# Space-and-time-synchronized simultaneous vehicle tracking/formation using
cascaded prescribed-time control

Peng Wang<EMAIL_ADDRESS>Ziyin Chen<EMAIL_ADDRESS>Xiaobing
Zhang<EMAIL_ADDRESS>School of Energy and Power Engineering, Nanjing
University of Science and Technology, Nanjing 210094, China  Beijing Institute
of Space Mechanics and Electricity, China Academy of Space Technology, Beijing
100093, China

###### Abstract

In this paper, we present a space-and-time-synchronized control method with
application to the simultaneous tracking/formation. In the framework of polar
coordinates, through correlating and decoupling the reference/actual
kinematics between the self vehicle and target, time and space are separated,
controlled independently. As such, the specified state can be achieved at the
predetermined terminal time, meanwhile, the relative trajectory in space is
independent of time. In addition, for the stabilization before the predesigned
time, a cascaded prescribed-time control theorem is provided as the
preliminary of vehicle tracking control. The obtained results can be directly
extended to the simultaneous tracking/formation of multiple vehicles. Finally,
numerical examples are provided to verify the effectiveness and superiority of
the proposed scheme.

###### keywords:

Synchronization of space and time, cascaded prescribed-time control,
simultaneous arrival

##  1 Introduction

In practice, positioning and tracking problems are of special interest to many
important applications which involve the trajectory/path following of
autonomous vehicles [1, 2], formation/containment control of multiple
autonomous vehicles [3, 4] and so on. It is noted that there exist two
important scales in positioning and tracking: space and time. Existing
researches usually focus on when to reach a designated location, or what kind
of trajectory in space, but few considerations involve both of them at the
same time, which inspires this work.

First of all, a concept named ’synchronization of space and time’ is proposed.

_Synchronization of space and time:_ We summarize the following three
characteristics for the synchronization of space and time:

  1. (i)

Time and space are separated and can be controlled independently;

  2. (ii)

In the time control, the specified position can be reached at the
predetermined instant;

  3. (iii)

In the space control, the trajectory in space is independent of time, that is,
the trajectory in space is fixed regardless of the terminal time.

In the following, two aspects are discussed.

_Prescribed-time/distance control:_ Finite/fixed /prescribed-time control has
drawn increasing attention in recent years in view of the fact that many
practical applications require severe time response constraints. Homogeneity
property gives a useful analysis for HOSM (high-order sliding mode) control,
and if an asymptotically stable system is homogeneous of a negative degree,
then it is finite-time stable to a sliding manifold (see [5, 6, 7]). However,
the settling time in these finite-time methods grows unbounded when initial
conditions tend to infinity, and the initial conditions are hard to acquire in
some practical situations [8]. Fixed-time stability requires a controller
(observer) to provide some desired control (observation) precision at a given
time, independent of the initial conditions, and the settling time is subject
to an upper bound but varies due to uncertainties and nonlinearities [9, 10].
The reviews upon the fixed-time stability and its applications can be found in
[11]. However, the fixed-time settling time can not be preassigned arbitrarily
since its upper bound is subject to certain restrictions [12]. Moreover,
prescribed-time control employs a scaling of state by a function of time that
grows unbounded to the terminal time and yields the regulation in prescribed
finite time(see [13, 14]). Furthermore, regarding the prescribed-distance
control problem, the system regulation can be achieved before the prescribed
distance is reached. The conversion between the prescribed-distance control
and the existing prescribed-time control only requires the distance-driven
transformation for the differential dynamic system model, which will be
answered in this paper.

_Simultaneity in time:_ The finite-time distributed consensus has been a
popular topic in recent years, which is known as one of the most important
issues in cooperative control[15, 16, 14]. However, for some classes of
practical systems, this finite- time convergence is not enough operating, for
example, for a service robot hand, its joints are usually required to reach
the desired angle at the same time; the proximity operations require that the
pursuer perform the translational and rotational maneuvers with respect to the
target simultaneously [17]; for the multi-missile interception active defense
[18, 19], multiple missiles are expected to strike the target simultaneously.
In this regard, simultaneity in time for multi-agent systems are of great
importance. The works in [20, 21] proposed the time-synchronized consensus
control, focusing on how to design a controller which drives all the elements
of all the agent states to consensus at the same time; and fixed-time-
synchronized consensus, where the upper bound of the synchronized settling
time is independent of the initial states of multi-agent systems.

_Contributions of this work:_ The main contributions are stated as following

  1. (i)

In the framework of polar coordinates, given a reference relative trajectory,
the developed distance-driven controller enables the prescribed-distance
stability and achieves the separation between space and time. In comparison,
few of the existing works consider the distance-driven transformation about
the tracking problem. Moreover, the work in [19] employs a different distance-
driven transformation and obtains the travelling range in advance, while the
velocity value is set fixed and the target is assumed stationary;

  2. (ii)

Through correlating and decoupling the reference/actual kinematics between the
self vehicle and target, the control variables related to velocity and
inclination angle are controlled respectively to form synchronization between
the arrival time and path trajectory. This kind of correlating and decoupling
operations give the freedom to the velocity control in polar coordinates,
which is proposed for the first time, to the best of the authors’ knowledge;

  3. (iii)

A cascaded prescribed-time control theorem is proposed which enables the
prescribed-time stabilization for the general strict-feedback systems.
Compared with the prescribed-time control [13, 14], the proposed theorem
eliminates restrictions on the system model and allows the uncertainties and
the remaining-time-related interconnections in each subsystem; the cascaded
prescribed-time tracking problem can be directly solved through the separately
prescribed-time control design of each subsystem;

  4. (iv)

The synchronization of space and time control enables the simultaneous arrival
of multiple vehicles which reach the designated position or achieve formation
simultaneously. Compared with the existing simultaneous control work, the
target in the simultaneous tracking/formation problem does not have to be set
stationary as in [18, 19], and the simultaneous property can be directly
derived from the proposed space-and-time-synchronized control.

_Notation_ : Throughout this paper, ${\mathcal{R}}$ denotes the set of real
numbers, ${\mathcal{R}}^{n\times m}$ represents the set of $n\times m$ real
matrix, $\mathcal{N}_{+}$ is the set of positive integers. $(\cdot)^{T}$ is
the transpose of one matrix, diag$(\cdot)$ represents the diagonal matrix.
$\left|\cdot\right|$ is the absolute value of a scalar and
$\left\|\cdot\right\|$ is defined as the Euclidean norm of a vector. Given a
matrix $M$, $M>0$ (or $M<0$) means that $M$ is a positive definite (or
negative definite) matrix, $\lambda_{\text{min}}(M)$ and
$\lambda_{\text{max}}(M)$ denote the the minimum and maximum eigenvalue of
matrix $M$, respectively.

##  2 Formulation of space-and-time-synchronized problem

As the geometry of planer scenario depicted in Fig. 1, the subscripts $s$ and
$t$ denote the self vehicle and target, respectively, the subscripts $0$ and
$f$ represent the initial time and the predetermined time, respectively.
$R\in\mathcal{R}$ represents their relative distance along the line of sight
(LOS) angle $q\in\mathcal{R}$. ${{V}_{s}}\in\mathcal{R}$ and
${{V}_{t}}\in\mathcal{R}$ denote the velocities of the vehicle and target,
respectively, ${{\theta}_{s}}\in\mathcal{R}$ and
${{\theta}_{t}}\in\mathcal{R}$ are their corresponding path angles,
${{\eta}_{s}}\in\mathcal{R}$ and ${{\eta}_{t}}\in\mathcal{R}$ represent the
heading angles. The tracking kinematics between the vehicle and target in the
framework of polar coordinates are modeled as

$\displaystyle\dot{R}=-{{V}_{s}}\cos{{\eta}_{s}}+{{V}_{t}}\cos{{\eta}_{t}},$
(1) $\displaystyle
R\dot{q}={{V}_{s}}\sin{{\eta}_{s}}-{{V}_{t}}\sin{{\eta}_{t}},$
$\displaystyle{{\eta}_{s}}=q-{{\theta}_{s}},\ {{\eta}_{t}}=q-{{\theta}_{t}}$

with the initial condition ${{R}}(0)=R_{0}$, ${{q}}(0)=q_{0}$,
${{\eta}}_{s}(0)=\eta_{s0}$ and ${{V}}_{s}(0)=V_{s0}$. When it comes to a
special case $V_{t}=0$ which means a stationary target point, the kinematics
model (1) is reduced to a positioning problem.

Figure 1: Geometry description

Define the pair $(t,R,q)$: at time $t$, the distance and LOS angle between the
vehicle and target are $R$ and $q$, respectively. We can now formulate the
problem as follows:

###### Problem 1

(Synchronization of space and time) For a vehicle tracking problem (1), how to
ensure the vehicle starting from $(t_{0},R_{0},q_{0})$ eventually realizes
$(t_{f},R_{f},q_{f})$ and gets maintained in $(t,R_{f},q_{f})$ for $t>t_{f}$?

###### Algorithm 1

For the sake of generality, the solution with respect to Problem 1 can be
summarized as

  1. (i)

In the framework of polar coordinates, introduce the reference trajectory
$(R_{d},q_{d})$ for the vehicle-target relative motion, which achieves the
separation of time and space, see Section II.A;

  2. (ii)

Correlate and decouple the reference/actual relative kinematics to obtain the
separated control variables related to velocity $V_{s}$ and inclination angle
$\eta_{s}$, see Section II.B;

  3. (iii)

The travelling time and path trajectory are controlled respectively to form
synchronization between space and time.

###  2.1 Reference relative tracking trajectory

The motion equations for the reference tracking problem are given by

$\displaystyle{{{\dot{R}}}_{d}}=-{{V}_{d}}\cos{{\eta}_{d}},$ (2)
$\displaystyle{{R}_{d}}{{{\dot{q}}}_{d}}={{V}_{d}}\sin{{\eta}_{d}}$

where $R_{d}$ and $q_{d}$ denote the distance and LOS angle of the reference
trajectory, respectively, $V_{d}$ and $\eta_{d}$ are the corresponding
velocity and heading angle, respectively. Their initial conditions are set
${{R}_{d}}(0)=R_{0}$, ${{q}_{d}}(0)=q_{0}$, ${{\eta}_{d}}(0)=\eta_{d0}$ and
${{V}_{d}}(0)=V_{d0}$.

Define a new distance-scaled variable ${{{r}}_{d}}$ and a prescribed value
${{{r}}_{f}}$

$\displaystyle{{{r}}_{d}}=\ln(R_{0}-{R}_{f1})-\ln({{R}_{d}}(t)-{R}_{f1}),\ \
R_{d}\in\left(R_{f},R_{0}\right],$
$\displaystyle{{{r}}_{f}}=\ln(R_{0}-{R}_{f1})$ (3)

where ${R}_{f1}=R_{f}-1$. It is obvious that $r_{d}=r_{f}$ when $R_{d}=R_{f}$.
The relationship between the distance-scaled variable ${r}_{d}$ and the actual
time $t$ is subject to

${{{{\dot{r}}}_{d}}}={{{V}_{d}}\cos{{\eta}_{d}}}/({{{R}_{d}}-{R}_{f1}}),\ \
R_{d}\in\left(R_{f},R_{0}\right].$ (4)

Taking derivative with respect to the distance-scaled variable $r_{d}$, rather
than the time variable $t$, yields the rewritten dynamics: when
$r_{d}\in\left[0,r_{f}\right)$,

$\displaystyle\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}=\frac{{{R}_{d}}-{R}_{f1}}{{{R}_{d}}}\tan{{\eta}_{d}},$
(5)
$\displaystyle\frac{{{\text{d}}^{2}}{{q}_{d}}}{\text{d}r_{d}^{2}}=\frac{R_{d}-{R}_{f1}}{{{V}_{d}}{{\cos}}{{\eta}_{d}}}\left[\frac{R_{d}-{R}_{f1}}{R_{d}\cos^{2}{\eta_{d}}}\frac{\text{d}{{\eta}_{d}}}{\text{d}t}-\frac{{R}_{f1}V_{d}\sin{\eta_{d}}}{R_{d}^{2}}\right].$

###### Theorem 1

Consider the system in form of (2).

(i) When $R_{d}\in[0,r_{f})$, the control law of $\eta_{d}$ is designed
according to (1). The system state pair $(q_{d},\eta_{d})$ asymptotically
converges to the desired value $(q_{f},0)$ before ${{r}_{d}}$ approaching the
prescribed value ${{r}_{f}}$.

$\displaystyle{\dot{\eta}_{d}}=$
$\displaystyle-\frac{R_{d}{{V}_{d}}{{\cos}^{3}}{{\eta}_{d}}}{({{R}_{d}}-{R}_{f1})^{2}}\left[\frac{k_{d2}}{{{{r}}_{f}}-{{{r}}_{d}}}\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}+\frac{k_{d1}({{q}_{d}}-{{q}_{f}})}{{{({{{r}}_{f}}-{{{r}}_{d}})}^{2}}}\right]$
$\displaystyle+\frac{{R}_{f1}V_{d}\cos^{2}{\eta_{d}}\sin{\eta_{d}}}{(R_{d}-{R}_{f1})R_{d}}.$
(6)

When letting ${{k}_{d1}}=N_{1}N_{2},\ {{k}_{d2}}=N_{1}+N_{2}-1$, $N_{1}<N_{2}$
and $N_{1},N_{2}\in\mathcal{N}_{+}$, the travel range can be calculated as

$\displaystyle
S_{d}=\int_{R_{f}}^{{{R}_{0}}}{\sqrt{1+{{(\tan{{\eta}_{d}})}^{2}}}}\text{d}R_{d}.$
(7)

In this sequel, $S_{d}$ is irrespective of $V_{d}$, the travel time $t_{f}$ is
only related to the velocity $V_{d}$. For any constant velocity ${{V}_{d}}$,
the travel time is ${{t}_{f}}={S_{d}}/{V_{d}}$.

(ii) When $R_{d}$ approaches $R_{f}$, through $\dot{V}_{d}=0$ and
$\dot{\eta}_{d}=0$, the system state pair $(R_{d},q_{d},\eta_{d})$ gets
maintained in the desired value $(R_{f},q_{f},0)$.

