# Interpretable DRL-based Maneuver Decision of UCAV Dogfight

Haoran Han1, Jian Cheng1, and Maolong Lv2 1Haoran Han and Jian Cheng are with
the School of Information and Communication Engineering, University of
Electronic Science and Technology of China, Chengdu 611731, China. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Lv is with the Air Traffic
Control and Navigation College, Air Force Engineering University, Xi’an
710051, China, and also with College of Aeronautics Engineering, Air Force
Engineering University, Xi’an 710038, China. Email<EMAIL_ADDRESS>
###### Abstract

This paper proposes a three-layer unmanned combat aerial vehicle (UCAV)
dogfight frame where Deep reinforcement learning (DRL) is responsible for
high-level maneuver decision. A four-channel low-level control law is firstly
constructed, followed by a library containing eight basic flight maneuvers
(BFMs). Double deep Q network (DDQN) is applied for BFM selection in UCAV
dogfight, where the opponent strategy during the training process is
constructed with DT. Our simulation result shows that, the agent can achieve a
win rate of 85.75% against the DT strategy, and positive results when facing
various unseen opponents. Based on the proposed frame, interpretability of the
DRL-based dogfight is significantly improved. The agent performs yo-yo to
adjust its turn rate and gain higher maneuverability. Emergence of ”Dive and
Chase” behavior also indicates the agent can generate a novel tactic that
utilizes the drawback of its opponent.

##  I Introduction

Over the last few years, unmanned combat air vehicle (UCAV) has become
increasingly important in the battlefield. Most conventional aerial combat
methods, such as decision tree (DT) [1] or state machine [2], are easy to
implement and interpret. This process can be enhanced with target maneuver
recognition [3] technique. However, they are impossible to cover all combat
situations due to the complicated nonlinear nature of aerial combat.
Therefore, more researchers focused on the decision of basic flight maneuver
(BFM) with optimization techniques [4, 5, 6].

However, the strong non-linearity of aerial combat cause significant
difficulty for optimization. Therefore, deep reinforcement learning (DRL), a
data-driven optimization technique, has been attached more attention. The
first DRL algorithm cluster is to let the agent generate the control command
directly. Pope et al. [7] proposed a hierarchical DRL dogfight frame, where a
network selected three controllers with different combat styles. Chai et al.
[8] utilized two cascade DRL controllers to output the desired attitude and
control command respectively. Moreover, techniques such as transformer [9] and
inverse DRL [10] were also applied.

In addition, DRL are also utilized for maneuver decision. The first category
of BFM library consists of turning with different directions and loads [11].
Another library design consists of meaningful BFM, such as tracking, turn and
split-s. Sun et al. [12] applied self-play technique and observed emergence of
multiple interpretable tactics. Piao et al. [13] combined DRL with expert
knowledge, and Zhu et al. [14] designed an interruption mechanism where the
agent increased decision frequency when facing emergency.

However, most works concerning DRL-based dogfight [9, 10, 11, 12, 13, 14] only
focused on simplified UCAV dynamics with three degrees of freedom (DOF), which
is far from the real situation. Moreover, most works only claimed their
algorithms outperformed conventional methods. Some simple reasons were given,
such as ”more aggressive” [4, 7, 8, 9], but a deep interpretation of how DRL
agent performs is still lacking. In response to these problems, this paper
aims to enhance the interpretability of DRL behaviors when facing dogfights
under realistic 6-DOF dynamics. Therefore, this paper contributes to the field
of dogfight in three aspects.

Firstly, this paper proposes a three-layer UCAV dogfight decision frame. We
firstly construct a four-channel low-level control law, followed bt a library
consisting of eight common-apply BFMs. A bank-to-turn tracking algorithm is
adopted, and then generalized to other BFMs. Lastly, the double deep Q network
(DDQN) [15] is used to achieve the maneuver decision. Our simulation shows
that, the agent can achieve a success rate of 85.75% against its DT opponent.

Then, this paper gives an in-depth post-hoc interpretation on how the agent
behaves, mainly in two aspects. (1) This paper observes that, the agent can
use yo-yo to adjust its rate of turn, so that it can gain higher
maneuverability during a double-loop dogfight. (2) Furthermore, we also
observe the emergence of ”Dive and Chase” strategy, where the agent firstly
decrease its height, and then chase its opponent who climbs to avoid crashing
into the ground.

Lastly, this paper provides a gym-style environment for the DRL-based maneuver
decision at https://github.com/Han-Adam/Dogfight, where F16 [16] is adopted.

##  II Preliminaries

###  II-A UCAV Dynamical Model

This paper adopts north-west-down fame as the earth frame. The translation
dynamics is considered as

$\begin{matrix}\dot{\boldsymbol{p}}=R_{\text{b}}^{\text{e}}\boldsymbol{v}^{\text{b}},&\dot{\boldsymbol{v}}^{\text{b}}=(\boldsymbol{f}^{\text{t}}-\boldsymbol{f}^{\text{a}})/m+R_{\text{e}}^{\text{b}}\boldsymbol{g}-\boldsymbol{\Theta}\boldsymbol{v}^{\text{b}},\end{matrix}$
(1)

where $m$ is the mass, $\boldsymbol{p}=[p_{1},p_{2},p_{3}]^{T}$ is the
position, $\boldsymbol{v}^{\text{b}}=[u,v,w]^{T}$ is the velocity under the
body frame, $\boldsymbol{g}=[0,0,g]^{T}$ is the gravity,
$\boldsymbol{\Theta}=[\phi,\theta,\psi]^{T}$ is the Euler angle,
$R_{\text{b}}^{\text{e}}$ is the rotation matrix from body to earth frame,
$R_{\text{e}}^{\text{b}}=R_{\text{b}}^{\text{e},T}$, and
$\boldsymbol{f}^{\text{t}}=[f,0,0]$ and
$\boldsymbol{f}^{\text{a}}=[D,L,C]^{T}$ are the thrust and air force.
Moreover, the rotation dynmaics is considered as

$\begin{matrix}\dot{\boldsymbol{\Theta}}=H(\boldsymbol{\Theta})\boldsymbol{\omega},&\dot{\boldsymbol{\omega}}=J^{-1}(\boldsymbol{\tau}-\boldsymbol{\omega}\times
J\boldsymbol{\omega})\end{matrix}$ (2)

where $J$ is the inertia matrix, $H(\boldsymbol{\Theta})$ is the matrix
transforming the angular velocity $\boldsymbol{\omega}=[p,q,r]^{T}$ into the
change rate of Euler angle, $\boldsymbol{\tau}=[l,m,n]^{T}$ is the external
torque. The ground speed $V$, attack angle $\alpha$, and sideslip angle
$\beta$ are

$\begin{matrix}V=||\boldsymbol{v}^{\text{b}}||,&\alpha=\arctan(w/u),&\beta=\arcsin(v/V).\end{matrix}$
(3)

Additionally, the mach number $Ma$ is defined as the ratio between the ground
speed and the air speed. The force and the torque generated by the air effect
is calculated as

$\begin{matrix}D=\bar{q}SC_{D},&L=\bar{q}SC_{d},&C=\bar{q}SC_{C},\\\
l=\bar{q}SbC_{l},&m=\bar{q}S\bar{c}C_{m},&n=\bar{q}SbC_{n},\end{matrix}$ (4)

where $\bar{q}$ is the dynamic press, $S$ and $b$ are the area and span for
the wing, $\bar{c}$ is the reference length. The aerodynamic coefficients
$C_{\xi}$, where $\xi=D,L,C,l,m,n$, are determined by the deflection of
elevator $\delta^{\text{e}}$, aileron $\delta^{\text{a}}$, rudder
$\delta^{\text{r}}$, $\alpha$, $\beta$, and $Ma$. In addition, the thrust $f$
is determined by the throttle $\delta^{\text{t}}$. According to [16], the
amplitude and the change rate of the control input
$\boldsymbol{\delta}=[\delta^{\text{t}},\delta^{\text{e}},\delta^{\text{a}},\delta^{\text{r}}]$
are all restricted.

