that v 0 ≤ w ms , by Lemma 5.5 and (3. then, by differentiating (5.21), we infer thaẗ ]. After some lengthy computations we see that If h 0 = P 0 (v 0 ), the above arguments show that D(t) < D(t M ) = 0 for all t = t M , proving item (iv). , both representing instants of time: the former is the ratio between a mass and a variation of mass, while the latter is a ratio between the velocity and and the gravity acceleration.
In Lemma 5.9 we provide a condition ensuring that the trajectory (v(t), h(t)) intersects the critical parabola corresponding to an initial amount of fuel m 0 − kt in an admissible region.
Proof. It is straightforward that η(0) > 0 andη(t) = − 2 m0−kt < 0. Then we definē We first prove (5.22) 2 and, hence, we assume that v 0 > |v − ms |. To this end, we define the function We have that and, by (1.5), Hence, for any v 0 ∈ (|v − ms |, w ms ), F (v 0 ) =t − t M > 0; recall that w ms < v + ms by Lemma 5.5. This proves (5.22) 2 . To obtain (5.22) 1 and (5.22) 3 , by the properties of D outlined in Lemma 5.7, it is enough to prove that D(t) < 0. Notice that These two facts and some tedious computations enable us to show that , ending the proof.

Proof of Theorem 3.1
We assume that m 0 > m s is fixed and we argue on the remaining initial data (v 0 , h 0 ). By Lemma 5.4 the map m → v − m is negative and strictly increasing over the interval [m s , m 0 ] and v − m0 = 0. Therefore, this map is invertible and we denote by its inverse map (also strictly increasing). Then consider the map which is positive and strictly decreasing over [m s , m 0 ] with h(m 0 ) = 0, see [10], Theorem 1. These facts enable us to define the strictly decreasing composite function Notice that v − ms (t), h − ms (t) corresponds to the extreme point v − ms (t), Γ m0−kt (v − ms (t)) of the switching curve, when the initial amount of fuel is m 0 − kt and, ). The proof of Theorem 3.1 is performed by steps, we analyze the optimal control (if any) in dependence on the initial data, following Theorem 2.3. We first determine conditions ensuring the existence of a {1}-control.  As a consequence of Theorem 5 in [10], we have the following statement.
By definition of V m0 it suffices to prove (5.27) 1 . To this end, we use (1.5) and (5.15) to deduce Next, we analyze the behavior of the extremal point v − ms , Γ m0 (v − ms ) as m 0 varies.
Let v − ms be as in (3.1) and let Γ m0 be as in (5.23). Then Proof. To prove (5.28) 1 , we notice that, by (1.5), For (5.28) 2 we analyze the behavior of the extremal point of the switching curve. This is the initial point of safe landing with {1}-control and no fuel left at landing. By (5.11) 1 we then see that m 0 = m(0) = kτ + m s , that is, τ = m0−ms k . By replacing this into (5.11) 3 we infer that Therefore, where the inequality follows from (5.15). This proves (5.28) 2 .
In particular, (5.28) 1 shows that the interval of definition of the map v → Γ m0 (v) enlarges as m 0 increases. Further, by their explicit expression in (3.1), we deduce immediately that, m 0 → |v − ms | and m 0 → v + ms are strictly increasing.
Finally, we notice that the very same proof of Lemma 5.14 shows that, for any residual mass m(τ ) ≥ m s , if the initial mass m 0 is increased then the terminal point (corresponding to t = 0 in (5.11)) moves "up and left" as in Figure 5. Consider now the function Ψ introduced in Lemma 5.4. To analyze the monotonicity of Ψ(m s , v) with respect to m 0 (for a fixed v ∈ (v − ms , v + ms )), we study the function Note that, for any fixed m 0 ∈ m s , 1 g , we have F (m 0 , v) = k 2 Ψ(m s , v).
Lemma 5.15. Let F = F (m 0 , v) be as above, then Proof. For any fixed v ∈ (v − ms , v + ms ), we have that Notice that see Lemma 5.4 and (5.15). Furthermore, for any v ∈ (v − ms , v + ms ), by using (1.5) we infer that The three above inequalities show that there exists a uniqueṽ ms ∈ v − ms , v + ms such that and the proof of the lemma will be complete if we show that v ms <ṽ ms . By Lemma 5.

Final comments
In this final section we state some (marginal) further results. The first deals with Theorem 3.1. Remark 6.2. If in case (v) of Theorem 3.2 we have that h 0 = P 0 (v 0 ), then the unique solution of (3.10) is given by Concerning Figure 2, we show that both the situations described may occur. In the following lemma we discuss the mutual position of v ms and V m0 in dependence on m s and m 0 . In turn, through a further application of the Lagrange Theorem, the latter shows that The physical interpretation of the assumption gm s ≶ 2( √ 2 − 1) is that the empty drone has small/large mass. In both cases, m 0 close to m s means that there is little initial fuel. Under the assumptions of Lemma 6.3, both the inequalities v ms ≶ V m0 may occur, see Figure 9. For small m s , the graph of the function h = Ψ(m s , v) remains below the graph of h = R(m 0 ) − v 2 2g and intersects the h-axis at v ms < V m0 , for large m s the two graphs intersect and cross at some point in (0, V m0 ). These pictures should then be completed as in Figure 2.
We conclude this paper with two possible future developments of our results. We saw that a full comprehension of the safe landing strategy in minimal time requires a large amount of computations. There are many parameters involved and establishing their roles and their hierarchy is mandatory to have a complete picture of the phenomena involved. Since the precise rules have been established in this paper, one could seek numerical codes able to guide the pilot in real time.
A further problem is to consider an asymmetric interval for the control α. Recall that α represents the thrust and it is quite realistic to assume different strengths while accelerating and braking. How do the results and pictures in the present paper change if α ∈ [−1, 2] or α ∈ [−2, 1]?