Proof. See Appendix A. $\square$

###### Remark 1

Here comes the motivation of introducing the proposed distance-scaled variable
$r$ and doing the $r$-related dynamics transformation:

  1. (i)

The system dynamics (5) with respect to the introduced distance-scaled
variable $r_{d}$ is independent of time and directly linked to the terminal
distance $R_{f}$. As a result, this proposed distance-related transformation
is more intuitive to the distance tracking problem like the prescribed-
distance control;

  2. (ii)

This kind of transformation (2.1) together with the controller (1) enables the
achievement of regulation objective before $r_{d}=r_{f}$. The resultant
irrelevance between $S_{d}$ and $V_{d}$ creates a separation between space and
time, which provides the possibility for the space-and-time-synchronized
control;

  3. (iii)

Such a state transformation brings $\eta_{d}$ into a state-related form, i.e.
$\tan\eta_{d}=\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}\frac{R_{d}}{R_{d}-R_{f1}}$,
such that the specific expression $\eta_{d}$ in space can be directly obtained
in (36). Some heading-angle-constrained problems can be handled through
appropriate parameter selection when considering the field-of-view limits.

$\displaystyle
M=\left[\begin{matrix}-\cos{{\eta}_{d}}&{{V}_{s}}\left(\cos{{\eta}_{d}}\int_{0}^{1}{\sin(\tau{{\eta}_{e}})\text{d}\tau}+\sin{{\eta}_{d}}\int_{0}^{1}{\cos(\tau{{\eta}_{e}})\text{d}\tau}\right)\\\
\sin{{\eta}_{d}}&{{V}_{s}}\left(\cos{{\eta}_{d}}\int_{0}^{1}{\cos(\tau{{\eta}_{e}})\text{d}\tau}-\sin{{\eta}_{d}}\int_{0}^{1}{\sin(\tau{{\eta}_{e}})\text{d}\tau}\right)\\\
\end{matrix}\right],\ \ G=\left[\begin{matrix}{{V}_{t}}\cos{{\eta}_{t}}\\\
-{{V}_{t}}\sin{{\eta}_{t}}-{{V}_{d}}\sin{{\eta}_{d}}({{{R}_{e}}}/{R_{d}})\\\
\end{matrix}\right].$

###  2.2 Kinematic modelling

To correlate the reference/actual relative kinematics, we define the following
error variables, i.e., distance error ${{R}_{e}}=R-{{R}_{d}}$, LOS angle error
${{q}_{e}}=q-{{q}_{d}}$, angle ${{\eta}_{e}}={{\eta}_{s}}-{{\eta}_{d}}$,
velocity error ${{V}_{e}}={{V}_{s}}-{{V}_{d}}$.

Taking time derivative of ${{R}_{e}}$ along the solutions of (1) and (2) leads
to

$\displaystyle{{{\dot{R}}}_{e}}$
$\displaystyle=-{{V}_{s}}\cos{{\eta}_{s}}+{{V}_{s}}\cos{{\eta}_{d}}-{{V}_{s}}\cos{{\eta}_{d}}$
$\displaystyle\ \ \ +{{V}_{d}}\cos{{\eta}_{d}}+{{V}_{t}}\cos{{\eta}_{t}}$
$\displaystyle=-{{V}_{s}}(\cos({{\eta}_{d}}+{{\eta}_{e}})-\cos{{\eta}_{d}})-{{V}_{e}}\cos{{\eta}_{d}}+{{V}_{t}}\cos{{\eta}_{t}}$
$\displaystyle=-{{V}_{e}}\cos{{\eta}_{d}}+{{V}_{t}}\cos{{\eta}_{t}}$
$\displaystyle\ \ \
+{{V}_{s}}\left[\cos{{\eta}_{d}}\frac{1-\cos{{\eta}_{e}}}{{{\eta}_{e}}}+\sin{{\eta}_{d}}\frac{\sin{{\eta}_{e}}}{{{\eta}_{e}}}\right]{{\eta}_{e}}.$
(8)

A similar operation for ${{q}_{e}}$ brings

$\displaystyle R{{{\dot{q}}}_{e}}=$
$\displaystyle{{{V}_{s}}\sin{{\eta}_{s}}}-{{{V}_{s}}\sin{{\eta}_{d}}}+{{{V}_{s}}\sin{{\eta}_{d}}}-{{{V}_{d}}\sin{{\eta}_{d}}}$
$\displaystyle-{{{V}_{d}}\sin{{\eta}_{d}}}({R_{e}}/{R_{d}})-{{{V}_{t}}\sin{{\eta}_{t}}}$
$\displaystyle=$
$\displaystyle{{{V}_{e}}\sin{{\eta}_{d}}}+{V_{s}}\left[\sin{{\eta}_{d}}\frac{\cos{{\eta}_{e}}-1}{\eta_{e}}+\cos{{\eta}_{d}}\frac{\sin{{\eta}_{e}}}{\eta_{e}}\right]{\eta_{e}}$
$\displaystyle-{{{V}_{d}}\sin{{\eta}_{d}}}({R_{e}}/{R_{d}})-{{{V}_{t}}\sin{{\eta}_{t}}}.$
(9)

Invoking the fact

$\displaystyle\frac{1-\cos{{\eta}_{e}}}{{\eta}_{e}}=\int_{0}^{1}{\sin(\tau{{\eta}_{e}})\text{d}\tau},\
\
\frac{\sin{{\eta}_{e}}}{{\eta}_{e}}=\int_{0}^{1}{\cos(\tau{{\eta}_{e}})\text{d}\tau}$
(10)

and combing the above dynamics of ${{R}_{e}}$ and ${{q}_{e}}$, we readily
obtain the kinematic model of vehicle tracking problem

$\displaystyle\left[\begin{matrix}{{{\dot{R}}}_{e}}\\\ R{{{\dot{q}}}_{e}}\\\
\end{matrix}\right]=$ $\displaystyle
M({{\eta}_{s}},{{\eta}_{e}},V_{s})\left[\begin{matrix}{{V}_{e}}\\\
{{\eta}_{e}}\\\
\end{matrix}\right]+G(R_{e},R_{d},{{V}_{t}},{{\eta}_{t}},{{V}_{d}},{{\eta}_{d}})$
(11)

where the detailed expressions of $M({{\eta}_{s}},{{\eta}_{e}},V_{s})$ and
$G(R_{e},R_{d},{{V}_{t}},{{\eta}_{t}},{{V}_{d}},{{\eta}_{d}})$ are at the top
of next page.

###### Remark 2

With regard to (10), a special case is when $\eta_{e}=0$, through the
L’Hospital rule and limit of indeterminate form, one obtains
$(1-\cos\eta_{e})/\eta_{e}=0$ and $\sin\eta_{e}/\eta_{e}=1$.

###### Remark 3

As for the parameter matrix $M({{\eta}_{s}},{{\eta}_{e}},V_{s})$, its
determinant is
$\left|M({{\eta}_{s}},{{\eta}_{e}},V_{s})\right|=-{{V}_{s}}(\sin\eta_{e}/\eta_{e})<0$,
which contributes to the full rank of $M({{\eta}_{s}},{{\eta}_{e}},V_{s})$ and
enables the later matrix inversion operation.

###  2.3 Dynamic modelling

Define the control variable
$u={{\left[\begin{matrix}{{u}_{V}}&{{u}_{\theta}}\\\
\end{matrix}\right]}^{T}}$ where ${{u}_{V}}$ is the tangential acceleration
and ${{u}_{\theta}}$ is the lateral acceleration. The vehicle’s dynamic model
is expressed by

$\displaystyle{{{\dot{V}}}_{s}}={{u}_{V}}+d_{V},$ (12)
$\displaystyle{{{\dot{\theta}}}_{s}}={u_{\theta}}/{{{V}_{s}}}+d_{\theta}$

where $d_{V}$ and $d_{\theta}$ are the external disturbances.

As such, the compact system model combing (11) and (12) is written as

$\displaystyle\left[\begin{matrix}{\dot{R}_{e}}\\\ {\dot{q}_{e}}\\\
\end{matrix}\right]=\bar{M}\left[\begin{matrix}V_{s}\\\ {\eta}_{s}\\\
\end{matrix}\right]-\bar{M}\left[\begin{matrix}V_{d}\\\ \eta_{d}\\\
\end{matrix}\right]+\bar{G},$ (13a)
$\displaystyle\left[\begin{matrix}{{{\dot{V}}}_{s}}\\\
{{{\dot{{\eta}}}}_{s}}\\\
\end{matrix}\right]={B}\left[\begin{matrix}{{u}_{V}}\\\ {{u}_{\theta}}\\\
\end{matrix}\right]+H+{F}$ (13b)

where $\bar{M}(R,{{\eta}_{s}},{{\eta}_{e}},V_{s})=\text{diag}(1,1/R)M$,
$\bar{G}(R,R_{e},R_{d},{{V}_{t}},{{\eta}_{t}},{{V}_{d}},{{\eta}_{d}})=\text{diag}(1,1/R)G$
, $B(V_{s})=\text{diag}(1,-1/V_{s})$ ,
$H(\dot{q},\dot{R},R,\eta_{s})=\left[\begin{matrix}0&\dot{q}\end{matrix}\right]^{T},{F}(d_{V},d_{\theta})=\left[\begin{matrix}d_{V}&-d_{\theta}\end{matrix}\right]^{T}$
.

##  3 Prescribed-time Control Design

For the prescribed-time control design, we reach the following consensus: as
long as $R_{e}$ and $q_{e}$ converge to 0 before the final time $t_{f}$
(before $R$ converges to ${R}_{f}$), the vehicle $(R,q)$ reaches the specified
state $(R_{f},q_{f})$ at the time $t_{f}$.

###### Problem 2

(Prescribed-time stabilization) For some kind of strict feedback systems like
system (13a)(13b), how to ensure the system stability within the prescribed
time $t\in\left[0,t_{f}\right)$ ?

###### Algorithm 2

The corresponding solution with respect to Problem 2 is summarized as

  1. (i)

The prescribed-time stability of a single first-order system can be achieved
by employing a scaling of the state by a function of $t_{f}-t$;

  2. (ii)

During the backstepping design procedure, the above first-order prescribed-
time control design is repeated for each subsystem. In this sequel, a cascaded
system consisting of two prescribed-time stable subsystems is obtained;

  3. (iii)

The prescribed-time stability of the cascaded system is addressed and extended
to cascade cases of multiple subsystems.

###  3.1 Prescribed-time stabilization of cascaded systems

Before processing the prescribed-time control, a cascaded prescribed-time
theorem is presented as a preliminary. Cascades-based control essentially uses
the designed control law to make the closed-loop system in a cascaded
structure, which usually has the advantage of reducing the complexity of the
controller and the difficulty of stability analysis.

Consider a kind of cascaded system described by

$\displaystyle{{\Sigma}_{i}}:\dot{x}_{i}(t)={{f}_{i}}(t,x_{i},{{\Delta}_{x,i}})+g_{i}(t,x_{i},x_{i+1})x_{i+1},$
(14)
$\displaystyle{{\Sigma}_{n}}:\dot{x}_{n}(t)={{f}_{n}}(t,x_{n},{{\Delta}_{x,n}}),i=1,\cdots,n-1$

where $x_{i}(t)\in{\mathcal{R}^{n_{i}}}$ is the system state of subsystem
${{\Sigma}_{i}}$. ${{f}_{i}}(t,x_{i},{{\Delta}_{x,i}})$, $g(t,x_{i},x_{i+1})$
are continuous in their arguments, and locally Lipschitz in $x_{i}$,
$(x_{i},x_{i+1})$ respectively. For a simplified presentation, we view the
cascaded system as the subsystem ${{\hat{\Sigma}}_{i}}$ perturbed by the
output of subsystem ${{\Sigma}_{i+1}}$.

${{\hat{\Sigma}}_{i}}:\dot{x}_{i}(t)={{f}_{i}}(t,x_{1},{{\Delta}_{x,i}}).$
(15)

###### Definition 1

[22] A continuous function $\alpha:\left[0,a\right)\to\left[0,\infty\right)$
belongs to class $\mathcal{K}$ if it is increasing and $\alpha(0)=0$, if
moreover $\alpha(a)\to\infty$ as $a\to\infty$, it belongs to class
$\mathcal{K}_{\infty}$. A continuous function
$\beta:\left[0,a\right)\to\left[0,\infty\right)$ belongs to class
$\mathcal{L}$ if it is decreasing and $\beta(s)\to 0$ as $s\to\infty$. Again,
a continuous function
$\gamma:\left[0,a\right)\times\left[0,a\right)\to\left[0,\infty\right)$
belongs to class $\mathcal{KL}$ if the mapping $\gamma(r,s)$ belongs to class
$\mathcal{K}$ with respect to $r$ when fixed $s$, and the mapping
$\gamma(r,s)$ decreases along $s$, i.e., $\gamma(r,s)\to 0$ as $s\to\infty$,
when fixed $r$.

###### Definition 2

Consider the system $\dot{x}=f(t,x)$, where $f(t,x)$ is piecewise continuous
in $t$ and locally Lipschitz in $x$. The system is globally uniformly
prescribed-time stable (GUPTS), if there exist a class $\mathcal{KL}$ function
$\gamma(\cdot,\cdot)$ and a settling-time $t_{f}$ such that for any initial
values $x({{t}_{0}})$:

$\left\|x(t)\right\|\leq\gamma(\left\|x({{t}_{0}})\right\|,\tau(t,t_{f})),\ \
\ {{t}_{0}}\leq t<{{t}_{f}}$ (16)

where $\tau(t,t_{f})\to\infty$ as $t\to t_{f}$, from which one obtains the
strictly prescribed-time convergence of $x$: $x(t)\to 0$ as $t\to{{t}_{f}}$.