###  II-B Simplified 3-DOF Model

In the simplified model, the heading is equivalent to the direction of the
velocity. Therefore, the 3-DOF model is

$\begin{matrix}\dot{p}_{1}=V\cos\chi\cos\zeta,&\dot{p}_{2}=V\cos\chi\sin\zeta,&\dot{p}_{3}=-V\sin\chi,\\\
\dot{V}=g(n^{\text{l}}-\sin\chi),&\dot{\chi}=\frac{g(n^{\text{n}}\cos\phi-\cos\chi)}{V},&\dot{\zeta}=\frac{gn^{\text{n}}\sin\phi}{V\cos\chi},\end{matrix}$
(5)

where $\chi$ and $\zeta$ represent the path pitch and path yaw, $n^{\text{l}}$
and $n^{\text{n}}$ represent the longitudinal and normal load. Here, the roll
angle $\phi$, also known as bank angle, can be instantly changed.

Figure 1: Geometric relation of UCAVs in the dogfight, where the blue one is
controlled by the agent, and the red one is regarded as the opponent.

###  II-C Dogfight Setting

the line of sight (LOS) is defined as
$\boldsymbol{l}=\boldsymbol{p}^{\text{o}}-\boldsymbol{p}=[\Delta p_{1},\Delta
p_{2},\Delta p_{3}]^{T}$, where the superscript of o represents the value of
the opponent. The distance, heading crossing angle (HCA), antenna train angle
(ATA), and aspect angle (AA) [17] between a UCAV and its opponent are defined
as

$\begin{matrix}d=||\boldsymbol{l}||,&HCA=\arccos(\boldsymbol{h}^{T}\boldsymbol{h}^{\text{o}}),\\\
ATA=\arccos(\boldsymbol{l}^{T}\boldsymbol{h}/d),&AA=\arccos(\boldsymbol{l}^{T}\boldsymbol{h}^{\text{o}}/d),\end{matrix}$
(6)

where $\boldsymbol{h}$ is the heading of the UCAV. The Geometic relation is
illustrated in Fig. 1. The damage caused by weapons is constant, and the
engagement zone [17] is designed as

$\begin{matrix}100\text{ m}\leq d\leq 1000\text{ m},&ATA\leq
1^{\circ}.\end{matrix}$ (7)

The initial blood of each UCAV is assigned as one, which means it can afford
damage for only one second. The UCAV is regarded as crashed, if its distance
to the opponent is less than 10 m, or its height is less than 10 m.

###  II-D Deep Reinforcement Learning

The reinforcement learning is modeled as Markov decision process (MDP),
$(\mathcal{S},\mathcal{A},\mathcal{R},\mathcal{P},\gamma)$. At each time $t$,
the agent receives a state $s_{t}\in\mathcal{S}$ and takes an action
$a_{t}\in\mathcal{A}$. The state of the agent then changes based on
$s_{t+1}\sim\mathcal{P}(\cdot|s_{t},a_{t})$, and the instantaneous reward is
generated based on $r_{t}=\mathcal{R}(s_{t},s_{t+1})$. The discounted reward
is defined as $G_{t}=\sum_{i=0}^{\infty}\gamma^{i}r_{t+i}$, where the discount
factor $\gamma\in(0,1)$. The agent samples actions based on its policy $\pi$,
and the Q function for a given policy is defined as
$Q_{\pi}(s_{t},a_{t})=\mathbb{E}_{a_{i}\sim\pi(\cdot|s_{i}),s_{i+1}\sim\mathcal{P}(\cdot|s_{i},a_{i}),i>t}[G_{t}|s_{t},a_{t}]$.

DRL uses neural networks to approximate the Q function. This paper uses $\eta$
and $\hat{\eta}$ to denote the parameter of real and target networks
respectively. The policy is induced by

$\pi(\cdot|s_{t})=\left\\{\begin{matrix}\arg\max(Q_{\eta}(s_{t}))&\text{with
probability of }\epsilon.\\\ U(\mathcal{A})&\text{with probability of
}1-\epsilon.\end{matrix}\right.$ (8)

where $U(\cdot)$ is uniform distribution, and the greedy factor $\epsilon$
belongs to $(0,1)$ during the train process, and equals to $1$ during the test
process. This paper adopts DDQN to update the parameter, where the loss
function is defined as

$\displaystyle L(\eta)=\mathbb{E}_{\mathcal{D}}($ $\displaystyle
r_{t}+(1-\mathbb{I}(done_{t}))$ (9) $\displaystyle
Q_{\hat{\eta}}(s_{t},\max_{a}Q_{\eta}(s_{t+1},a))-Q_{\eta}(s_{t},a_{t}))^{2}.$

Here, $\mathcal{D}$ is the replay buffer, from which all trajectories are
uniformly sampled, $\mathbb{I}(\cdot)$ equals one if the statement is True or
zero otherwise, $done_{t}$ represent whether the state is terminal. The target
network parameters $\hat{\eta}$ is updated for every $\widetilde{C}$ steps.

##  III Methodology

A three-layer hierarchical dogfight frame, as illustrated in Fig. 2 is
described in this section.

Figure 2: Overview of the proposed UCAV dogfight frame.

###  III-A Low-Level Control Law

This paper uses widely-applied PID, which is defined as

$\operatorname{PID}(\xi,\xi^{\text{d}},\boldsymbol{k}_{\xi})=k^{\text{P}}_{\xi}e_{\xi}+\int
k^{\text{I}}_{\xi}e_{\xi}-k^{\text{D}}_{\xi}\dot{e}_{\xi},$ (10)

for low-level control, where $\xi^{\text{d}}$ is the desired value of a
variable $\xi$, $e_{\xi}=\xi^{\text{d}}-\xi$ is the error, and
$\boldsymbol{k}_{\xi}=[k^{\text{P}}_{\xi},k^{\text{I}}_{\xi},k^{\text{D}}_{\xi}]^{T}$.

This paper divides the UCAV into four channels. First, the UCAV is set to
chase a predefined mach number of 0.9, because a high velocity will cause
performance degradation. Then, the elevator and aileron are adjusted based on
the angle of attack $\alpha$ and roll $\phi$ respectively. This paper adopts
bank-to-turn control mode. Thus, we directly use rudder to maintain $\beta$ at
zero. In summary, the low level control law is

$\displaystyle\delta^{\text{t, d}}$
$\displaystyle=\operatorname{clip}(\operatorname{PID}(Ma,0.9,\boldsymbol{k}_{Ma}),\delta^{\text{t}}_{\text{min}},\delta^{\text{t}}_{\text{max}}),$
(11) $\displaystyle\delta^{\text{e, d}}$
$\displaystyle=\operatorname{clip}(\operatorname{PID}(\alpha,\alpha^{\text{d}},\boldsymbol{k}_{\alpha}),\delta^{\text{e}}_{\text{min}},\delta^{\text{e}}_{\text{max}}),$
$\displaystyle\delta^{\text{a, d}}$
$\displaystyle=\operatorname{clip}(\operatorname{PID}(\phi,\phi^{\text{d}},\boldsymbol{k}_{\phi}),\delta^{\text{a}}_{\text{min}},\delta^{\text{a}}_{\text{max}}),$
$\displaystyle\delta^{\text{r, d}}$
$\displaystyle=\operatorname{clip}(\operatorname{PID}(\beta,0,\boldsymbol{k}_{\beta}),\delta^{\text{r}}_{\text{min}},\delta^{\text{r}}_{\text{max}}).$

###  III-B Basic Flight Maneuver Library Construction

The BFM library constructed in this paper consists of eight operations, namely
position tracking, attitude tracking, straight flight, climb, somersault,
split-s, high yo-yo and low yo-yo. They are built based on the simplified
model.