###### Theorem 2

Consider the cascaded system (14) with $n$ subsystems. The globally uniformly
prescribed-time stability (GUPTS) of the subsystem $\Sigma_{i}$,
$i=1,\cdots,n$ can be obtained, if the following assumptions hold:

(i) Assumption on subsystem ${{\hat{\Sigma}}_{i}}$, $i=1,\cdots,n$: the
subsystem ${{\hat{\Sigma}}_{i}}$ is GUPTS, and there exists a continuously
differentiable Lyapunov function ${{V}_{i}}(t,x_{i})$ such that

$\displaystyle{{\alpha}_{i1}}(\tau){{\left\|x_{i}\right\|}^{2}}\leq{{V}_{i}}(t,x_{i})\leq{{\alpha}_{i2}}(\tau){{\left\|x_{i}\right\|}^{2}},$
$\displaystyle{{{\dot{V}}}_{i}}={\partial{{V}_{i}}}/{\partial
t}\;+({\partial{{V}_{i}}}/{\partial
x_{i}}\;){{f}_{i}}(t,x_{i},{{\Delta}_{x,i}})$ $\displaystyle\ \ \ \
\leq-{{\alpha}_{i3}}(\tau){{\left\|x_{i}\right\|}^{2}}+{{b}_{i}}({{\alpha}^{2}_{i4}}(\tau)/\alpha_{i3}(\tau)){{\left\|{{\Delta}_{x,i}}\right\|}^{2}},$
$\displaystyle\left\|{\partial{{V}_{i}}}/{\partial
x_{i}}\;\right\|\leq{{\alpha}_{i4}}(\tau)\left\|x_{i}\right\|,$
$\displaystyle{{\alpha}_{i3}}(\tau)={{\alpha}_{i4}}(\tau)\times{{\alpha}_{i5}}(\tau)$
(17)

where ${\alpha}_{ij}(\cdot)$, $j=1,\ldots,5$ are class $\mathcal{K}_{\infty}$
functions, $b_{i}>0$.

(ii) Assumption on interconnection $g_{i}(t,x_{i},x_{i+1})$, $i=1,\cdots,n-1$:
there exist a class $\mathcal{K}_{\infty}$ function $\alpha_{i6}(\cdot)$ and a
positive constant $c_{g,i}$ satisfying

$\displaystyle\left\|({\partial{{V}_{i}}}/{\partial
x_{i}}\;)g_{i}(t,x_{i},x_{i+1})\right\|\leq{{\alpha}_{i6}}(\tau)c_{g,i}\left\|x_{i}\right\|,$
(18)
$\displaystyle{{\lim}_{t\to{{t}_{f}}}}{{{\alpha}_{i6}}(\tau)/(\alpha_{i3}(\tau)\alpha_{i+1,5}(\tau))}=0.$

Proof. See Appendix B. $\square$

Here we give an example which has been introduced in [13] for justification of
assumption (2). Consider the first-order system governed by
$\dot{x}=b(x,t)u+f(x,t)$ where $x,u\in\mathcal{R}$ and
$0<\underline{b}\leq\left|b(x,t)\right|$, $\left|f(x,t)\right|\leq
d(t)\psi(x)$. Employing the residual-time-based scaling function
$\mu(t-{{t}_{0}})=\frac{{{T}^{1+m}}}{{{(T+{{t}_{0}}-t)}^{1+m}}},t\in[{{t}_{0}},{{t}_{0}}+T]$
with positive integer $m$, the controller was derived
$u=-\frac{1}{{\underline{b}}}(k+\lambda\psi{{(x)}^{2}}+\frac{1+m}{T})\mu(t-{{t}_{0}})x$
with $k,\lambda>0$. Following the Lyapunov function candidate
$V={{{\omega}^{2}}}/{2}\;$, $\omega=\mu(t-{{t}_{0}})x$ and the presentation in
[13], upon applying the control law, one obtains $\dot{V}\leq-2k\mu
V+\frac{\mu}{4\lambda}{{d}^{2}}$. In accordance with the proposed assumption,
it is obvious ${{\alpha}_{i1}}(\tau)={{a}_{1}}{{\mu}^{2}}$$(0<{{a}_{1}}<0.5)$,
${{\alpha}_{i2}}(\tau)={{a}_{2}}{{\mu}^{2}}$$({{a}_{2}}>0.5)$,
${{\alpha}_{i3}}(\tau)=k{{\mu}^{3}}$, ${{\alpha}_{i4}}(\tau)=\mu^{2}$,
${{\alpha}_{i5}}(\tau)=k{{\mu}}$, ${{\alpha}_{i6}}(\tau)=\mu$,
${{b}_{i}}={1}/{(4\lambda}\;)$, all of which satisfy the condition in the
applied assumption.

###### Remark 4

A strict feedback form of the cascaded system (14) can be obtained through the
backstepping procedure

$\displaystyle{{\Sigma}_{i}}:\dot{x}_{i}(t)=-k_{i}(\tau)x_{i}+g_{i}(t,x_{i},x_{i+1})x_{i+1}+{{\Delta}_{x,i}},$
(19)
$\displaystyle{{\Sigma}_{n}}:\dot{x}_{n}(t)=-k_{n}(\tau)x_{n}+{{\Delta}_{x,n}},i=1,\cdots,n-1$

where ${k}_{i}(\cdot)$, $i=1,\ldots,n$ are class $\mathcal{K}_{\infty}$
functions. In principle, each subsystem can achieve the prescribed-time
convergence due to the high-gain nature of $k_{i}(\tau)$ at the prescribed
time $t_{f}$. Under the premise of (18), since the convergence of $\Sigma_{i}$
is influenced by $\alpha_{i4}\|x_{i}\|\|x_{i+1}\|$ or
$\alpha_{i4}\|x_{i}\|^{2}\|x_{i+1}\|$, an ideal manner would be for $x_{i+1}$
to converge to zero before $\alpha_{i4}$ approaches infinity, which inspires
the design $k_{i+1}(\tau)>k_{i}(\tau)$.

###### Remark 5

In the high-order prescribed-time control proposed in [13], the system model
is in a standard integral form which limits that no uncertainties occur in the
$x_{i}$’s dynamics. The reason lies in the repeated differentiation of the
time-scaling transformation $\mu(t)x_{1}$ where $\mu(t)$ is a time-varying
function which goes to infinity when $t$ approaches $t_{f}$, as the form.
However, the proposed cascaded prescribed-time control scheme in Theorem 2 has
the distinguished features:

  1. (i)

Eliminate restrictions on the system model, and allow the uncertainties in
$x_{i}$’ dynamics;

  2. (ii)

Allow the remaining-time ($t_{f}-t$)-related interconnection
$g_{i}(t,x_{i},x_{i+1})$ appear in $x_{i}$’ dynamics;

  3. (iii)

Each subsystem can also be individually designed for its prescribed-time
stability performance.

###  3.2 Prescribed-time Controller design

For the sake of clarity, the following prescribed-time control design borrows
from the techniques of backstepping and the overall dynamics are organized
into a cascaded form. Invoking the definition of $\tau(t,t_{f})$ in Definition
2, let a user-defined $\tau(t,t_{f})$ satisfy $\tau(0,t_{f})=1$. Introduce now
a time-varying scaling function of $t_{f}-t$ with ${{h}}>0$

${{\mu}}(t,h)=\left\\{\begin{matrix}\tau^{h}(t,t_{f}),\ \ \
t\in\left[0,{{t}_{f}}\right)\\\ 1,\ \ \ \ \ \ \ \ \ \ \ \
t\in\left[{{t}_{f}},\infty\right)\\\ \end{matrix}\right.$ (20)

whose derivative with respect to $t$ yields

$\displaystyle{\dot{\mu}}(t,h)=\left\\{\begin{matrix}h{{\mu}}(t,h)(\dot{\tau}/\tau),\
\ t\in\left[0,{{t}_{f}}\right)\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
t\in\left[{{t}_{f}},\infty\right)\\\ \end{matrix}\right.$ (21)
$\displaystyle\dot{\tau}(t,t_{f})=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
t\in\left[{{t}_{f}},\infty\right)$

For simplicity, define $\mu_{1}(t)=\mu(t,h_{1})$, $\mu_{2}(t)=\mu(t,h_{2})$
with $h_{1},h_{2}>0$.

###### Theorem 3

Consider the kinematic model (13a) (13b). Assume there exists a positive
$\sigma$ satisfying ${{{F}}^{T}}{F}\leq\sigma^{2}$, $\sigma>0$ due to the
boundedness of $d_{V}$ and $d_{\theta}$. Choose the controller

$\displaystyle\left[\begin{matrix}\alpha_{V}^{*}\\\ \alpha_{\eta}^{*}\\\
\end{matrix}\right]=-({{k}_{1}}\mu_{1}+\frac{h_{1}\dot{\tau}}{\tau})\left[\begin{matrix}{{R}_{e}}\\\
R{{q}_{e}}\\\ \end{matrix}\right]-{G}+{M}\left[\begin{matrix}{{V}_{d}}\\\
{{\eta}_{d}}\\\ \end{matrix}\right],$ (22a)
$\displaystyle\left[\begin{matrix}{V}_{s}^{*}\\\ {\eta}_{s}^{*}\\\
\end{matrix}\right]=\left\\{\begin{matrix}{{M}^{-1}}\left[\begin{matrix}\alpha_{V}^{*}&\alpha_{\eta}^{*}\\\
\end{matrix}\right]^{T},\ \ \ t\in\left[0,t_{f}\right)\\\
f_{M}(\left[\begin{matrix}\alpha_{V}^{*}&\alpha_{\eta}^{*}\\\
\end{matrix}\right]),\ \ \ \ \ \ t\in\left[t_{f},\infty\right)\\\
\end{matrix}\right.$ (22b) $\displaystyle\left[\begin{matrix}{{u}_{V}}\\\
{{u}_{\theta}}\\\
\end{matrix}\right]={{{B}}^{-1}}\left\\{-(({{k}_{2}}+{{\lambda}_{2}}){{\mu}_{2}}+\frac{h_{2}\dot{\tau}}{\tau})\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right]-\left[\begin{matrix}\dot{V}_{s}^{*}\\\
\dot{{\eta}}_{s}^{*}\\\ \end{matrix}\right]-H\right\\}$ (22c)

where ${{k}_{1}}>0$, ${{k}_{2}}>0,{{\lambda}_{2}}>0$ and the function

$f_{M}(\left[\begin{matrix}\alpha_{V}^{*}&\alpha_{\eta}^{*}\\\
\end{matrix}\right])=\left[\begin{matrix}\|\left[\begin{matrix}\alpha_{V}^{*}&\alpha_{\eta}^{*}\\\
\end{matrix}\right]\|\\\ \text{atan}(-\alpha_{V}^{*}/\alpha_{\eta}^{*})\\\
\end{matrix}\right].$

Define the tracking errors ${{e}_{V}}={{{V}}_{s}}-{V}_{s}^{*}$,
${{e}_{\eta}}={{{\eta}}_{s}}-{\eta}_{s}^{*}$. Consequently, the closed-loop
system is prescribed-time stable, which means,
$({{R}_{e}},{{q}_{e}},e_{V},e_{\eta})$ converges to the origin before
$t=t_{f}$. Furthermore, one obtains

  1. (i)

when $R_{f}=0$, the choices of $\mu_{1}(t)$ and $\mu_{2}(t)$ should satisfy
${{\lim}_{t\to{{t}_{f}}}}(t_{f}-t){\mu_{1}\mu_{2}}=\infty$;

  2. (ii)

when $R_{f}>0$, the formation gets maintained in $(R_{f},q_{f})$ through
$V_{df}=0$, $\dot{\eta}_{d}=0$ and $\dot{V}_{d}=0$ for $t\geq t_{f}$.

Proof. The proof is divided into four step:

_Step 1._ Prescribed-time convergences of $R_{e}$ and $q_{e}$ under
$V_{s}^{*}$ and $\eta_{s}^{*}$ in (22b) before $t=t_{f}$.

With the controller (22b), the error dynamics is obtained as

$\left[\begin{matrix}{\dot{R}_{e}}\\\ {\dot{q}_{e}}\\\
\end{matrix}\right]=-({{k}_{1}}\mu_{1}+\frac{h_{1}\dot{\tau}}{\tau}))\left[\begin{matrix}{{R}_{e}}\\\
{{q}_{e}}\\\ \end{matrix}\right].$ (23)

Consider the Lyapunov candidate ${{V}_{1}}(r)=x_{1}^{T}{{x}_{1}}$ and
${{W}_{1}}(r)=\mu_{1}^{2}x_{1}^{T}{{x}_{1}}=\omega_{1}^{T}{{\omega}_{1}}$ with
${{x}_{1}}={{\left[\begin{matrix}{{R}_{e}}&{{q}_{e}}\\\
\end{matrix}\right]}^{T}}$ and
${{\omega}_{1}}={{\mu}_{1}}{{\left[\begin{matrix}{{R}_{e}}&{{q}_{e}}\\\
\end{matrix}\right]}^{T}}$. The derivative of ${{W}_{1}}(r)$ along (23) is

$\displaystyle{\dot{W}_{1}}(t)$
$\displaystyle=2\omega_{1}^{T}\left[{\dot{\mu}_{1}}{{x}_{1}}-({{k}_{1}}\mu_{1}+{\dot{\tau}}/{\tau}){{\mu}_{1}}{{\omega}_{1}}\right]$
(24)
$\displaystyle=2\omega_{1}^{T}\left[h_{1}(\dot{\tau}/\tau){{\omega}_{1}}-({{k}_{1}}\mu_{1}+h_{1}{\dot{\tau}}/{\tau}){{\omega}_{1}}\right]$
$\displaystyle\leq-2{{k}_{1}}{{\mu}_{1}}{{W}_{1}}.$

Solving this differential inequality and invoking the fact
$\int_{0}^{t}{\mu_{1}(\nu)\text{d}\nu}=-\phi_{1}(t)$ give
$\mu_{1}^{2}(t){{V}_{1}}(t)<{{\exp}^{2{{k}_{1}}{{\phi}_{1}}}}{{W}_{1}}(0)$
which yields

$\displaystyle\left\|\left[\begin{matrix}{{R}_{e}}\\\ {{q}_{e}}\\\
\end{matrix}\right]\right\|\leq\frac{1}{{{\mu}_{1}}(t)}{{\exp}^{{{k}_{1}}{{\phi}_{1}}}}(R_{e}^{2}(0)+q_{e}^{2}(0)).$
(25)

where
${{\phi}_{1}}(t)=\frac{{{{t}}_{f}}}{{{h}_{1}}-1}(1-\mu_{1}^{{({{h}_{1}}-1)}/{{{h}_{1}}}\;})$.
Thus, the origin of system (23) is globally prescribed-time stable within the
prescribed time ${{t}_{f}}$.

_Step 2._ Prescribed-time convergences of ${V}_{s}$ to ${V}_{s}^{*}$ and
${\eta}_{s}$ to ${\eta}_{s}^{*}$ under the proposed $u_{V}$ and $u_{\theta}$
in (22c) before $t=t_{f}$.

With the controller (22c), the dynamic model of ${{e}_{V}}$, ${{e}_{\eta}}$ is
obtained as

$\left[\begin{matrix}{\dot{e}_{V}}\\\ {\dot{e}_{\eta}}\\\
\end{matrix}\right]=-(({{k}_{2}}+{{\lambda}_{2}}){{\mu}_{2}}+\frac{h_{2}\dot{\tau}}{\tau})\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right]+{F}.$ (26)

Consider the Lyapunov candidate ${{V}_{2}}(t)=x_{2}^{T}{{x}_{2}}$ and
${{W}_{2}}(t)=\mu_{2}^{2}x_{2}^{T}{{x}_{2}}=\omega_{2}^{T}{{\omega}_{2}}$ with
${{x}_{2}}={{\left[\begin{matrix}{{e}_{V}}&{{e}_{\eta}}\\\
\end{matrix}\right]}^{T}}$ and
${{\omega}_{2}}={{\mu}_{2}}{{\left[\begin{matrix}{{e}_{V}}&{{e}_{\eta}}\\\
\end{matrix}\right]}^{T}}$. By applying Young’s inequality with
${{\lambda}_{2}}>0$, one obtains
$\omega_{2}^{T}{{\mu}_{2}}{F}\leq{{\lambda}_{2}}{{\mu}_{2}}\omega_{2}^{T}{{\omega}_{2}}+{{{\mu}_{2}}{{{{F}}}^{T}}{F}}/({4{{\lambda}_{2}}})$.
In this way, the derivative of ${{W}_{2}}(t)$ with respect to the variable $t$
yields

$\displaystyle{\dot{W}_{2}}(t)$
$\displaystyle=2\omega_{2}^{T}\left[\dot{\mu}_{2}{{x}_{2}}-(({{k}_{2}}+{{\lambda}_{2}}){{\mu}_{2}}+h_{2}\dot{\tau}/\tau){{\omega}_{2}}+{{\mu}_{2}}{F}\right]$
$\displaystyle\leq-2{{k}_{2}}{{\mu}_{2}}{{W}_{2}}+{{{\mu}_{2}}{{{{F}}}^{T}}{F}}/({2{{\lambda}_{2}}}).$
(27)

In view of ${{{F}}^{T}}{F}\leq\sigma^{2}$ and invoking Lemma 1 in [13], it is
direct to deduce
$\mu_{2}^{2}(t){{V}_{2}}(t)<\left({{\exp}^{2{{k}_{2}}{{\phi}_{2}}}}{{W}_{2}}(0)+{\sigma^{2}}/({4k_{2}\lambda_{2}})\right)$
which yields

$\displaystyle\left\|\left[\begin{matrix}{{e}_{V}}\\\ {{e}_{\eta}}\\\
\end{matrix}\right]\right\|\leq\frac{1}{{{\mu}_{2}}}\left({{\exp}^{{{k}_{2}}{{\phi}_{2}}}}(e_{V}^{2}(0)+e_{\theta}^{2}(0))+\frac{\sigma}{2\sqrt{k_{2}\lambda_{2}}}\right)$
(28)

where
${{\phi}_{2}}(t)=\frac{{{{t}}_{f}}}{{{h}_{2}}-1}(1-\mu_{2}^{{({{h}_{2}}-1)}/{{{h}_{2}}}\;})$.
Thus, it follows the prescribed-time stability of $(e_{V},e_{\eta})$.

_Step 3:_ Prescribed-time convergence of the cascaded system.

Combining the two system models into a cascaded form, we obtain

$\displaystyle\left[\begin{matrix}{\dot{R}_{e}}\\\ {\dot{q}_{e}}\\\
\end{matrix}\right]=-({{k}_{1}}\mu_{1}+\frac{h_{1}\dot{\tau}}{\tau}))\left[\begin{matrix}{{R}_{e}}\\\
{{q}_{e}}\\\ \end{matrix}\right]+\bar{M}\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right],$ (29a)
$\displaystyle\left[\begin{matrix}{\dot{e}_{V}}\\\ {\dot{e}_{\eta}}\\\
\end{matrix}\right]=-(({{k}_{2}}+{{\lambda}_{2}}){{\mu}_{2}}+\frac{h_{2}\dot{\tau}}{\tau})\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right]+{F}.$ (29b)

  1. (i)

Under the case $R_{f}>0$, it is obvious that $\|\bar{M}\|$ is bounded. In
addition, $\alpha_{13}(\tau)=2k_{1}\mu_{1}^{3}$,
$\alpha_{14}(\tau)=2\mu_{1}^{2}$, $\alpha_{15}(\tau)=k_{1}\mu_{1}$;
$\alpha_{23}(\tau)=2k_{2}\mu_{2}^{3}$, $\alpha_{24}(\tau)=2\mu^{2}_{2}$,
$\alpha_{25}(\tau)=k_{2}\mu_{2}$. Invoking the proposed Theorem 2, the
cascaded system is globally prescribed-time stable in the sense that the
reference trajectory gets tracked before $t_{f}$;

  2. (ii)

Under the case $R_{f}=0$, it is obvious that $\|\bar{M}\|$ is related to
$1/(t_{f}-t)$ such that $\alpha_{16}(\tau)=\mu^{2}_{1}/(t_{f}-t)$. Invoking
the proposed Theorem 2, the cascaded system is globally prescribed-time stable
if only ${{\lim}_{t\to{{t}_{f}}}}\frac{1/(t_{f}-t)}{\mu_{1}\mu_{2}}=0$.

_Step 4:_ Formation keeping over $[t_{f},\infty)$ when $R_{f}>0$.

When $t\in[t_{f},\infty)$, $\mu_{1}=\mu_{2}=1$, $\dot{\tau}=0$ such that the
cascaded system is rewritten into

$\displaystyle\left[\begin{matrix}{\dot{R}_{e}}\\\ {\dot{q}_{e}}\\\
\end{matrix}\right]=-{{k}_{1}}\left[\begin{matrix}{{R}_{e}}\\\ {{q}_{e}}\\\
\end{matrix}\right]+\bar{M}_{m}\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right],$ (30a)
$\displaystyle\left[\begin{matrix}{\dot{e}_{V}}\\\ {\dot{e}_{\eta}}\\\
\end{matrix}\right]=-({{k}_{2}}+{{\lambda}_{2}})\left[\begin{matrix}{{e}_{V}}\\\
{{e}_{\eta}}\\\ \end{matrix}\right]+{F}$ (30b)

where $V_{d}(t)=0$, $\eta_{d}(t)=0$, $t\geq t_{f}$ and
$\bar{M}_{m}=\text{diag}(1,1/R){M}(R,{{\eta}_{s}},e_{\eta},V_{s})$. Define the
Lyapunov candidate $V=x_{1}^{T}x_{1}+x_{2}^{T}x_{2}$ whose time derivative
yields

$\dot{V}\leq-(2k_{1}-\|\bar{M}_{m}\|)x_{1}^{T}x_{1}-(2k_{2}-\|\bar{M}_{m}\|)x_{2}^{T}x_{2}$
(31)

where $k_{1},k_{2}>0.5\|\bar{M}_{m}\|$. The corresponding stabilization is
obvious established according to the above proof. $\square$

###### Remark 6

It is obvious that the larger $k_{i}$, $h_{i}$ ($i=1,2$) are, the faster
$R_{e}$, $q_{e}$ converge. Invoking Remark 4, it is better to design
$k_{2}\mu_{2}>k_{1}\mu_{1}$ to make the states of different subsystems
converge in a reasonable order. In addition, the high-order scaling of state
by a function of time in [13] requires the parameter $h$ to be positive
integers greater than 1. Our results do not require such limitations.

##  4 Extensions to simultaneous tracking/formation of multi-vehicles

For the multi-agent tracking/formation scenario, a distinguishing requirement
is the simultaneous arrival. Suppose that $n$ vehicles, $M_{1},\cdots,M_{n}$,
participates in a tracking/formation with respect to a single target. In the
following, the variable $X_{i},1\leq i\leq n$ stands for the corresponding $X$
of the $i$-th vehicle. With the above design and analysis, each vehicle
realizes the prescribed-time stabilization, that is, it realizes the tracking
of $R_{i}$ to $R_{d,i}$ and $q_{i}$ to $q_{d,i}$ before the terminal time
$t_{f,i}$.

According to the proposed reference relative tracking trajectory theorem in
Section II.A, for the simultaneous tracking/formation with $t_{f,i}=T_{d}$,
the sufficient condition is
$\int_{0}^{T_{d}}{{{V}_{d,i}}(t)\text{d}t}=S_{d,i}$, $i=1,\cdots,n$.

Assume a centralized coordination strategy exists between vehicles and the
desired terminal time can be determined at the initial time. Thus, it is a
direct solution to allocate the velocity reasonably to obtain simultaneous
arrival. Below is an adjustment strategy which can achieve the smooth arrival
to the terminal situation.

Construct a $m$-order polynomial function to describe the law of
${{V}_{d,i}}(t)$

${{V}_{d,i}}(t)=I_{1}t^{m}+I_{2}t^{m-1}+\cdots+I_{m}t+I_{m+1}$ (32)

which satisfies $m+1$ equality constraints, including the following 3
necessary conditions

${{V}_{d,i}}(0)=V_{d0,i},{{V}_{d,i}}(T_{d})=V_{df,i},\int_{0}^{T_{d}}{{{V}_{d,i}}}(\tau)\text{d}\tau=S_{d,i}.$

Figure 2: Trajectories of four vehicles

Figure 3: States of four vehicles

##  5 Numerical examples

Table 1: Scenarios for simultaneous formation. | Initial condition |  | Terminal condition
---|---|---|---
Distance $R_{0}$ |  LOS $q_{0}$ |  Path angle $\theta_{0}$ |  Distance $R_{f}$ |  LOS $q_{f}$
Vehicle 1 | 7000 m | 220 deg | 190 deg | 1000 m | 230 deg
Vehicle 2 | 6000 m | -60 deg | -10 deg |  | 1000 m | -40 deg
Vehicle 3 | 5000 m | 30 deg | 0 deg |  | 1000 m | 50 deg
Vehicle 4 | 5000 m | 150 deg | 180 deg |  | 1000 m | 140 deg

Suppose that four vehicles track a single target with the initial and terminal
condition shown in Table 1. Each vehicle has the same initial velocity, 50
m/s.

Figure 4: Tracking errors of four vehicles

Figure 5: Control input of four vehicles

For a target moving at a speed of $V_{t}=100$m/s in a 45-degree direction, the
scenario at $T_{d}=t_{f,i}=20s$ expects the four vehicles to be at a 90-degree
angle to each other and a distance of 1000m from the target, meanwhile, it
requires $V_{df,i}=0$m/s for the gentle transition to the phase of formation
keeping $t>T_{d}$.

In accordance with (32), use a third-order polynomial for velocity planning.
The function $\tau(t,t_{f})$ is chosen in form of
$\tau(t,t_{f})=1+\text{ln}(\frac{t_{f}}{t_{f}-t})$. The control parameters are
as follows: $N_{1}=6$, $N_{2}=10$, $k_{1}=1$, $k_{2}=2$, $h_{1}=1$, $h_{2}=2$
and $\lambda_{2}=1$. The corresponding results for Case 2 can be found in Fig.
2-5. From the curves of $R_{i}$ and $q_{i}$ in Fig. 2, the simultaneous
formation and formation keeping are obvious achieved at $T_{d}$, and the
transmission process to formation keeping is smooth due to the small value of
the desired relative velocity $V_{df,i}$. The prescribed-time convergence of
tracking errors can be obtained in Fig. 4. It is observed from Fig. 3 that the
changes of both velocities and angles are slow towards the prescribed time
$T_{d}$, which promotes the limited amount of control input near the time
$T_{d}$ as shown in Fig. 5. After the prescribed time $T_{d}$, the task
becomes formation keeping relative to the target. The controller switching
brings the controller mutation at time $T_{d}$ in Fig. 5. As shown in the
short period after $T_{d}$ in Fig. 4, there is a regulation stage for the
angle $\eta_{s,i}$ such that the speed of each vehicle gets to be parallel to
that of the target.

It is worth emphasizing that the realization of simultaneous tracking and
formation is completely dependent on the proposed space-and-time-synchronized
strategy.

##  6 Conclusion

In this paper, we presented a space-and-time-synchronized control method for
simultaneous tracking/ formation. The resultant control is able to achieve the
predetermined state at the prescribed terminal time with a fixed relative
space trajectory independent of time. As a result, simultaneous
tracking/formation of multiple vehicles can be directly implemented. Extending
this method to the simultaneous tracking/formation with no predetermined
terminal time under communication topologies is an interesting topic for
future research.

##  Appendix A Proof of Theorem 1

_Step 1._ Prescribed-distance stability before $r_{d}=r_{f}$.