Position Tracking: The velocity vector of the UACV should point to the
opponent. Therefore, the desired path pitch and path yaw are firstly
calculated as

$\displaystyle\chi^{\text{d}}$ $\displaystyle=\arctan(-\Delta
p_{3}/\sqrt{\Delta p_{1}^{2}+\Delta p_{2}^{2}}),$ (12)
$\displaystyle\zeta^{\text{d}}$ $\displaystyle=\arctan(\Delta p_{2}/\Delta
p_{1}).$

Then, the load under the trajectory frame is calculated as

$\begin{matrix}n_{2}=\frac{V}{g}k_{\zeta}e_{\zeta}\cos\chi,&n_{3}=\frac{V}{g}k_{\chi}e_{\chi}+\cos\chi.\end{matrix}$
(13)

The desired roll angle and normal load are generated as

$\begin{matrix}\phi^{\text{d}}=\arctan(n_{3}/n_{2}),&n^{\text{n}}=\operatorname{clip}(n_{2}\sin\phi+n_{3}\cos\phi).\end{matrix}$
(14)

The normal load is transformed with

$\alpha^{\text{d}}=\operatorname{clip}(k_{n}^{\alpha}n^{\text{n}},\alpha^{\text{d}}_{\text{min}},\alpha^{\text{d}}_{\text{max}})$
(15)

where $k_{n}^{\alpha}$ can be approximately set as a constant, and
$\alpha^{\text{d}}_{\text{min}},\alpha^{\text{d}}_{\text{max}}$ are bounds for
the desired value.

Attitude Tracking: The heading of UACV should point to the opponent, which
means the Euler angle rather than the path angle should track the desired
angle value in Eq. (12). Therefore, when applying Eq. (13) to calculate the
load, all errors are replaced by $e_{\chi}=\chi^{d}-\theta$ and
$e_{\zeta}=\zeta^{\text{d}}-\psi$.

Straight Flight: The $\phi^{\text{d}}$ and $\chi^{\text{d}}$ are set as
$0^{\circ}$.

Climb: The $\phi^{\text{d}}$ and $\chi^{\text{d}}$ are set as zero and a
positive value.

Somersault: The $\phi^{\text{d}}$ is set as $0^{\circ}$ or $180^{\circ}$
according to whether the body is reversed or not. The desired angle of attack
should be maintained as $\alpha^{\text{d}}=\alpha^{\text{full}}_{\text{max}}$.

Split-S: This is a composite maneuver consisting of three steps. First, the
UCAV reverses so that its normal direction points to the ground. Then, it
pulls the load to maintain
$\alpha^{\text{d}}=\alpha^{\text{full}}_{\text{max}}$ and decrease its height.
Lastly, it changes the maneuver to straight flight after its pitch return to
zero degree.

Yo-Yo: This maneuver is used to exchange height and speed, and thus the
tracking should have a vertical offset $\Delta h$. In this paper, the
$\chi^{\text{d}}$ in Eq. (12) is rewritten as

$\chi^{\text{d}}=\arctan((-\Delta p_{3}\pm\Delta h)/\sqrt{\Delta
p_{1}^{2}+\Delta p_{2}^{2}}),$ (16)

where the sign is set as positive for high yo-yo and negative for low yo-yo.
Additionally, the offset is calculated as a certain ratio of the kinetic
energy of the UCAV, which means

$\Delta h=k_{h}V^{2}/g.$ (17)

###  III-C Maneuver Decision Based on Decision Tree

This paper constructs a DT including the baseline of tracking and three
typical options. Self-protection is used to prevent the UCAV from falling into
extreme conditions. We also design strategy for UCAV to escape from being
chased from behind. Yo-yo is introduced to adjusts the approaching speed. The
decision sequence is protection, escape, yo-yo, and lastly tracking. The three
options are denoted with $SP$, $ES$, $YY$, which can be set as True or False
to determine whether to use corresponding strategies. Details are as follows.

Self-Protection: To prevent the UCAV from crashing, it selects the climb
maneuver to increase its height if $-p_{3}<h^{\text{protect}}$. Additionally,
it selects the straight flight to recover its speed and maneuverability if
$Ma<Ma^{\text{protect}}$.

Escape: The UCAV escapes from disadvantage conditions. If
$d<d^{\text{close}}$, $ATA>ATA^{\text{escape}}$, and $AA>AA^{\text{escape}}$,
this paper assumes the UCAV is about to be chased from behind. Then, if
$p_{3}<p_{3}^{\text{o}}$, it chooses somersault to escape from above.
Otherwise, it selects split-s to escape from below.

Yo-Yo: When $d>d^{\text{close}}$ and $ATA<ATA^{\text{aim}}$, this paper
assumes the UCAV aims at the opponent from a distance. According to [18], the
UCAV chooses yo-yo when $AA^{\text{yo-yo}}_{\text{min}}<AA<AA^{\text{yo-
yo}}_{\text{max}}$. Moreover, it selects a maneuver of high yo-yo to reduce
speed and avoid overshoot when $V>V^{\text{o}}$, and low yo-yo to increase
speed when $V<V^{\text{o}}$.

Tracking: The UCAV needs to track its opponent. If $d<d^{\text{close}}$ and
$ATA<ATA^{\text{aim}}$, it selects attitude tracking to aim its nose at the
opponent and cause damage. Otherwise, it selects position tracking to approach
the opponent.

###  III-D Maneuver Decision Based on DDQN

During training, this paper let the agent control the blue UCAV, while the red
one controlled with the DT strategy serves as the opponent. We modify the DDQN
to train the agent to make maneuver decision for the dogfight, which is shown
in Algorithm 1. Details are described as below.

State: The state space of the agent is designed as

$s_{t}=(p_{3},V,\phi,\theta,\chi,\psi-\zeta^{\text{d}},\xi-\zeta^{\text{d}},HCA,ATA,AA,d,V^{\text{o}}).$
(18)

Then, we elaborate the conciseness of the state design. The absolute
coordinate of the UCAV is omitted because only the relative position makes
sense. Considering the isotropy of space, the relationship between two UCAVs
when they rotate with the same angle has no changes. Therefore, the yaw is
reduced to $\psi-\zeta^{\text{d}}$ and $\zeta-\zeta^{\text{d}}$. The position
and attitude information of the opponent can be induced by $HCA$, $ATA$, $AA$,
and $d$. Thus, they are not listed explicitly.

Action: The action space is discrete and $|\mathcal{A}|=8$, where each one
represents a BFM in Section III.B.

Reward: The reward function in this paper is designed as

$\displaystyle r_{t}$
$\displaystyle=\lambda^{\text{f}}r^{\text{f}}_{t}+\lambda^{\text{d}}r^{\text{d}}_{t}+\lambda^{\text{a}}r^{\text{a}}_{t},$
(19) $\displaystyle r^{\text{f}}_{t}$ $\displaystyle=\mathbb{I}(\text{red UCAV
fails})-\mathbb{I}(\text{blue UCAV fails}),$ $\displaystyle r^{\text{d}}_{t}$
$\displaystyle=(blood^{\text{red}}_{t-1}-blood^{\text{red}}_{t})-(blood^{\text{blue}}_{t-1}-blood^{\text{blue}}_{t}),$
$\displaystyle r^{\text{a}}_{t}$ $\displaystyle=180-ATA-AA.$

Here, $\lambda$ means constant coefficients, $r^{\text{s}}_{t}$ represents the
sparse reward determined by whether the agent or opponent fails,
$r^{\text{d}}_{t}$ is the difference between the damage caused by two UCAVs,
$r^{\text{a}}_{t}$ is the angle advantage. If the agent is defeated, damaged,
or at a disadvantaged angle, it will receive a negative penalty. Otherwise, it
will obtain a positive reward.

Transition: Once the agent and its opponent select their BFM, the dynamical
model updates for $\Delta T$ steps. If one of the UCAVs fails, then the
simulation stops and the agent starts another episode.

Algorithm 1 DDQN for UCAV Dogfight

0: Initial network parameter $\eta$.

0: Optimized network parameter.

1: for $\text{episode}=1,2,...$ do

2: Randomly initialize states of two UCAVs.

3: for $t=0,1,...$ do

4: Sample maneuvers for red and blue UCAVs based on the DT in Section III-C
and Eq. (8), respectively.

5: Set $done_{t}$ as False.

6: for $t^{\prime}=1,2,...,\Delta T$ do

7: Calculate inputs based on Section III-A and B.

8: Update dynamics based on Section II-A

9: if One of UCAVs fails according to Section II-C then

10: Set $done_{t}$ as True, and break.

11: end if

12: end for

13: Calculate the next state $s_{t+1}$ and reward $r_{t}$.

14: Store the trajectory $(s_{t},a_{t},s_{t+1},r_{t},done_{t})$ into
$\mathcal{D}$.

15: Uniformly sample trajectories from $\mathcal{D}$.

16: Update $\eta$ with the loss function of Eq. (9).

17: Update parameters of the target network with $\hat{\eta}\leftarrow\eta$
for every $\widetilde{C}$ training steps.

18: end for

19: end for

##  IV Simulation

###  IV-A Simulation Setup

This paper selects F16s as the UCAV model, where all parameters follow [16].
PID for the low-level control are set as $\boldsymbol{k}_{Ma}=[10,0,0]^{T}$,
$\boldsymbol{k}_{\alpha}=[0.8,1.8,10]^{T}$,
$\boldsymbol{k}_{\phi}=[0.07,0,0]^{T}$, and
$\boldsymbol{k}_{\beta}=[12,0,4]^{T}$. Here, roll and sidelip channels reduce
to P and PD control respectively, because their trims are zero.

Parameters of tracking are set as $k_{\zeta}=k_{\chi}=0.02$,
$k_{n}^{\alpha}=4$. Bounds of $\alpha$ are set as
$\alpha^{\text{d}}_{\text{min}}=-4^{\circ}$ and
$\alpha^{\text{d}}_{\text{max}}=20^{\circ}$. Moreover, we set
$\alpha_{\text{max}}^{\text{full}}=30^{\circ}$ to give escape BFM higher
maneuverability, and $k_{h}=0.1$ for yo-yo.

The protection bounds are set as $h^{\text{protect}}=1000$ m and
$Ma^{\text{protect}}=0.3$. In addition, this paper set $d^{\text{close}}=3000$
m and $ATA^{\text{aim}}=30^{\circ}$ for tracking,
$ATA^{\text{escape}}=AA^{\text{escape}}=120^{\circ}$ for escape, and
$AA^{\text{yo-yo}}_{\text{min}}=30^{\circ}$, $AA^{\text{yo-
yo}}_{\text{max}}=60^{\circ}$ for yo-yo [18]. The desired path pitch for climb
is $20^{\circ}$.

When applying the DRL algorithm, the simulation step is set as $0.01$ s. The
decision period is set as $1$ s, namely $\Delta T=100$. The initial position,
speed belong to $[-3,3]\times[-3,3]\times[-3,-8]$ km and $[0.3,0.9]$ mach.
Initial pitch and roll are set as zero, while yaw belongs to $[-180,180]$
degree. Each episode lasts for $300$ time steps. As for the setting of DDQN,
this paper set $|\mathcal{D}|=1\times 10^{5}$, $\epsilon=0.95$, $\gamma=0.95$,
and $\widetilde{C}=512$. The sparse reward is set as $\lambda^{\text{f}}=20$.
Dense ones are set as $\lambda^{\text{d}}=1$ and $\lambda^{\text{a}}=1/180$,
which normalize each term to $[-1,1]$.

Moreover, [7] indicates that wide shallow network is more suitable for DRL
algorithm. Therefore, this paper adopts networks with only two hidden layers,
each containing $512$ and $256$ nodes. Adam is applied as the optimizer, where
the learning rate and batch size are set as $1\times 10^{-4}$ and $512$.

Figure 3: (a) Accumulated number and (b) rate for win, loss, tie among the
training process. The solid lines mean the average calculated over 10 trials,
while the shadow ones represent the maximum and the minimum.

###  IV-B Training Process

Ten trails containing $5\times 10^{5}$ training steps are tested. The time
consumption of each trail is approximately $11$ hours, tested on a computer
with i9-13900K CPU. The accumulated number of win, loss, and tie for the agent
is shown in Fig. 3 (a). Then, we randomly initialize $400$ circumstance to
test the rate of win, loss, and tie of agents against the DT, which is shown
in Fig. 3 (b). In general, all trails converges and agents outperform the DT
after around $5\times 10^{4}$ steps of training. However, similar to numerous
previous works, the training process of DRL has significant randomness and
fluctuation. Some trails could achieve the highest success rate of 90% under
$100$ randomly initialize testing circumstances, while others could only
achieve 80%. Therefore, this paper selects the best-performed agent for the
following analysis.

###  IV-C Robustness Analysis

The strategy of opponents during the training process is a full DT, which may
vary in real dogfight conditions. Therefore, this section test the robustness
of the agent facing with different unseen opponents. We randomly initialize
$400$ circumstances. The performance of the best-performed agent facing eight
strategies with different settings of protection, escape, and yo-yo are
tested, which is shown in Table I.

TABLE I: Rate of win, loss and tie of the best-performed agent facing with different unseen opponents. Strategy | $SP$ | $ES$ | $YY$ | Win | Loss | Tie
---|---|---|---|---|---|---
1 |  |  |  | 54.00 | 30.50 | 15.50
2 | ✓ |  |  | 70.75 | 15.25 | 14.00
3 |  | ✓ |  | 56.50 | 30.50 | 13.00
4 |  |  | ✓ | 53.00 | 31.50 | 15.50
5 |  | ✓ | ✓ | 55.75 | 31.25 | 13.50
6 | ✓ |  | ✓ | 70.00 | 15.25 | 14.75
7 | ✓ | ✓ |  | 85.50 | 9.50 | 5.00
8 | ✓ | ✓ | ✓ | 85.75 | 9.25 | 5.00

In general, the agent can achieve positive results when facing all opponent.
Nevertheless, its performance dramatically declined within some conditions,
which indicate the need of further study on increasing DRL robustness.

Additionally, statistical results indicate following phenomena. Firstly,
whether the opponent apply yo-yo has little impact on the agent performance.
That is because the distance between two UCAVs are almost always lower than
$d^{\text{close}}=3000$ m, which means the opponent rarely conduct yo-yo BFM.
Then, the protection mechanism has the most important impact on the agent
performance. Specifically, this paper claims that the agent learns how to
leverage the height protection to defeat the opponent. Furthermore, the escape
strategy can bring improvement to the agent only when the opponent adopts
protection simultaneously.

###  IV-D Case Analysis: Double Loop

To illustrate how the agent trained by the proposed method defeat the opponent
in detail, this paper constructs two typical cases. The initial condition of
the first case is set as $\boldsymbol{p}^{\text{red}}=[0,-2000,-5000]$,
$\boldsymbol{p}^{\text{red}}=[0,2000,-5000]$, $\psi^{\text{red}}=0$,
$\psi^{\text{blue}}=180$, $Ma^{\text{red}}=Ma^{\text{blue}}=0.9$. This case
lasts for $145$ s, and forms a typical double-loop dogfight condition shown in
Fig. 4.

Figure 4: (a) 3D trajectories, and response of (b) ATA, distance, (c) mach,
height of two UCAVs in the first case with double-loop dogfight condition.

At the very beginning, the agent performs low yo-yo to decrease its height,
and then somersault (at $t=12$) to gain being damaged during merging. Compared
with tracking, somersault can provide the agent with higher maneuverability,
and thus build advantage after merging. At approximately $t=37$ s, the agent
has built sufficient angle advantage, and then it begins to perform tracking
BFM. Both UCAVs track each other, and therefore they form a double loop.
Furthermore, the agent always chooses low yo-yo after each merging. It uses
height to exchange kinetic energy, and thus gains higher turn rate to build
angle advantage. After two rounds of engagements, it begins to cause damage to
the opponent at $t=101,123,145$ s, and wins the dogfight without being
attacked. Therefore, this case illustrates that, the agent trained with DRL
can perform higher maneuverability compared with DT algorithm.

###  IV-E Case Analysis: Dive and Chase

The second case illustrates a learned strategy named as ”Dive and Chase” by
which the agent leverages the protection and escape mechanism of the opponent.
The initial mach is set as 0.8, where other conditions are the same with the
first case. Fig. 5 shows trajectories and response of this case.

Figure 5: 3D trajectories during (a) $[0,110)$, (b) $[110,140)$, (c)
$[140,169]$ s, and response of (d) ATA, distance, and (c) height in the second
case.