Similar to the prescribed-time design philosophy in [23], we choose the
controller (1) as a prescribed-distance controller by treating $r_{f}-r_{d}$
as a measure of the remaining distance to go. Substitution of (1) into (5)
leads to second-order differential equation

$\frac{{\text{d}^{2}}{{q}_{d}}}{\text{d}r_{d}^{2}}+\frac{{{k}_{d2}}}{{{{r}}_{f}}-{{{r}}_{d}}}\frac{\text{d}{{q}_{d}}}{\text{d}{{{r}}_{d}}}+\frac{{{k}_{d1}}}{{{({{{r}}_{f}}-{{{r}}_{d}})}^{2}}}({{q}_{d}}-{{q}_{f}})=0.$
(33)

This is a kind of second-order Cauchy equation. Here, we introduce a scaling
transformation $\zeta=\ln({{r}_{f}}-{{r}_{d}})$ with which the differential
equation is transformed into

$\frac{{\text{d}^{2}}{{q}_{d}}}{\text{d}{{\zeta}^{2}}}-({{k}_{d2}}+1)\frac{\text{d}{{q}_{d}}}{\text{d}\zeta}+{{k}_{d1}}({{q}_{d}}-{{q}_{f}})=0.$
(34)

The corresponding characteristic equation is obtained by letting
${{q}_{d}}-{{q}_{f}}={{e}^{\lambda\zeta}}$ as
${{\lambda}^{2}}-({{k}_{d2}}+1)\lambda+{{k}_{d1}}=0$ whose discriminant is
$\Delta={{({{k}_{d2}}+1)}^{2}}-4{{k}_{d1}}$. In order to obtain an elegant
description of the solution, we make
${{\lambda}_{1}}=N_{1}\in\mathcal{N}_{+}$,
${{\lambda}_{2}}=N_{2}\in\mathcal{N}_{+}$, $N_{1}<N_{2}$ which brings about
${{k}_{d1}}=N_{1}N_{2},\ {{k}_{d2}}=N_{1}+N_{2}-1$, and the discriminant
$\Delta>0$. Hence, with the initial condition ${{q}_{d}}(0)={{q}_{0}}$ and
$\tan({{\eta}_{d}}(0))=\tan{{\eta}_{d0}}$, the solution of this Cauchy
equation can be obtained

${{q}_{d}}({{r}_{d}})={{C}_{1}}({{r}_{f}}-{{r}_{d}})^{N_{1}}+{{C}_{2}}{{({{r}_{f}}-{{r}_{d}})}^{N_{2}}}+{{q}_{f}}$
(35)

where

$\displaystyle{{C}_{1}}=\frac{N_{2}({{q}_{0}}-{{q}_{f}})+{{{r}}_{f}}\tan{{\eta}_{d0}}(R_{0}-{R}_{f1})/R_{0}}{(N_{2}-N_{1}){{{r}}^{N_{1}}_{f}}},$
$\displaystyle{{C}_{2}}=\frac{N_{1}({{q}_{f}}-{{q}_{0}})-{{{r}}_{f}}\tan{{\eta}_{d0}}(R_{0}-{R}_{f1})/R_{0}}{(N_{2}-N_{1})r_{f}^{N_{2}}}.$

Differentiating ${{q}_{d}}(r_{d})$ with respect to $r_{d}$ yields

$\displaystyle\tan{{\eta}_{d}}=-\frac{R_{d}}{R_{d}-{R}_{f1}}$
$\displaystyle\left[N_{1}{{C}_{1}}{{({{r}_{f}}-{{r}_{d}})}^{N_{1}-1}}\right.$
(36) $\displaystyle\ \ \
\left.+N_{2}{{C}_{2}}{{({{r}_{f}}-{{r}_{d}})}^{N_{2}-1}}\right].$

It is thus clear that, given the positive integers ${{N}_{1}}$, ${{N}_{2}}$,
as $r_{d}$ approaching $r_{f}$, ${{q}_{d}}({{r}_{d}})$ and
${{\eta}_{d}}({{r}_{d}})$converges to $q_{f}$ and 0, respectively.

_Step 2._ Position maintaining after $R_{d}$ approaching $R_{f}$.

Since $\eta_{d}(r_{f})=0$ in (36), $\eta_{d}(r_{d})\equiv 0$ through
$\dot{\eta}_{d}=0$ for $r_{d}\geq r_{f}$. When ${V}_{df}=0$ and
$\dot{V}_{d}=0$ for $r_{d}\geq r_{f}$, it is apparent $R_{d}\equiv R_{f}$
after $R_{d}$ approaching $R_{f}$. Invoking
${{{\text{d}}^{2}}{{q}_{d}}}/{\text{d}r_{d}^{2}}=0$ in (5), one obtains
$q_{d}(r_{d})\equiv 0$.

##  Appendix B Proof of Theorem 2

We start from the simplified cascaded system with $n=2$ subsystems. Evidently
the prescribed-time stability of $x_{2}(t)$ in system ${{\Sigma}_{2}}$ can be
obtained by taking time derivative of ${{V}_{2}}(t,x_{2})$

$\displaystyle{{{\dot{V}}}_{2}}$
$\displaystyle\leq-{{\alpha}_{23}}(\tau){{\left\|x_{2}\right\|}^{2}}+{{b}_{2}}({{\alpha}^{2}_{24}}(\tau)/\alpha_{23}(\tau)){{\left\|{{\Delta}_{x,2}}\right\|}^{2}}$
(37)
$\displaystyle=-{{\delta}_{2}}{{\alpha}_{23}}{{\left\|x_{2}\right\|}^{2}}$
$\displaystyle\ \ \
-{{\alpha}_{23}}\left[(1-{{\delta}_{2}}){{\left\|x_{2}\right\|}^{2}}-{{b}_{2}}{{{\left\|{{\Delta}_{x,2}}\right\|}^{2}}}/{{{\alpha}^{2}_{25}}}\;\right]$

with ${{\delta}_{2}}\in(0,1)$. Define the region
${{\Omega}_{x_{2}}}(r)=\left\\{x_{2}|{{\alpha}^{2}_{25}}{{\left\|x_{2}\right\|}^{2}}\leq{{b}_{2}}{{\left\|{{\Delta}_{x,2}}\right\|}^{2}}/(1-{{\delta}_{2}})\right\\}$
and we get
${{\Omega}_{{x_{2}}}}({{t}_{f}})={{\lim}_{t\to{{t}_{f}}}}{{\Omega}_{{x_{2}}}}(t)=0$,
following from ${{\lim}_{t\to{{t}_{f}}}}{{\alpha}_{25}}(\tau)=\infty$. When
$x_{2}$ is outside of ${{\Omega}_{x_{2}}}(r)$, the increasing property of
${{\alpha}_{23}}(\tau)$ will force $x_{2}$ to ${{\Omega}_{x_{2}}}(t)$; once
$x_{2}$ is inside of ${{\Omega}_{x_{2}}}(t)$, $x_{2}$ will never escape
${{\Omega}_{x_{2}}}(t)$ and converge to 0 as $t$ approaching ${{t}_{f}}$.

Then, the time derivative of ${{V}_{1}}(t,x_{1})$ along ${{\Sigma}_{1}}$
becomes

$\displaystyle{{{\dot{V}}}_{\text{1}}}$
$\displaystyle\leq-{{\alpha}_{13}}{{\left\|x_{1}\right\|}^{2}}+{{b}_{1}}({{\alpha}^{2}_{14}}/\alpha_{13}){{\left\|{{\Delta}_{x,1}}\right\|}^{2}}+{{\alpha}_{16}}{{c}_{g,1}}\left\|x_{1}\right\|\left\|x_{2}\right\|$
$\displaystyle=-0.5{{\delta}_{1}}{{\alpha}_{13}}{{\left\|x_{1}\right\|}^{2}}-{{\alpha}_{13}}\left[(1-{{\delta}_{1}}){{\left\|x_{1}\right\|}^{2}}\right.$
$\displaystyle\ \ \ \
\left.-\gamma\alpha^{2}_{16}{{{\left\|x_{2}\right\|}^{2}}}/{\alpha^{2}_{13}}-{{b}_{1}}{{{\left\|{{\Delta}_{x,1}}\right\|}^{2}}}/{{{\alpha}^{2}_{15}}}\;\right]$
(38)

where $\gamma={0.5c_{\theta}^{2}}/{{{\delta}_{1}}}\;$. In view of the
relationship among $\alpha_{13}$, $\alpha_{16}$ and $\alpha_{25}$, one obtains

$\displaystyle{{\lim}_{t\to{{t}_{f}}}}{\alpha^{2}_{16}{{{\left\|x_{2}\right\|}^{2}}}/{\alpha^{2}_{13}}}$
$\displaystyle=\alpha^{2}_{16}/(\alpha^{2}_{13}\alpha^{2}_{25})\left\|{{\Delta}_{x,2}}\right\|^{2}{{{b}_{2}}/(1-{{\delta}_{2}})}$
$\displaystyle=0.$ (39)

Invoking the boundedness of ${{\left\|{{\Delta}_{x,1}}\right\|}^{2}}$, the
prescribed-time convergence of $x_{1}$ in system ${{\Sigma}_{1}}$ can be
proved.

The prescribed-time convergence of cascaded system with $n=2$ is thus
concluded. From the prescribed-time stability of subsystems $\Sigma_{n}$ and
$\hat{\Sigma}_{n-1}$, the prescribed-time stability of ${\Sigma}_{n-1}$ system
is obtained; from the prescribed-time stability of subsystems ${\Sigma}_{n-1}$
and $\hat{\Sigma}_{n-2}$, the prescribed-time stability of system
${\Sigma}_{n-2}$ is obtained; and so on, the prescribed-time stability of
${\Sigma}_{1}$ system can be obtained. The proof is thus completed.

## References

  * Aguiar and Hespanha [2007] A. P. Aguiar, J. P. Hespanha, Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty, IEEE transactions on automatic control 52 (8) (2007) 1362–1379.
  * Wang et al. [2021a] B. Wang, H. Ashrafiuon, S. Nersesov, Leader–follower formation stabilization and tracking control for heterogeneous planar underactuated vehicle networks, Systems $\&$ Control Letters 156 (2021a) 105008\.
  * Wang et al. [2018a] Y.-W. Wang, X.-K. Liu, J.-W. Xiao, Y. Shen, Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control, Automatica 93 (2018a) 26–32.
  * Liu et al. [2018] X. Liu, S. S. Ge, C. Goh, Y. Li, Event-triggered coordination for formation tracking control in constrained space with limited communication, IEEE transactions on cybernetics 49 (3) (2018) 1000–1011.
  * Davila [2013] J. Davila, Exact Tracking Using Backstepping Control Design and High-Order Sliding Modes, IEEE Transactions on Automatic Control 58 (8) (2013) 2077–2081.
  * Levant [2005] A. Levant, Homogeneity approach to high-order sliding mode design, Automatica 41 (2005) 823 – 830.
  * Bhat and Bernstein [2005] S. Bhat, D. Bernstein, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Syst. 17 (2005) 101 – 127.
  * Hu et al. [2019] Q. Hu, W. Chen, Y. Zhang, Concurrent Proximity Control of Servicing Spacecraft With an Uncontrolled Target, IEEE/ASME Transactions on Mechatronics 24 (6) (2019) 2815–2826.
  * Tian et al. [2017] B. Tian, Z. Zuo, X. Yan, H. Wang, A fixed-time output feedback control scheme for double integrator systems, Automatica 80 (2017) 17 – 24.
  * Polyakov et al. [2015] A. Polyakov, D. Efimov, W. Perruquetti, Finite-time and fixed-time stabilization: Implicit Lyapunov function approach, Automatica 51 (2015) 332 – 340.
  * Zuo et al. [2018] Z. Zuo, Q.-L. Han, B. Ning, X. Ge, X.-M. Zhang, An Overview of Recent Advances in Fixed-Time Cooperative Control of Multiagent Systems, IEEE Transactions on Industrial Informatics 14 (6) (2018) 2322–2334.
  * Wang et al. [2018b] Y. Wang, Y. Song, D. J. Hill, M. Krstic, Prescribed-time consensus and containment control of networked multiagent systems, IEEE transactions on cybernetics 49 (4) (2018b) 1138–1147.
  * Song et al. [2017] Y. Song, Y. Wang, J. Holloway, M. Krstic, Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica 83 (2017) 243–251.
  * Wang and Song [2018] Y. Wang, Y. Song, Leader-following control of high-order multi-agent systems under directed graphs: Pre-specified finite time approach, Automatica 87 (2018) 113–120.
  * Cao and Ren [2014] Y. Cao, W. Ren, Finite-time consensus for multi-agent networks with unknown inherent nonlinear dynamics, Automatica 50 (10) (2014) 2648–2656.
  * Zhao et al. [2017] L. Zhao, Y. Jia, J. Yu, Adaptive finite-time bipartite consensus for second-order multi-agent systems with antagonistic interactions, Systems & Control Letters 102 (2017) 22–31.
  * Shao et al. [2020] X. Shao, Q. Hu, Y. Shi, Adaptive Pose Control for Spacecraft Proximity Operations With Prescribed Performance Under Spatial Motion Constraints, IEEE Transactions on Control Systems Technology .
  * Zhou and Yang [2016] J. Zhou, J. Yang, Distributed guidance law design for cooperative simultaneous attacks with multiple missiles, Journal of Guidance, Control, and Dynamics 39 (10) (2016) 2439–2447.
  * Chen and Wang [2019] X. Chen, J. Wang, Optimal control based guidance law to control both impact time and impact angle, Aerospace Science and Technology 84 (2019) 454–463.
  * Li et al. [2021a] D. Li, H. Yu, K. P. Tee, Y. Wu, S. S. Ge, T. H. Lee, On Time-Synchronized Stability and Control, IEEE Transactions on Systems, Man, and Cybernetics: Systems (2021a) 1–14.
  * Li et al. [2021b] D. Li, S. S. Ge, T. H. Lee, Fixed-Time-Synchronized Consensus Control of Multiagent Systems, IEEE Transactions on Control of Network Systems 8 (1) (2021b) 89–98.
  * Panteley and Loria [2001] E. Panteley, A. Loria, Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems, Automatica 37 (3) (2001) 453 – 460.
  * Wang et al. [2021b] P. Wang, X. B. Zhang, S. S. Ge, Prescribed-time control with explicit reference governor for a class of constrained cascaded systems, International Journal of Robust and Nonlinear Control (2021b) 1–16.