During the period of $[0,11000]$ s, the agent mixes numerous low yo-yo and
split-s. This behavior causes the height of two UCAVs rapidly drop. After
$t=110$ s, the agent performs a compound maneuver of ”spli-s, somersault,
tracking” at a low altitude. This behavior triggers the height protection of
the opponent, and builds angle advantage for the agent. After $t=130$ s, the
opponent tracks the agent, but falls below $h^{\text{protect}}=1000$ m for the
second time. Then, the agent tracks its opponent who climbs due to height
protection, and builds a significant angle advantage. The opponent then
performs somersault to escape from being chased. This paper claims that, the
agent can recognize the high maneuverability escape BFM through the state.
Hence, the agent abandons tracking and selects somersault at the same time, by
which it can maintain its angle advantage even its opponent intends to escape,
as shown in Fig. 5 (c). At last, the agent wins. The emergence of ”Dive and
Chase” strategy during the DRL training process indicates that, the agent can
defeat opponents built by expertise from a tactical perspective.

##  V Conclusion

In this paper, we propose a DRL-based maneuver decision frame for the
dogfight. A four-channel low-level control law is firstly designed, and then
eight commonly used BFMs are then constructed. A DT consisting of four typical
strategies is built. Lastly, we introduce DDQN for the UCAV maneuver decision.
Our simulation result shows that, DRL can enormously outperform the
conventional DT method from the perspective of both maneuverability and
tactic.

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# On solutions of the Diophantine equation $L_{n}+L_{m}=3^{a}$

Pagdame TIEBEKABE $\&$ Ismaïla DIOUF

###### Abstract

Let $(L_{n})_{n\geq 0}$ be the Lucas sequence given by $L_{0}=2,L_{1}=1$ and
$L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 0$. In this paper, we are interested in
finding all powers of three which are sums of two Lucas numbers, i.e., we
study the exponential Diophantine equation $L_{n}+L_{m}=3^{a}$ in nonnegative
integers $n,m,$ and $a$. The proof of our main theorem uses lower bounds for
linear forms in logarithms, properties of continued fractions, and a version
of the Baker-Davenport reduction method in Diophantine approximation.

Keywords and phrases: Linear forms in logarithm; Diophantine equations;
Fibonacci sequence; Lucas sequence; perfect powers.
2010 Mathematics Subject Classification. 11B39, 11J86

##  1 Introduction

The determination of perfect powers of Lucas and Fibonacci sequences does not
date from today. The real contribution of determination of perfect powers of
Lucas and Fibonacci sequences began in 2006. By classical and modular
approaches of Diophantine equations, Bugeaud, Mignotte, and Siksek [5] defined
all perfect powers of Lucas and Fibonacci sequences by solving the equations
$F_{n}=y^{p}$ and $L_{n}=y^{p}$ respectively. From there, many researchers
tackled similar problems. It is in the same thought that, others have
determined the powers of $2$ of the sum/difference of two Lucas numbers [3],
powers of $2$ of the sum/difference of Fibonacci numbers [4], powers of $2$
and of $3$ of the product of Pell numbers and Fibonacci numbers.

We move our interest on the powers of $3$ as a sum of two Lucas numbers. This
paper follows the following steps : We first give the generalities on binary
linear recurrence, then we demonstrate an important inequality on Lucas
numbers and finally determine and reduce a coarse bound by section $3$. The
section $4$ is devoted to the reduction of the obtained bound in section $3$
and discussion of possible different cases. We know from Bravo and Lucas [3]
that the only solutions of the Diophantine equation $F_{n}+F_{m}=2^{a}$ in
positive integers $n$, $m$ and $a$ with $n\geq m$ are given by

$2F_{1}=2,\quad 2F_{2}=2,\quad 2F_{3}=4,\quad 2F_{6}=16,$

and

$F_{2}+F_{1}=2,\quad F_{4}+F_{1}=F_{4}+F_{2}=4,\quad F_{5}+F_{4}=8,\quad
F_{7}+F_{4}=16.$

and in [4] that all solutions of the Diophantine equation $L_{n}+L_{m}=2^{a}$
in nonnegative integers $n\geq m$ and $a$, are

$2L_{0}=4,\quad 2L_{1}=2,\quad 2L_{3}=8,\quad L_{2}+L_{1}=4,\quad
L_{4}+L_{1}=8,\quad\text{and}\quad L_{7}+L_{2}=32.$

Here in this paper, we determine all the solutions of the following
Diophantine equation:

$L_{n}+L_{m}=3^{a}$ (1)

in nonnegative integers $n\geq m$ and $a$.

We are interested in finding all powers of three which are sums of two Lucas
numbers, i.e., we study the exponential Diophantine equation
$L_{n}+L_{m}=3^{a}$ in nonnegative integers $n$, $m,$ and $a$. The proof of
our main theorem uses lower bounds for linear forms in logarithms, properties
of continued fractions, and a version of the Baker-Davenport reduction method
in Diophantine approximation.

We notice that many authors have already tackled this type of problems.

##  2 Preliminaries

###  2.1 Generalities

######  Definition 2.1.

Let $k\geq 1$. The sequence $\\{H_{n}\\}_{n\geq 0}\subseteq\mathbb{C}$ is
called a recurrent linear sequence of order $k$ if the sequence satisfies

$H_{n+k}=a_{1}H_{n+k-1}+a_{2}H_{n+k-2}+\cdots+a_{k}H_{n}$

for all $n\geq 0$ with $a_{1},\ldots,a_{k}\in\mathbb{C}$, fixed.

We suppose that $a_{k}\neq 0$ (otherwise, the sequence $\\{H_{n}\\}_{n\geq 0}$
satisfies a recurrence of order less than $k$). If
$a_{1},\ldots,a_{k}\in\mathbb{Z}$ and $H_{0},\ldots,H_{k-1}\in\mathbb{Z}$,
then we can easily prove by induction on $n$ that $H_{n}$ is an integer for
all $n\geq 0$. The polynomial

$f(X)=X^{k}-a_{1}X^{k-1}-a_{2}X^{k-2}-\cdots-a_{k}\in\mathbb{C},$

is called the characteristic polynomial of $(H_{n})_{n\geq 0}$. We suppose
that

$f(X)=\prod_{i=1}^{m}(X-\alpha_{i})^{\sigma_{i}},$

where $\alpha_{1}$, …, $\alpha_{m}$ are distinct roots of $f(X)$ with
respectively $\sigma_{1}$, …, $\sigma_{m}$ their multiplicities.

######  Definition 2.2.

We define the sequences $(A_{n})_{n\geq 0}$ and $(B_{n})_{n\geq 0}$ for all
positive integers $\mathbb{N}$ by

$\left\\{\begin{array}[]{lllll}A_{n+2}&=&aA_{n+1}+A_{n},&A_{0}=0,&A_{1}=1\\\
B_{n+2}&=&aB_{n+1}+B_{n},&B_{0}=2,&B_{1}=a.\\\ \end{array}\right.$

For $a=1$, $(A_{n})_{n\geq 0}=(F_{n})_{n\geq 0}$ and $(B_{n})_{n\geq
0}=(L_{n})_{n\geq 0}$ , which are Fibonacci and Lucas sequences respectively,
defined above.

######  Remark 2.1.

If $k=2$, the sequence $(H_{n})_{n\geq 0}$ is called a binary recurrent
sequence. In this case, the characteristic polynomial is of the form

$f(X)=X^{2}-a_{1}X-a_{2}=(X-\alpha_{1})(X-\alpha_{2}).$

Suppose that $\alpha_{1}\neq\alpha_{2}$, then
$H_{n}=c_{1}\alpha_{1}^{n}+c_{2}\alpha_{2}^{n}$ for all $n\geq 0$.

######  Definition 2.3.

The binary recurrent sequence $\\{H_{n}\\}_{n\geq 0}$ is said to be non
degenerated if $c_{1}c_{2}\alpha_{1}\alpha_{2}\neq 0$ and
$\alpha_{1}/\alpha_{2}$ is not a root of unity.

Binet’s formula for the general term of Fibonacci and Lucas sequences is
obtained using standard methods for solving recurrent sequences, which are
given by :

$F_{n}=\dfrac{\alpha^{n}-\beta^{n}}{\alpha-\beta}\quad\text{and}\quad
L_{n}=\alpha^{n}+\beta^{n}$

where
$\left(\alpha,\beta\right)=\left(\dfrac{1+\sqrt{5}}{2},\dfrac{1-\sqrt{5}}{2}\right)$
are the zeros of the characteristic polynomial $X^{2}-X-1$.

######  Definition 2.4.

For all algebraic numbers $\gamma$, we define its measure by the following
identity :

${\rm M}(\gamma)=|a_{d}|\prod\limits_{i=1}^{d}\max\\{1,|\gamma_{i}|\\},$

where $\gamma_{i}$ are the roots of
$f(x)=a_{d}\prod\limits_{i=1}^{d}(x-\gamma_{i})$ is the minimal polynomial of
$\gamma$.

Let us define now another height, deduced from the last one, called the
absolute logarithmic height. It is the most used one.

######  Definition 2.5.

( Absolute logarithmic height)

For a non-zero algebraic number of degree $d$ on $\mathbb{Q}$ where the
minimal polynomial on $\mathbb{Z}$ is
$f(x)=a_{d}\prod\limits_{i=1}^{d}(x-\gamma_{i})$, we denote by

$h(\gamma)=\dfrac{1}{d}\left(\log|a_{d}|+\sum\limits_{i=1}^{d}\log\max\\{1,|\gamma_{i}|\\}\right)=\dfrac{1}{d}\log{\rm
M}(\gamma).$

the usual logarithmic absolute height of $\gamma$.

The following properties of the logarithmic height, will also be used in the
next section:

  * •

$h(\gamma\pm\eta)\leq h(\gamma)+h(\eta)+\log 2$.

  * •

$h(\gamma\eta^{\pm 1})\leq h(\gamma)+h(\eta)$.

  * •

$h(\gamma^{s})=|s|h(\gamma)$.

###  2.2 Inequalities involving the Lucas numbers

In this section, we state and prove important inequalities associated with the
Lucas numbers that will be used in solving the equation (1).

######  Proposition 2.1.

For $n\geq 2$, we have

$0.94\,\alpha^{n}<(1-\alpha^{-6})\alpha^{n}\leq
L_{n}\leq(1+\alpha^{-4})\alpha^{n}<1.15\,\alpha^{n}$ (2)

###### Proof.

This follows directly from the formula $L_{n}=\alpha^{n}+(-1)^{n}\alpha^{-n}.$
∎

###  2.3 Linear forms in logarithms and continued fractions

In order to prove our main result, we have to use a Baker-type lower bound
several times for a non-zero linear forms of logarithms in algebraic numbers.
There are many of these methods in the literature like that of Baker and
Wüstholz in [1]. We recall the result of Bugeaud, Mignotte, and Siksek which
is a modified version of the result of Matveev [8]. With the notation of
section $2$, Laurent, Mignotte, and Nesterenko [7] proved the following
theorem:

######  Theorem 2.1.

Let $\gamma_{1}$, $\gamma_{2}$ be two non-zero algebraic numbers, and let
$\log\gamma_{1}$ and $\log\gamma_{2}$ be any determination of their
logarithms. Put
$D=[\mathbb{Q}(\gamma_{1},\gamma_{2}):\mathbb{Q}]/[\mathbb{R}(\gamma_{1},\gamma_{2}):\mathbb{R}]$,
and

$\Gamma:=b_{2}\log\gamma_{2}-b_{1}\log\gamma_{1},$

where $b_{1}$ and $b_{2}$ are positive integers. Further, let $A_{1},A_{2}$ be
real numbers $>1$ such that

$\log
A_{i}\geq\max\left\\{h(\gamma_{i}),\dfrac{|\log\gamma_{i}|}{D},\dfrac{1}{D}\right\\},\quad(i=1,2).$

Then, assuming that $\gamma_{1}$ and $\gamma_{2}$ are mutiplicatively
independent, we have

$\log|\Gamma|>-30.9\cdot D^{4}\left(\max\left\\{\log
b^{\prime},\dfrac{21}{D},\dfrac{1}{2}\right\\}\right)^{2}\log A_{1}\cdot\log
A_{2},$

where

$b^{\prime}=\dfrac{b_{1}}{D\log A_{2}}+\dfrac{b_{2}}{D\log A_{1}}.$

We shall also need the following theorem due to Matveev, Lemma due to Dujella
and Pethő and Lemma due to Legendre [8, 6].

######  Theorem 2.2.

(Matveev [8])

Let $n\geq 1$ an integer. Let $\mathbb{L}$ a field of algebraic number of
degree $D$. Let $\eta_{1}$, …, $\eta_{l}$ non-zero elements of $\mathbb{L}$
and let $b_{1}$, $b_{2}$, …, $b_{l}$ integers,

$B:=\max\\{|b_{1}|,...,|b_{l}|\\},$

and

$\Lambda:=\eta_{1}^{b_{1}}\cdots\eta_{l}^{b_{l}}-1=\left(\prod\limits_{i=1}^{l}\eta_{i}^{b_{i}}\right)-1.$

Let $A_{1}$, …, $A_{l}$ reals numbers such that

$A_{j}\geq\max\\{Dh(\eta_{j}),|\log(\eta_{j})|,0.16\\},1\leq j\leq l.$

Assume that $\Lambda\neq 0$, So we have

$\log|\Lambda|>-3\times 30^{l+4}\times(l+1)^{5.5}\times d^{2}\times
A_{1}...A_{l}(1+\log D)(1+\log nB)$

Further, if $\mathbb{L}$ is real, then

$\log|\Lambda|>-1.4\times 30^{l+3}\times(l)^{4.5}\times d^{2}\times
A_{1}...A_{l}(1+\log D)(1+\log B).$

During our calculations, we get upper bounds on our variables which are too
large, so we have to reduce them. To do this, we use some results from the
theory of continued fractions. In particular, for a non-homogeneous linear
form with two integer variables, we use a slight variation of a result due to
Dujella and Pethő, (1998) which is in itself a generalization of the result of
Baker and Davemport [2].

For a real number $X$, we write $\left\lVert X\right\rVert:=\min\\{\mid
X-n\mid:n\in\mathbb{Z}\\}$ for the distance of $X$ to the nearest integer.

######  Lemma 2.1.

(Dujella and Pethő, [6])

Let $M$ a positive integer, let $p/q$ the convergent of the continued fraction
expansion of $\kappa$ such that $q>6M$ and let $A$, $B$, $\mu$ real numbers
such that $A>0$ and $B>1$. Let $\varepsilon:=\left\lVert\mu
q\right\rVert-M\left\lVert\kappa q\right\rVert$.
If $\varepsilon>0$ then there is no solution of the inequality

$0<m\kappa-n+\mu<AB^{-m}$

in integers $m$ and $n$ with

$\dfrac{\log(Aq/\varepsilon)}{\log B}\leqslant m\leqslant M.$

######  Lemma 2.2.

(Legendre)

Let $\tau$ real number such that $x$, $y$ are integers such that

$\left|\tau-\dfrac{x}{y}\right|<\dfrac{1}{2y^{2}}.$

then $\dfrac{x}{y}=\dfrac{p_{k}}{q_{k}}$ is the convergence of $\tau$.

Further,

$\left|\tau-\dfrac{x}{y}\right|>\dfrac{1}{(q_{k+1}+2)y^{2}}.$

##  3 Main result

Our main result can be stated in the following theorem.