# Rotating regular black holes in AdS spacetime and its shadow

Balendra Pratap Singha<EMAIL_ADDRESS>Md Sabir Alib
<EMAIL_ADDRESS>Sushant G. Ghoshc;d<EMAIL_ADDRESS><EMAIL_ADDRESS>a Tulas Institute, Dehradun, Uttarakhand 248197, India  b
Indian Institute of Science Education and Research Kolkata, West Bengal
741246, India  c Centre for Theoretical Physics, Jamia Millia Islamia, New
Delhi 110025, India  d Astrophysics and Cosmology Research Unit, School of
Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,
Private Bag 54001, Durban 4000, South Africa

###### Abstract

The presence of a photon region around the black hole is an essential feature
in receiving the emitted spectrum from the vicinity of the black hole by the
distant observer. In this paper, we investigate the optical properties of
rotating regular anti-de Sitter (AdS) black holes, which characterized by its
mass $(M)$, spin parameter $(a)$, deviation parameter $(k)$ and the
cosmological constant $(\Lambda)$ related to the curvature radius via
$(\Lambda=-3/l^{2})$. We derive the complete null geodesic equations of motion
and study the unstable circular orbits for an observer at given Boyer-
Lindquist coordinates ($r_{O},\;\vartheta_{O}$) in the domain of outer
communication and trace the photon rings. For the analytical part of our
study, we investigate the observables, namely, shadow radius $(R_{s})$ and
distortion parameter $(\delta_{s})$. These rotating regular AdS black holes
have smaller shadows when compare to the Kerr and regular spacetimes. We also
estimate the energy emission rate of the black hole.

##  I Introduction

Observing the rotating black hole shadow is one of the possible method to
measure the black hole parameters. The incoming photons from the source having
larger angular momentum get scattered from the black hole and reach the
faraway observer and those having smaller angular momentum fall inside the
black hole and form a dark shadow region. The black hole shadow forms in the
close vicinity of the event horizon and thereby helps us to have the idea of
the basic geometrical structure of horizons. Recent astronomical observations
from Event Horizon Telescope (EHT) confirm the existence of the black hole
Sagittarius A* at the center of our galaxy Milky way Akiyama:2022L12 ;
Akiyama:2022L13 ; Akiyama:2022L14 ; Akiyama:2022L15 ; Akiyama:2022L16 ;
Akiyama:2022L17  and a supermassive black hole at the galactic center of
Messier 87 Akiyama:2019cqa ; Akiyama:2019brx ; Akiyama:2019sww ;
Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . The observed image of
the black hole is a two-dimensional dark disk surrounded by luminous rings.
The dark region is black hole shadow and the luminous rings are photon orbits.

The apparent shape of the shadow also depends on the matter fields surrounding
the black hole. Hence, the shadow of the black hole is treated as a useful
tool to detect black hole spin and other deformation parameters because its
shape and size carry the fingerprint of the geometry around the black hole.
The astrophysical black hole candidates are having an appreciable amount of
rotation. This consideration motivates us to study the rotating black holes
and the shadow cast by them. Since the regular black holes contain Kerr-metric
as a special case, we consider them as a prototype for a large class of non-
Kerr families where the metric would have the same form with mass parameter
$M$ simply replaced by $m(r)$. The analysis of the shadows will be a useful
tool for better understanding of astrophysical black holes and also for
comparing modified theories of gravity (MGs) with general theory of gravity
(GR). In GR the black hole shadow is completely characterized by its Mass $M$
and spin parameter $a$ Bardeen:1973gb ; Young:1976  but in MGs, there are also
some additional free parameters which affect the shape and size of the black
hole shadow. Hence, it is useful to study further the shadow of black holes in
various available MGs. In the last few years, the study of black hole shadow
in various spacetimes has been investigated Johannsen:2015qca ; De ;
Amarilla:2010zq ; Amarilla:2013sj ; Yumoto:2012kz ; Abdujabbarov:2016hnw ;
Amir:2016cen ; Tsukamoto:2014tja ; Hioki:2009na ; Bambi:2010hf ; Goddi:2016jrs
; Takahashi:2005hy ; Wei:2013kza ; Abdujabbarov:2012bn ; Amarilla:2011fx ;
Bambi:2008jg ; Atamurotov:2013sca ; Wang:2017hjl ; Schee:2008kz ;
Grenzebach:2014fha ; Singh:2017xle ; Kumar:2019ohr ; Kumar:2017tdw ;
Kumar:2020owy ; Narzilloev:2022bbs ; Wei:2019pjf ; Papnoi:2021rvw ;
Atamurotov:2021hck . The effect of the plasma environment on the black hole
shadow has been also discussed in Perlick:2015vta ; Atamurotov:2015nra ;
Abdujabbarov:2015pqp . The exotic feature of the black hole’s shadow has also
been explored in the higher-dimensional spacetime Papnoi:2014aaa ;
Singh:2017vfr ; Amir:2017slq ; Abdujabbarov:2015rqa ; Atamurotov:2021cgh .
Cunha et. al. Cunha:2015yba ; Cunha:2016bpi  developed a ray tracing method
for a better understanding of photon rays in the vicinity of Kerr spacetime
with and without scalar hair.

Here, we shall adopt the formalism presented in Grenzebach:2014fha  for an
observer located at some given Boyer-Lindquist coordinates
($r_{O},\;\vartheta_{O}$) instead of infinity and study the shadow images of
the regular anti-de Sitter (AdS) black holes. The shadow images for Kerr-
Newman-NUT-AdS black holes and rotating AdS black holes in the braneworld
scenario have been studied Grenzebach:2014fha ; Eiroa:2017uuq . Belhaj et. al.
Belhaj:2021rae ; Belhaj:2022rmc , investigated shadows of superentropic and
Bardeen black holes in AdS spacetime. Shadows of rotating black holes with
cosmological constant have been studied in Belhaj:2020kwv . The authors of
Afrin:2021ggx , estimated the cosmological constant from Kerr-de sitter black
holes.

In this work, we analytically estimate the effects of the spin and the related
deformation parameters on the size and shape of the rotating regular AdS black
hole. The size of the black hole shadow decreases and gets more distorted in
comparison with the Kerr Bardeen  and nonsingular Amir:2016cen  black hole in
AdS spacetime.

The paper is arranged as follows: In Sec. II, we describe the rotating regular
AdS black hole. The geodesics structure of a test particle in this spacetime
has been discussed in Sec. III. Further, in Sec. IV, we derive an analytical
formula for the shadow and plot them for different values of black hole
parameters. In Sec. V, we estimate the energy emission rate and finally, we
conclude our results in Sec. VI.

##  II Rotating regular ads black hole

In this section, we study the general solution to static spherically symmetric
regular black holes and their rotating counterparts in AdS spacetime due to
the presence of nonlinear electrodynamics. We start with a four-dimensional
static black hole with spherical symmetry described by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{2}^{2},$ (1)

where $f(r)$ is the metric function and
$d\Omega_{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the line element on a
two-dimensional unit sphere. The action for regular AdS spacetimes has the
following form

$\displaystyle\mathcal{I}=\int{d^{4}x}\sqrt{-g}[R-\mathcal{L\left(F\right)}+6\l^{-2}],$
(2)

where $R$ is the Ricci scalar, $\l$ is the curvature radius related to the
cosmological constant $\Lambda=-3/\l^{2}$, and $\mathcal{L\left(F\right)}$ is
the Lagrangian of the non-linear electrodynamics source having the form

$\displaystyle\mathcal{L\left(F\right)}=e^{\left[{-s\left(2g^{2}\mathcal{F}\right)^{\frac{1}{4}}}\right]}\mathcal{F},$
(3)

with $\mathcal{F}=F_{\mu\nu}F^{\mu\nu}/4$, where
$F_{\mu\nu}=2\nabla_{[\mu}A_{\nu]}$. We take $s=q/2M$, where $q$ and $M$
correspond to the charge and the mass of the regular black hole. The variation
of the action $\mathcal{I}$, with respect to $g_{\mu\nu}$ and $A_{\mu}$ leads
to the field equations

$\displaystyle
G_{\mu\nu}-\frac{3}{\l^{2}}=T_{\mu\nu},\;\;\;\nabla_{\mu}\left(\frac{\partial\mathcal{L\left(F\right)}}{\partial{F}}{F^{\nu\mu}}\right)=0,$
(4)

where $T_{\mu\nu}$ is energy-momentum tensor given as

$\displaystyle
T_{\mu\nu}=2\left[\frac{\partial\mathcal{L\left(F\right)}}{\partial{F}}F_{\mu\alpha}F_{\nu}^{\alpha}-g_{\mu\nu}\mathcal{L\left(F\right)}\right],$
(5)

The form of $F_{\mu\nu}$ for a nonlinear charged follows the ansatz

$\displaystyle
F_{\mu\nu}=2\delta^{\theta}_{[\mu}\delta^{\phi}_{\nu]}q\left(r\right)\sin\theta.$
(6)

Eq. (4) implies $dF=0$ , which in turn reads $q^{\prime}(r)dr\wedge
d\theta\wedge d\phi=0$, leading to $q(r)=constant=q$. The constant $q$ is
identified as nonlinear charge arising from nonlinear electrodynamics.
Therefore, the resulting Maxwell tensor has the only one component

$\displaystyle F_{\theta\phi}=2q\sin\theta,$ (7)

and

$\displaystyle\mathcal{F}=\frac{q^{2}}{2r^{4}},$ (8)

The Lagrangian (3) of the nonlinear electrodynamics thus reads

$\displaystyle\mathcal{L\left(F\right)}=\frac{q^{2}}{2r^{4}}e^{-\frac{q^{2}}{2Mr}}.$
(9)

The $(t,t)$-component of the field equations (4) along with the energy-
momentum tensor (5) and the Lagrangian (9) leads the metric function with
$q^{2}=2Mk$, having the form

$\displaystyle f(r)=1-\frac{2M\;e^{{-\frac{k}{r}}}}{r}+\frac{r^{2}}{\l^{2}}.$
(10)

In summary the metric for a non-singular AdS black holes in $4$-dimensional
spacetime reads

$\displaystyle\mathrm{ds^{2}}=-\left(1-\frac{2M\;e^{{-\frac{k}{r}}}}{r}+\frac{r^{2}}{\l^{2}}\right)dt^{2}+\frac{1}{\left(1-\frac{2M\;e^{{-\frac{k}{r}}}}{r}+\frac{r^{2}}{\l^{2}}\right)}dr^{2}+r^{2}d\Omega_{2}^{2}$
(11)

where $M\;\text{and}\;k$ are, respectively, the Arnowitt-Deser-Misner (ADM)
mass and the nonlinear parameter of the black hole. Next, we turn to discuss
the rotating counterpart of the regular AdS black hole (11). Before going into
the main discussion, we write the gauge potential for a magnetically charged
black hole. The electromagnetic potential for the magnetically charged
rotating black hole was recently studied and derived using the usual Newman-
Janis procedure, which leads to having the form Toshmatov:2017zpr ;
Erbin:2016lzq

$\displaystyle
A_{\mu}=-\frac{g\;a\;\cos\theta}{\Sigma}\delta_{\mu}^{t}+\frac{g(r^{2}+a^{2})\cos\theta}{\Sigma}\delta_{\mu}^{\phi}.$
(12)

The gauge potential in the absence of the rotation ($a=0$), reduces to the
expression for spherically symmetric spacetime is written as
$A_{\mu}=g\cos\theta\delta_{\mu}^{\phi}$. With this gauge potential in hand,
we have Maxwell field invariant to be Toshmatov:2017zpr

$\displaystyle\mathcal{F}=\frac{2g^{2}\left(\left(\Sigma-2r^{2}\right)^{2}-4a^{2}r^{2}\cos^{2}\theta\right)}{\Sigma^{4}}.$
(13)

The Einstein’s field equations lead to have the expressions of Lagrangian
Toshmatov:2017zpr  in the form

$\displaystyle\mathcal{L}(r)$ $\displaystyle=$
$\displaystyle\frac{4r^{2}\left(2r^{2}\Sigma\;m^{\prime\prime}+5\;m^{\prime}\left(\left(\Sigma-2r^{2}\right)^{2}-8r^{2}\Sigma+4r^{2}\right)\right)}{\Sigma^{4}},$
$\displaystyle\mathcal{L}_{\mathcal{F}}(r)$ $\displaystyle=$
$\displaystyle\frac{r\Sigma\;m^{\prime\prime}+2m^{\prime}\left(\Sigma-2r^{2}\right)}{2g^{2}},$
(14)

where ′ and ′′ denotes, respectively, the first and second derivatives with
respect to the radial coordinate $r$. The rotating regular black holes are
described by four parameters, e.g., the mass $M$, the rotation parameter $a$,
the nonlinear parameter $q$. The rotating regular metric belongs to prototype
family of non-Kerr solutions, which in Boyer-Lindquist coordinates has same
form as that of Kerr metric with mass $m$ replaced by some mass function
$\tilde{m}$ which contains an additional deviation parameter $q$ from the NED.
The regular rotating AdS black holes are stationary and axially symmetric
spacetimes with a negative cosmological constant. The exact solution of the
Einstein field equations with a negative cosmological constant that describes
the rotating black holes in four-dimensional spacetime with asymptotic AdS
behavior in the standard Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ takes
the following form

$\displaystyle\mathrm{ds^{2}}=-\frac{\Delta_{r}}{\Sigma}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right)^{2}+\frac{\Sigma}{\Delta_{r}}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\Sigma}\left(adt-\frac{a^{2}+r^{2}}{\Xi}d\phi\right)^{2},$
(15)

with

$\Sigma=r^{2}+a^{2}\cos^{2}\theta,\;\;\Delta_{r}=\left(r^{2}+a^{2}\right)\left(1+\frac{r^{2}}{l^{2}}\right)-2Mre^{-k/r},$
(16)

and

$\Delta_{\theta}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta,\;\;\Xi=1-\frac{a^{2}}{l^{2}}.$
(17)

The metric (15) depends on four parameters, namely the mass $M$, the spin $a$,
the free parameter $k$ and the curvature radius $l$. The cosmological constant
($\Lambda$) is related to the radius of curvature $l$ via, $\Lambda=-3/l^{2}$.
The metric (15) reduces to the Schwarzschild ($a=k=1/l^{2}=0$), the
Schwarzschild-AdS ($a=k=0$), Kerr ($k=1/l^{2}=0$) and Kerr-AdS ($k=0$) black
holes as special cases. The metric (15) becomes singular apparently at
$\Sigma=0$ and $\Delta_{r}=0$, but $\Delta_{r}=0$ is a coordinate singularity.
It is seen that the metric (15), for $k<<r$ reduces to the Kerr-Newman-AdS
black hole such that Plebanski