######  Theorem 3.1.

The only solutions $(n,m,a)$ of the exponential Diophantine equation

$L_{n}+L_{m}=3^{a}$ in nonnegative integers $n\geq m$ and $a$, are :
$(1,0,1)\quad\text{and}\quad(4,0,2)$

$\text{i.e}\quad L_{1}+L_{0}=3,\quad\text{and}\quad L_{4}+L_{0}=9.$

###### Proof.

First, we study the case $n=m$, next we assume $n>m$ and study the case $n\leq
200$ with SageMath in the range $0\leq m<n\leq 200$ and finally we study the
case $n>200$. Assume throughout that equation (1) holds. First of all, observe
that if $n=m$, then the original equation (1) becomes

$L_{n}=\dfrac{3^{a}}{2}.$

This equation has no solution because, $\forall n>0,L_{n}\in\mathbb{Z}.$ So
from now, we assume $n>m$.
If $n\leq 200$, the search with SageMath in the range $0\leq m<n\leq 200$
gives the solutions $(n,m,a)\in\left\\{(1,0,1),(4,0,2)\right\\}$. Now for the
rest of the paper, we assume that $n>200$ . Let first get a relation between
$a$ and $n$ which is important for our purpose. Combining (1) and the right
inequality of (2), we get:

$3^{a}=L_{n}+L_{m}\leq
2\alpha^{n}+2\alpha^{m}<2^{n+1}+2^{m+1}=2^{n+1}(1+2^{n-m})\leq
2^{n+1}(1+1/2)<2^{n+2}.$

Taking $\log$ both sides, we obtain

$a\log 3\leq(n+2)\log 2\Longrightarrow a\leq(n+2)c_{1}\quad\text{where}\quad
c_{1}=\frac{\log 2}{\log 3}.$

Rewriting equation (1) as:

$L_{n}+L_{m}=\alpha^{n}+\beta^{n}+L_{m}=3^{a}\Longrightarrow\alpha^{n}-3^{a}=-\beta^{n}-L_{m}.$

Taking absolute value both sides, we get

$|\alpha^{n}-3^{a}|=|\beta^{n}+L_{m}|\leq|\beta|^{n}+L_{m}<\frac{1}{2}+2\alpha^{m}\quad\because|\beta|^{n}<\frac{1}{2},\quad\text{and}\quad
L_{m}<2\alpha^{m}.$

Dividing both sides by $\alpha^{n}$ and considering that $n>m$, we get:

$\left|1-\alpha^{-n}\cdot
3^{a}\right|<\dfrac{\alpha^{-n}}{2}+2\alpha^{m-n}<\dfrac{1}{\alpha^{n-m}}+\dfrac{2}{\alpha^{n-m}}\quad\because\frac{1}{2\alpha^{n}}<\frac{1}{\alpha^{n-m}};\quad
n>m$

Hence

$\left|1-\alpha^{-n}\cdot 3^{a}\right|<\dfrac{3}{\alpha^{n-m}}$ (3)

Let’s take

$\gamma_{1}:=\alpha,\quad\gamma_{2}:=3,\quad b_{1}:=n,\quad
b_{2}:=a,\quad\Gamma:=a\log 3-n\log\alpha$

in order to apply Theorem 2.1. Therefore equation (3) can be rewritten as:

$\left|1-e^{\Gamma}\right|<\dfrac{3}{\alpha^{n-m}}\quad\text{where}\quad
e^{\Gamma}=\alpha^{-n}3^{a}.$ (4)

Since $\mathbb{Q}(\sqrt{5})$ is the algebraic number field containing
$\gamma_{1},\gamma_{2}$; so we can take $D:=2$. Using equation (1) and Binet
formula for Lucas sequence, we have :

$\alpha^{n}=L_{n}-\beta^{n}<L_{n}+1\leq L_{n}+L_{m}=3^{a}$

which implies $1<3^{a}\alpha^{-n}$ and so $\Gamma>0$. Combining this with (4),
we get

$0<\Gamma<\dfrac{3}{\alpha^{n-m}}$ (5)

where we used the fact that $x\leq e^{x}-1,\quad\forall x\in\mathbb{R}.$
Applying $\log$ on right and left hand side of (5), we get

$\log\Gamma<\log 3-(n-m)\log\alpha.$ (6)

Logarithm height of $\gamma_{1}$ and $\gamma_{2}$ are:

$h(\gamma_{1})=\dfrac{1}{2}\log\alpha=0.2406\cdots,$ $h(\gamma_{2})=\log
3=1.09862\cdots,$ thus we can choose

$\log A_{1}:=0.5\quad\text{and}\quad\log A_{2}:=1.1.$

Finally, by recalling that $a\leq(n+2)c_{1};\quad c_{1}=0.63093,$ we get :

$b^{\prime}:=\dfrac{b_{1}}{D\log A_{2}}+\dfrac{b_{2}}{D\log
A_{1}}=\dfrac{n}{2.2}+a=0.45n+a<0.45n+(n+2)c_{1}<2n.$

It is easy to see that $\alpha$ and $3$ are multiplicatively independent. Then
by Theorem 2.1, we have

$\log\Gamma\geq-30.9\cdot
2^{4}\left(\max\left\\{\log(2n),\dfrac{21}{2},\dfrac{1}{2}\right\\}\right)^{2}\cdot
0.5\cdot 1.1$
$\log\Gamma>-272\left(\max\left\\{\log(2n),\dfrac{21}{2},\dfrac{1}{2}\right\\}\right)^{2}.$
(7)

Combining (6) and (7), we obtain the following important result

$(n-m)\log\alpha<276\left(\max\left\\{\log(2n),\dfrac{21}{2},\dfrac{1}{2}\right\\}\right)^{2}.$
(8)

Let us find a second linear form in logarithm. For this, we rewrite (1) as
follows:

$\alpha^{n}(1+\alpha^{n-m})-3^{a}=-\beta^{n}-\beta^{m}.$

Taking absolute values in the above relation, we get

$|\alpha^{n}(1+\alpha^{m-n})-3^{a}|<2,\quad\beta=(1-\sqrt{5})/2,\quad|\beta|^{n}<1\quad\text{and}\quad|\beta|^{m}<1;\forall
n>200,\quad m\geq 0.$

Dividing both sides of the above inequality by $\alpha^{n}(1+\alpha^{m-n})$,
we obtain

$\left|1-3^{a}\alpha^{-n}(1+\alpha^{m-n})^{-1}\right|<\dfrac{2}{\alpha^{n}}\quad\text{i.e}\quad|\Lambda|<\dfrac{2}{\alpha^{n}}.$
(9)

All the conditions are now met to apply a Matveev’s theorem (Theorem 2.2).

  * •

Data:

$t:=3;\quad\gamma_{1}:=3;\quad\gamma_{2}:=\alpha;\quad\gamma_{3}:=1+\alpha^{m-n}$

$b_{1}:=a;\quad\quad b_{2}:=-n,\quad b_{3}=-1.$

As before, $\mathbb{K}=\mathbb{Q}(\sqrt{5})$ contains
$\gamma_{1},\gamma_{2},\gamma_{3}$ and has $D:=[\mathbb{K}:\mathbb{Q}]=2.$
Before continuing with the calculations, let’s check whether $\Lambda\neq 0$.

$\Lambda\neq 0$ comes from the fact that if it was zero, we would have

$3^{a}=\alpha^{n}+\alpha^{m}$ (10)

Taking the conjugate of the above relation in $\mathbb{Q}(\sqrt{5})$, we get :

$3^{a}=\beta^{n}+\beta^{m}.$ (11)

Combining (10) and (11), we get :

$\alpha^{n}<\alpha^{n}+\alpha^{m}=|\beta^{n}+\beta^{m}|\leq|\beta|^{n}+|\beta|^{m}<2.$

Recall that $n>200$. This relation is impossible for $n>200$. Hence
$\Lambda\neq 0.$

  * •

Calculation of $h(\gamma_{3})$

Let us now estimate $h(\gamma_{3})$ where $\gamma_{3}=1+\alpha^{m-n}$

$\gamma_{3}=1+\alpha^{m-n}<2\quad\text{and}\quad\gamma^{-1}=\dfrac{1}{1+\alpha^{m-n}}<1$

so $|\log\gamma_{3}|<1.$ Notice that

$h(\gamma_{3})\leq|m-n|\left(\dfrac{\log\alpha}{2}\right)+\log 2=\log
2+(n-m)\left(\dfrac{\log\alpha}{2}\right).$

  * •

The calculation of $A_{1}$ and $A_{2}$ gives :

$A_{1}:=2.2$

and

$A_{2}:=0.5$

and we can take

$A_{3}:=2+(n-m)\log\alpha\quad\text{since}\quad h(\gamma_{3}):=\log
2+(n-m)\left(\dfrac{\log\alpha}{2}\right)$

  * •

Calculation of $B$

Since $a<(n+2)c_{1}$, it follows that, $B=\max\\{1,n,a\\}$. Thus we can take
$B=n+1$.

The Matveev’s theorem gives the lower bound on the left hand side of (9) by
replacing the data. We get :

$\exp\left(-C(1+\log(n+1))\cdot 2.2\cdot 0.5\cdot(2+(n-m)\log\alpha)\right)$

where

$C:=1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 2^{2}(1+\log 2)<9.7\times 10^{11}.$

Replacing in equation (9), we get:

$\exp\left(-C(1+\log(n+1))\cdot 2.2\cdot
0.5\cdot(2+(n-m)\log\alpha)\right)<|\Lambda|<\dfrac{2}{\alpha^{n}}$

which leads to

$n\log\alpha-\log 2<C((1+\log(n+1))\cdot 1.1\cdot(2+(n-m)\log\alpha)<2C\log
n\cdot 1.1\cdot(2+(n-m)\log\alpha)$

then

$n\log\alpha-\log 2<1.26\times 10^{12}\log n\cdot(2+(n-m)\log\alpha)$ (12)

where we used inequality $1+\log(n+1)<2\log n$, which holds for $n>200$.

Now, using (8) in the right term of the above inequality (12) and doing the
related calculations, we get

$n<7.3\times 10^{14}\log
n\left(\max\left\\{\log(2n),\dfrac{21}{2}\right\\}\right)^{2}.$ (13)

If $\max\\{\log(2n),21/2\\}=21/2,$ it follows from (13) that $n<8.04825\cdot
10^{16}\log n\Longrightarrow n<3.5\cdot 10^{18}.$ On the other hand, if
$\max\\{\log(2n),21/2\\}=\log(2n)$, then from (13), we get $n<7.3\cdot
10^{14}\log n\log^{2}(2n)$ and so $n<7.2\cdot 10^{19}$. We can easily see that
for the two possible values of $\max\\{\log(2n),21/2\\}$, $n<7.2\cdot
10^{19}$.

All the calculations done so far can be summarized in the following lemma.

######  Lemma 3.1.

If $(n,m,a)$ is a solution in positive integers of (1) with conditions $n>m$
and $n>200$, then inequalities

$a\leq n+2<1.2\times 10^{20}\quad\text{hold.}$

##  4 Reducing of the bound on $n$

Dividing across inequality (5) : $0<a\log
3-n\log\alpha<\dfrac{3}{\alpha^{n-m}}$ by $\log\alpha$, we get

$0<a\gamma-n<\dfrac{7}{\alpha^{n-m}};\quad\text{where}\quad\gamma:=\dfrac{\log
3}{\log\alpha}.$ (14)

The continued fraction of the irrational number $\gamma$ is :

$[a_{0},a_{1},a_{2},......]=[1,2,3,1,1,2,3,2,4,2,1,11,2,1,11,......]$

and let denote $p_{k}/q_{k}$ its convergent. An inspection using SageMath
gives the following inequality

$4977896525362041575=q_{41}<1.2\times 10^{20}<q_{42}=805929983250536127817.$

Furthermore, $a_{M}:=\max\left\\{a_{i}|i=0,1,...,42\right\\}=161$ Now applying
Lemma 2.2 and properties of continued fractions, we obtain

$|a\gamma-n|>\dfrac{1}{(a_{M}+2)a}.$ (15)

Combining equation (14) and (15), we get

$\dfrac{1}{(a_{M}+2)a}<|a\gamma-n|<\dfrac{7}{\alpha^{n-m}}\Longrightarrow\dfrac{1}{(a_{M}+2)a}<\dfrac{7}{\alpha^{n-m}}\Longrightarrow\alpha^{n-m}<7\cdot(161+2)a<1.3692\cdot
10^{23}.$

Applying $\log$ above and divide by $\log\alpha$, we get :

$(n-m)\leq\dfrac{\log\left(7\cdot 163\cdot a\right)}{\log\alpha}<111.$

To improve the upper bound on $n$, let consider

$z:=a\log 3-n\log\alpha-\log\rho(u)\quad\text{where}\quad\rho=1+\alpha^{-u}.$
(16)

From (9), we have

$|1-e^{z}|<\dfrac{2}{\alpha^{n}}.$ (17)

Since $\Lambda\neq 0$, then $z\neq 0$. Two cases arise : $z<0$ and $z>0$. For
each case, we will apply Lemma 2.1.

  * •

Case 1 : $z>0$

From (17), we obtain $0<z\leq e^{z}-1<\dfrac{2}{\alpha^{n}}.$ Replacing (16)
in the above inequality, we get:

$0<a\log 3-n\log\alpha-\log\rho(n-m)\leq
3^{a}\alpha^{-n}\rho(n-m)^{-1}-1<2\alpha^{-n}$

hence

$0<a\log 3-n\log\alpha-\log\rho(n-m)<2\alpha^{-n}$

and by diving above inequality by $\log\alpha$

$0<a\left(\dfrac{\log
3}{\log\alpha}\right)-n-\dfrac{\log\rho(n-m)}{\log\alpha}<5\cdot\alpha^{-n}.$
(18)

Taking, $\gamma:=\dfrac{\log
3}{\log\alpha},\quad\mu:=-\dfrac{\log\rho(n-m)}{\log\alpha},\quad A:=5,\quad
B:=\alpha$, inequality (18) becomes

$0<a\gamma-n+\mu<AB^{-n}.$

Since $\gamma$ is irrational, we are now ready to apply lemma 2.1 of Dujella
and Pethö on (18) for $n-m\in\\{1,...,111\\}$.

Since $a\leq 1.2\times 10^{20}$ from lemma 3.1, we can take $M=1.2\times
10^{20},$ and we get

$n<\dfrac{\log(Aq/\varepsilon)}{\log B}\quad\text{where}\quad q>6M$

and $q$ is the denominator of the convergent of the irrational number $\gamma$
such that $\varepsilon:=||\mu q||-M||\gamma q||>0$. With the help of SageMath,
with conditions $z>0,\quad\text{and}\quad(n,m,a)$ a possible zero of (1), we
get $n<112$ which contradicts our assumption $n>200$. Then it is false.

  * •

Case 2 : $z<0$

Since $n>200$, then $\frac{2}{\alpha^{n}}<\frac{1}{2}$. Hence (17) implies
that $|1-e^{|z|}|<2$. Also, since $z<0$, we have

$0<|z|\leq e^{|z|}-1=e^{|z|}|e^{|z|}-1|<\dfrac{4}{\alpha^{n}}.$

Replacing (16) in the above inequality and dividing by $\log 3$, we get:

$0<n\left(\dfrac{\log\alpha}{\log 3}\right)-a+\dfrac{\rho(n-m)}{\log
3}<\dfrac{4}{\log 3}\cdot\alpha^{-n}<4\cdot\alpha^{-n}$ (19)

In order to apply lemma 3.1 on (19) for $n-m\in\\{1,2,...,111\\}$, let’s take
again $M=1.2\times 10^{20}$. With the help of SageMath, with conditions
$z<0,\quad\text{and}\quad(n,m,a)$ a possible zero of (1), we get $n<111$ which
contradicts our assumption $n>200$. Then it is false.

This completes the proof of our main result (Theorem 3.1). ∎

## Acknowledgments

The authors thank the professor Maurice Mignotte for his remarks and
diponibility.

## References

  * [1] A. Baker and G. Wüstholz, Logarithmic and Diophantine Geometry, Vol 9 New Mathematical Monographs (Cambridge University Press), 2007.
  * [2] A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. 20 (1969), 129-137.
  * [3] J. J. Bravo and F. Lucas, Powers of Two as Sums of Two Lucas Numbers, Journal of Integers sequences, Vol.17(2014) Article 14.8.3
  * [4] Bravo, J. J. and Luca, F. On the Diophantine Equation $F_{n}+F_{m}=2^{a}$, Quaestiones Mathematicae, (2016) 39 (3), 391–400.
  * [5] Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), 969-1018.
  * [6] A. Dujella and A. Pethö, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. 49 (1998), no. 195, 291-306.
  * [7] M. Laurent, M. Mignotte, and Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation, (French) (Linear forms in two logarithms and interpolation determinants), J. Number Theory 55 (1995), 285-321.
  * [8] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180. Translation in Izv. Math. 64 (2000), 1217-1269.

Pagdame TIEBEKABE
Université Cheikh Anta Diop (UCAD), Laboratoire d’Algèbre, de Cryptologie, de
Géométrie Algébrique et Applications (LACGAA)
Dakar, Sénégal

Ismaïla DIOUF
Université Cheikh Anta Diop (UCAD),  Laboratoire d’Algèbre, de Cryptologie, de
Géométrie Algébrique et Applications (LACGAA)
Dakar, Sénégal