$\displaystyle\Delta_{r}=\left(r^{2}+a^{2}\right)\left(1+\frac{r^{2}}{l^{2}}\right)-2Mr+q^{2}+\mathcal{O}(k^{2}/r^{2}).$
(18)

The determinant of the metric (15) takes the form

$\displaystyle\sqrt{-g}=\frac{\Sigma\sin\theta}{\Xi}$ (19)

Clearly, Eq. 15 becomes singular if $a^{2}=l^{2}$. At this critical condition
the boundary of AdS spacetime rotates at the speed of light. Therefore one
must have to restrict $a^{2}<l^{2}$. The stationary and axially symmetric
spacetime (11) admits two commuting Killing vectors having the form
$\xi^{\mu}{{}_{(}t)}=\delta^{\mu}{{}_{t}}$, and
$\xi^{\mu}{{}_{(}\phi)}=\delta^{\mu}{{}_{\phi}}$ which leads to obtaining
various metric components as a scalar product such that

$\displaystyle\xi^{\mu}{{}_{(t)}}\xi_{(t){{}_{\mu}}}$ $\displaystyle=$
$\displaystyle
g_{tt}=-1+\frac{2Mre^{-k/r}}{\Sigma}-\frac{r^{2}+a^{2}\sin^{2}\theta}{l^{2}},$
(20) $\displaystyle\xi^{\mu}{{}_{(t)}}\xi_{(\phi){{}_{\mu}}}$ $\displaystyle=$
$\displaystyle
g_{t\phi}=-\frac{a\sin^{2}\theta}{\Xi}\left(\frac{2Mre^{-k/r}}{\Sigma}-\frac{r^{2}+a^{2}}{l^{2}}\right),$
(21) $\displaystyle\xi^{\mu}{{}_{(\phi)}}\xi_{(\phi){{}_{\mu}}}$
$\displaystyle=$ $\displaystyle
g_{\phi\phi}=\frac{\sin^{2}\theta}{\Xi}\left(\left(r^{2}+a^{2}\right)\Xi+\frac{2Mra^{2}\sin^{2}\theta
e^{-k/r}}{\Sigma}\right).$ (22)

Using Einstein equation with a cosmological constant

$G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi
T_{\mu\nu},$ (23)

We have the Einstein tensor as Xu:2016jod

$\displaystyle G_{tt}$ $\displaystyle=$
$\displaystyle\dfrac{2[r^{4}-2r^{3}m+a^{2}r^{2}-a^{4}\sin^{2}\theta\cos^{2}\theta]m^{{}^{\prime}}}{\Sigma^{3}}-\dfrac{ra^{2}\sin^{2}\theta
m^{{}^{\prime\prime}}}{\Sigma^{2}}+\Lambda\dfrac{a^{2}\sin^{2}\theta-\Delta_{r}}{\Sigma},$
$\displaystyle G_{rr}$ $\displaystyle=$
$\displaystyle-\dfrac{2r^{2}m^{{}^{\prime}}}{\Sigma\Delta_{r}}+\Lambda\dfrac{\Sigma}{\Delta_{r}},~{}~{}~{}~{}G_{\theta\theta}=-\dfrac{2a^{2}\cos^{2}\theta
m^{{}^{\prime}}}{\Sigma}-rm^{{}^{\prime\prime}}+\Lambda\Sigma,$ $\displaystyle
G_{t\phi}$ $\displaystyle=$
$\displaystyle\dfrac{2a\sin^{2}\theta[(r^{2}+a^{2})(a^{2}\cos^{2}\theta-r^{2})]m^{{}^{\prime}}}{\Sigma^{3}}-\dfrac{ra^{2}\sin^{2}\theta(r^{2}+a^{2})m^{{}^{\prime\prime}}}{\Sigma^{2}}+\Lambda\dfrac{a\sin^{2}\theta[\Delta_{r}-r^{2}-a^{2}]}{\Sigma},$
$\displaystyle G_{\phi\phi}$ $\displaystyle=$
$\displaystyle-\dfrac{a^{2}\sin^{2}\theta[(r^{2}+a^{2})(a^{2}+(2r^{2}+a^{2})\cos
2\theta)+2r^{3}\sin^{2}\theta
m)]m^{{}^{\prime}}}{\Sigma^{3}}-\dfrac{r\sin^{2}\theta(r^{2}+a^{2})^{2}m^{{}^{\prime\prime}}}{\Sigma^{2}}$
(24)
$\displaystyle+\Lambda\dfrac{\sin^{2}\theta[(r^{2}+a^{2})^{2}-a^{2}\Delta_{r}]}{\Sigma}.$

The two Killing vectors $\xi^{\mu}{{}_{(t)}}$ and $\xi^{\mu}{{}_{(\phi)}}$
denote the Killing symmetries corresponding to the time-translational symmetry
along $t$-axis and the rotational symmetry around $\phi$-axis, together give
rise to a Killing vector field
$\chi^{\mu}=\xi^{\mu}{{}_{(t)}}+\Omega_{+}\xi^{\mu}{{}_{(\phi)}}$, which a
null generator of the event horizon when $\Delta_{r}=0$, i.e., it is tangent
to the null surface of the event horizon. The quantity $\Omega_{+}$ is the
angular velocity at the event horizon of the black hole which reads as

$\displaystyle\Omega_{+}=\frac{a\Xi}{a^{2}+r_{+}^{2}}.$ (25)

One can also derive the angular velocity $\Omega$ of the observer moving on
the orbits of constant $r$ and $\theta$ with four-velocity $u^{\mu}$ such that
$u^{\mu}\xi_{(\phi){{}_{\mu}}}=0$ which turns out to be

$\displaystyle\Omega=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{a\Xi\left(\left(r^{2}+a^{2}\right)\Delta_{\theta}-\Delta_{r}\right)}{\left(r^{2}+a^{2}\right)^{2}\Delta_{\theta}-a^{2}\sin^{2}\theta\Delta_{r}}.$
(26)

In case of rotating regular AdS black hole the angular velocity (26) does not
vanish at asymptotic infinity rather it will give us a finite quantity so that

$\displaystyle\Omega_{\infty}=-\frac{a}{l^{2}}.$ (27)

Now the Eqs. (26) and (27) help us to define the angular velocity for the
black holes with respect to a frame that is static at asymptotic infinity
which gives

$\displaystyle\omega_{+}=\Omega_{+}-\Omega_{\infty}=\frac{a}{a^{2}+r_{+}^{2}}\left(1+\frac{r_{+}^{2}}{l^{2}}\right).$
(28)

The angular velocity is the most important characteristic of the rotating AdS
black holes since it enters in the first law of thermodynamics to have a
consistent relation. On the other hand, it can be shown that angular velocity
coincides with that of the boundary Einstein universe, thereby providing the
relevant basis for a AdS/CFT duality of rotating regular AdS black hole.

The metric (15) belongs to the family of a non-Kerr black hole which
encompasses Kerr as a special case ($k=1/l^{2}=0$). The non-Kerr black holes
put some bounds on the spin parameter and other deviation parameters to behave
like a Kerr black hole. The spacetime geometry around black hole candidates
with the available radio and X-ray data Bambi:2013sha ; Bambi:2013qj ;
Bambi:2011mj ; Cardenas-Avendano:2016zml  shows us possible deviations from
the black hole in GR. Proceeding along this line we consider the regular
rotating AdS black hole candidates which may discard or constrain possible
deviations from the Kerr background.

##  III Null geodesics

In order to describe the black hole shadow, we require the complete study of
geodesic equations of photons around the black hole. The shadow images can
provide the apparent shape of event horizon which is related to the geodesic
structure of massless particle in the vicinity of the black hole. To study the
geodesics of phpton around rotating regular AdS black hole, we choose
Hamilton-Jacobi equation due to Carter Carter:1968rr . The complete geodesic
equations of motion can be derived by solving Hamilton-Jacobi equations

$\displaystyle\frac{\partial{\mathcal{S}}}{\partial\sigma}=-\frac{1}{2}g^{\alpha\beta}\frac{\partial{\mathcal{S}}}{\partial
x^{\alpha}}\frac{\partial{\mathcal{S}}}{\partial x^{\beta}},$ (29)

where $\sigma$ is the affine parameter alongside the geodesics and
$g^{\alpha\beta}$ is the inverse metric tensor. The rotating regular AdS
spacetime (15) has two Killing vectors corresponding to two conserved
quantities, energy $E$ and the angular momentum $L$. We choose the following
separable solution Chandrasekhar:1992

$\displaystyle\mathcal{S}=\frac{1}{2}{m_{0}}^{2}\sigma-{E}t+{L}\phi+\mathcal{S}_{r}(r)+\mathcal{S}_{\theta}(\theta),$
(30)

where $m_{0}$ is the mass of the test particle which is zero for photon case.
Solving the Hamilton-Jacobi equation (29) using variable separable solution
(30), we derive the complete geodesic equations of motion

$\Sigma\frac{dt}{d\sigma}=\frac{r^{2}+a^{2}}{\Delta_{r}}\left[E\left(r^{2}+a^{2}\right)-a\Xi
L\right]-\frac{a}{\Delta_{\theta}}\left(aE\sin^{2}\theta-\Xi L\right),$ (31)
$\Sigma\frac{d\phi}{d\sigma}=\frac{a\Xi}{\Delta_{r}}\left[E\left(r^{2}+a^{2}\right)-a\Xi
L\right]-\frac{\Xi}{\Delta_{\theta}}\left(aE-\Xi L\csc^{2}\theta\right),$ (32)
$\Sigma\frac{dr}{d\sigma}=\sqrt{R},$ (33)

and

$\Sigma\frac{d\theta}{d\sigma}=\sqrt{\Theta},$ (34)

with

$R=\left[(a^{2}+r^{2})E-a\Xi
L\right]^{2}-\left[\left(L-aE\right)^{2}+\mathcal{K}\right]\Delta_{r}$ (35)

and

$\Theta=-\left(a\sin^{2}\theta-\xi\Xi\right)^{2}\csc^{2}\theta-\left[\left(\xi-a\right)^{2}+\mathcal{K}\right]\Delta_{\theta},$
(36)

where $\mathcal{K}$ is the Carter separable constant. The above geodesic
equations describes the motion of photons around the rotating regular AdS
black hole. To study the photon orbits we introduce two impact parameters in
term of constants ${E}$, ${L}$ and $\mathcal{K}$ as $\xi={L}/{E}$ and
$\eta=\mathcal{K}/{{E}}^{2}$. The radial equation of motion (33) can be
rewritten in terms of impact parameters $\eta$ and $\xi$

$R=\left[(a^{2}+r^{2})-a\xi\Xi\right]^{2}-\left[\left(\xi-a\right)^{2}+\eta\right]\Delta_{r}.$
(37)

The apparent shape of black hole shadow can be determined by some constant
values of the orbits $r=r_{p}$, which must satisfy the following conditions

$\mathcal{R}(r)=\frac{\partial\mathcal{R}(r)}{\partial r}=0,$ (38)

solving these equations simultaneously one can get the impact parameters
$\eta$ and $\xi$

$\displaystyle\xi(r_{p})$ $\displaystyle=$
$\displaystyle\frac{-4r_{p}\Delta_{r}+\left(r_{p}^{2}+a^{2}\right)\Delta_{r}{{}^{\prime}}}{a\Xi\Delta_{r}{{}^{\prime}}},$
(39) $\displaystyle\eta(r_{p})$ $\displaystyle=$
$\displaystyle\frac{-16r_{p}^{2}\Delta_{r}^{2}-\left(r_{p}^{2}+\frac{a^{4}}{l^{2}}\right)^{2}\Delta_{r}{{}^{\prime}}^{2}+8r_{p}\Delta_{r}\left(2a^{2}r_{p}\Xi^{2}+\left(r_{p}^{2}+\frac{a^{4}}{l^{2}}\right)\Delta_{r}{{}^{\prime}}\right)}{a^{2}\Xi^{2}\Delta_{r}{{}^{\prime}}^{2}},$

where the term $\Delta_{r}{{}^{\prime}}$ denotes the derivative of
$\Delta_{r}$ with respect to $r$ at point $r=r_{p}$ and it can be written as

$\displaystyle\Delta_{r}{{}^{\prime}}=-\frac{2}{3}\left(3\left(Me^{-k/r_{p}}(k+r_{p})/r_{p}\right)-3r_{p}-\frac{1}{3l^{2}}r_{p}(a^{2}+2r_{p}^{2})\right).$
(41)

The geometry of photon orbits around regular AdS black hole can be determined
by the impact parameters given in Eqs. (39) and (III). In the absence of
cosmological constant, these impact parameters reduce to the expressions for
nonsingular spacetime Amir:2016cen  as

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{3M\left(a^{2}(k+r_{p})+r_{p}^{2}(k-3r_{p})\right)+3r_{p}^{2}\left(a^{2}+r_{p}^{2}\right)e^{k/r_{p}}}{3aM(k+r_{p})-3ar_{p}^{2}e^{k/r_{p}}},$
(42) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle-\frac{r_{p}^{4}\left(4a^{2}Me^{k/r_{p}}(k-r_{p})+\left(r_{p}\left(r_{p}e^{k/r_{p}}-3M\right)+kM\right)^{2}\right)}{a^{2}\left(r_{p}\left(M-r_{p}e^{k/r_{p}}\right)+kM\right)^{2}},$
(43)

and which in addition, for $k=0$ exactly reduces to the Kerr spacetime such
that

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{a^{2}(M+r_{p})+r_{p}^{2}(r_{p}-3M)}{a(M-r_{p})},$ (44)
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle-\frac{r_{p}^{3}\left(r_{p}(r_{p}-3M)^{2}-4a^{2}M\right)}{a^{2}(M-r_{p})^{2}}.$
(45)

The expressions of $\eta$ and $\xi$ in Eqs (39) and (III) completely describe
the photon orbits. In order to determine the shadow images observed on the sky
at some finite distant one has to use the coordinates, say
($r_{O},\;\vartheta_{O}$), related to the orthonormal tetrads.

---
Figure 1:  Plot showing the black hole shadow in rotating regular AdS
spacetime at $r_{O}=50$ and for different values of $k$, $a$ and $1/l^{2}$.
---
Figure 2:  Plot showing the black hole shadow in rotating regular AdS
spacetime at $r_{O}=50$ and for different values of $k$, $a$, and $1/l^{2}$.

##  IV Rotating regular AdS black hole shadow

In this section, we construct the shadow of the rotating regular AdS black
holes. For this, we fix the observer position at some given Boyer-Lindquist
coordinates $(r_{O},\vartheta_{O})$ in the domain of outer communication. We
define the orthonormal tetrads at $(r_{O},\vartheta_{O})$ as
Grenzebach:2014fha ; Eiroa:2017uuq

$e_{0}=\left.\frac{\left(r^{2}+a^{2}\right)\partial_{t}+a\Xi\partial_{\phi}}{\sqrt{\Delta_{r}\Sigma}}\right|_{(r_{O},\vartheta_{O})},\;\;e_{1}=\left.\sqrt{\frac{\Delta_{\theta}}{\Sigma}}\partial_{\theta}\right|_{(r_{O},\vartheta_{O})},$
(46)

and

$e_{2}=-\left.\frac{\Xi\partial_{\phi}+a\sin^{2}\theta\partial_{t}}{\sqrt{\Delta_{\theta}\Sigma}\sin\theta}\right|_{(r_{O},\vartheta_{O})},\;\;e_{3}=-\left.\sqrt{\frac{\Delta_{r}}{\Sigma}}\partial_{r}\right|_{(r_{O},\vartheta_{O})},$
(47)

where $e_{0}$ and $e_{3}$ denotes the four velocity of the observer which
gives the spatial direction towards the black hole and the combination
$e_{0}\pm e_{3}$ is tangential to the outgoing and ingoing principal null
congruences of the metric. The coordinates
$\lambda(s)=(r(s),\theta(s),\phi(s),t(s))$ be assigned for each light ray
which in turn gives its tangent vector as

$\dot{\lambda}=\dot{r}\partial_{r}+\dot{\theta}\partial_{\theta}+\dot{\phi}\partial_{\phi}+\dot{t}\partial_{t},$
(48)

and the tangent vector in terms of the coordinates $\alpha$ and $\beta$
results

$\dot{\lambda}=\gamma\left(-e_{0}+\sin\alpha\cos\beta
e_{1}+\sin\alpha\sin\beta e_{2}+\cos\alpha e_{3}\right).$ (49)

For each light ray the scalar factor $\gamma$ can be calculated from Eqs. (48)
and (49) as

$\gamma=g(\dot{\lambda},e_{0})=g\left(\dot{\lambda},\frac{r^{2}+a^{2}}{\sqrt{\Delta_{r}\Sigma}}\partial_{t}+\frac{a\Xi}{\sqrt{\Delta_{r}\Sigma}}\partial_{\phi}\right)=\left.\frac{-E\left(r^{2}+a^{2}\right)+a\Xi
L}{\sqrt{\Delta_{r}\Sigma}}\right|_{(r_{O},\vartheta_{O})},$ (50)

and by comparing the corresponding coefficients in Eqs. (48) and (49), we have

$\sin\alpha=\left.\sqrt{1-\left[\frac{\dot{r}\Sigma}{E\left(r^{2}+a^{2}\right)-a\Xi
L}\right]^{2}}\right|_{(r_{O},\vartheta_{O})}$ (51)

and

$\sin\beta=\left.\frac{\sqrt{\Delta_{\theta}}\sin\theta}{\Xi\sqrt{\Delta_{r}}\sin\alpha}\left[\frac{\dot{\phi}\Sigma\Delta_{r}}{E\left(r^{2}+a^{2}\right)-a\Xi
L}-a\Xi\right]\right|_{(r_{O},\vartheta_{O})},$ (52)

by substituting the impact parameters (39) and (III) in Eqs. (32) and (33), we
find

$\sin\alpha=\left.\frac{\sqrt{\left[\left(\xi(r_{p})-a\right)^{2}+\eta(r_{p})\right]\Delta_{r}}}{r^{2}+a^{2}-a\Xi\xi(r_{p})}\right|_{(r_{O},\vartheta_{O})},$
(53)

and

$\sin\beta=\left.\frac{\sqrt{\Delta_{r}}\sin\theta}{\sqrt{\Delta_{\theta}}\sin\alpha}\left[\frac{a-\Xi\xi(r_{p})\csc^{2}\theta}{a\Xi\xi(r_{p})-\left(r^{2}+a^{2}\right)}\right]\right|_{(r_{O},\vartheta_{O})}.$
(54)

---
Figure 3:  Plots showing the non-rotating regular AdS black hole shadow and
its observable $R_{s}$ for different values of $k$ with $1/l^{2}$ at
$r_{O}=50$.

In order to obtain the apparent shape of the shadow, we have considered the
stereographic projection of celestial sphere onto a plane, where the Cartesian
coordinates in this plane takes the form

$\displaystyle
X=-2\tan\left(\frac{\alpha(r_{p})}{2}\right)\sin\left(\beta(r_{p})\right),$
$\displaystyle
Y=-2\tan\left(\frac{\alpha(r_{p})}{2}\right)\cos\left(\beta(r_{p})\right),$
(55)

which satisfies the relation

$\displaystyle X^{2}+Y^{2}=4\tan^{2}\left(\frac{\alpha(r_{p})}{2}\right).$
(56)

---
Figure 4:  Plots showing the variation of observables $R_{s}$ and $\delta_{s}$
with $1/l^{2}$ for different values of $a$ and $k$ at $r_{O}=50$.
---
Figure 5: The contours of $R_{s}$ (solid red lines) and $\delta_{s}$ (dashed
gray lines) in ($a,1/l^{2}$) plane. The point of intersection of $R_{s}$ and
$\delta_{s}$ lines gives the values of black hole parameters.

The shapes of regular AdS black hole shadow can be found by plotting the
contour of Eq. (56). The shape and size of the black hole shadow mainly depend
on its spin parameter $a$, the curvature radius $1/l^{2}$ and free parameter
$k$. We plot the contour of regular AdS black hole shadow in Fig. (1) and (2)
for various values of $a$, $1/l^{2}$, and $k$. The size of the black hole
shadow and distortion increases for the increasing values of the $1/l^{2}$
(cf. Fig. 1) while the size of black hole shadow decreases for the increasing
values of free parameter $k$ (cf. Fig. 2). We also plot the spherical ($a=0$)
regular AdS black hole shadow in Fig. (3) which appears as a perfect circle
and increases with the increasing values $1/l^{2}$.

Next, we define two observables, namely the shadow radius $R_{s}$ and the
distortion parameter $\delta_{s}$ Hioki:2009na . The observables $R_{s}$,
which measures, can be defined Hioki:2009na

$R_{s}=\frac{(X_{t}-X_{r})^{2}+{Y_{t}}^{2}}{2|{X_{t}}-{X_{r}}|},$ (57)

where $(X_{t},\;Y_{t})$ and $(X_{r},\;Y_{r})$ are the topmost and the
rightmost positions of the celestial coordinates from where the contour of the
black hole shadow passes. The second observable $\delta_{s}$ depend on the
black hole shadow radius $R_{s}$ and the dent $D_{s}$ which occur due to the
rotation of the black hole. The expression of $\delta_{s}$ reads

$\delta_{s}=\frac{D_{s}}{R_{s}}.$ (58)

$a=0.5M,k=0.1M$
---
radius of curvature $(M^{2}/l^{2})$ |  shadow radius $(R_{s}/M)$ |  distortion parameter $(\delta_{s})$
0.05 |  0.90 |  0.06
0.10 |  1.11 |  0.08
0.15 |  1.22 |  0.10
0.20 |  1.30 |  0.12

$a=0.6M,k=0.2M$
---
radius of curvature $M^{2}/l^{2}$ |  shadow radius $(R_{s}/M)$ |  distortion parameter $(\delta_{s})$
0.05 |  0.89 |  0.11
0.10 |  1.09 |  0.17
0.15 |  1.21 |  0.23
0.20 |  1.28 |  0.27

Table 1: Variation of shadow radius and distortion parameter with parameter
$M^{2}/l^{2}$.

In Fig. 4 we plot the variation of the black hole shadow radius $R_{s}$ and
the distortion parameter $\delta_{s}$. For fixed values of the spin parameter
and free parameter, the radius of the black hole shadow $R_{s}$ and the
distortion parameter $\delta_{s}$ increases with the radius of curvature
parameter $1/l^{2}$ (cf. Fig. 4). In Fig. 5 we plot the shadow radius $R_{s}$
and distortion parameter $\delta_{s}$ in $(a,1/l^{2})$ plane. The point of
intersection of $R_{s}$ and $\delta_{s}$ lines gives the spin parameter $a$
and radius of curvature parameter $1/l^{2}$. If we compare our results with
nonsigular black hole shadow Amir:2016cen , we find in ads spacetime the size
of black hole shadow appears larger and gets more distorted for the increasing
values of parameter $1/l^{2}$ (cf. Fig. 1). In comparison with the Kerr black
hole Chandrasekhar:1992 , our results shows that the size of shadow appears
bigger in regular ads spacetime. A detailed numerical values of shadow radius
and distortion parameter for different values of spin parameter, free
parameter and have shown in the Table 1.

##  V Energy emission rate

In the previous discussion, we derived the required derivation for the
nonsingular black hole silhouettes in asymptotically anti-de sitter
spacetimes. The silhouettes of the black holes are responsible for the high
energy absorption cross-section for the asymptotically flat or (Anti)-de
Sitter observers. The absorption cross section of the black holes slightly
modulates near a limiting constant value at high energies.

---
Figure 6:  Plot showing the energy emission rate of the photon region .

As a good approximation, we can use the the limiting constant value of the
absorption cross section for a nearly spherically symmetric black hole as

$\sigma_{\rm lim}\approx\pi R_{s}^{2}.$ (59)

We can use such limiting value of the geometric cross-section in order to
calculate the energy emission rate

$\displaystyle\frac{d^{2}{E(\omega)}}{d\omega
dt}=\frac{2\pi^{2}R_{s}^{2}}{e^{\omega/T}-1}\omega^{3},$ (60)

where $\omega$ is the frequency of the emitted photon and the temperature $T$
of the nonsingular-AdS black hole in terms of the event horizon radius is
expressed as

$\displaystyle T=T_{\rm Kerr-AdS}-\frac{k\left(1-r_{+}^{2}/\l^{2}\right)}{4\pi
r_{+}^{2}(r_{+}^{2}+a^{2})},$ (61)

where $T_{\rm Kerr-AdS}$ is the corresponding temperature for the Kerr-AdS
black holes

$\displaystyle T_{\rm Kerr-
AdS}=\frac{r_{+}^{2}\left(1+3r_{+}^{2}/l^{2}\right)-a^{2}\left(1-r_{+}^{2}/l^{2}\right)}{4\pi
r_{+}\left(r_{+}^{2}+a^{2}\right)}.$ (62)

In Fig. 6 we have plot the energy emission rate of the black hole with
$\omega$ for different values of curvature radius term. It is shown in the
figure that there exists a peak of energy emission rate, which decreases as we
increase $1/l^{2}$ term and other black hole parameters.

##  VI Conclusion

In rotating Kerr spacetimes the shapes of black hole shadow get distorted
mainly by the spin parameter $a$. In other theories of gravity $viz.$ the MGs,
some other parameters may enhance or decrease the shadow of the black hole. In
this paper, we have investigated the shadow of rotating regular black hole in
the presence of a cosmological constant. The size of the black hole shadow and
distortion parameter increases with the increasing values of curvature radius
parameter $1/l^{2}$. The black hole shadow appears smaller and more distorted
in comparison to the regular and Kerr spacetime (cf. Fig. 2). For a better
understanding of black hole shadow, we have also calculated observables the
shadow radius $R_{s}$ and the distortion parameter $\delta_{s}$. The
observables $R_{s}$ and $\delta_{s}$ increases with the increasing values and
$1/l^{2}$ and the effective size of the shadow radius decreases with the free
parameter $k$. From the study of energy emission rate we conclude that the
electromagnetic spectrum emitted by the shadow region of regular black holes
in AdS spacetime decreases for the increasing values of the parameter
$1/l^{2}$.

## Acknowledgement

The research of M. S. A. is supported by the National Postdoctoral Fellowship
of the Science and Engineering Research Board (SERB), Department of Science
and Technology (DST), Government of India, File No., PDF/2021/003491. B. P. S.
would like to thank Tulas Institute for providing research facilities.

## References

  * (1) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” Astrophys. J. 930, no. 2, L12 (2022).
  * (2) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration,” Astrophys. J. 930, no. 2, L13 (2022).
  * (3) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole,” Astrophys. J. 930, no. 2, L14 (2022).
  * (4) K. Akiyama et al. [Event Horizon Telescope Collaboration], ”First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass,” Astrophys. J. 930, no. 2, L15 (2022).
  * (5) K. Akiyama et al. [Event Horizon Telescope Collaboration], ”First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole,” Astrophys. J. 930, no. 2, L16 (2022).
  * (6) K. Akiyama et al. [Event Horizon Telescope Collaboration], ”First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric,” Astrophys. J. 930, no. 2, L17 (2022).
  * (7) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. 875, no. 1, L1 (2019)
  * (8) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. II. Array and Instrumentation,” Astrophys. J. 875, no. 1, L2 (2019)
  * (9) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” Astrophys. J. 875, no. 1, L3 (2019)
  * (10) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” Astrophys. J. 875, no. 1, L4 (2019)
  * (11) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” Astrophys. J. 875, no. 1, L5 (2019)
  * (12) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. 875, no. 1, L6 (2019).
  * (13) J. M. Bardeen, in Black Holes, ed. by C. De Witt and B.S. De Witt and B.S. De Witt (Gorden and Breach, New York, 1973), 215-239.
  * (14) P. J. Young, Phys. Rev. D 14, 3281 (1976).
  * (15) T. Johannsen, Astrophys. J. 777, 170 (2013).
  * (16) A. de Vries, Classical Quantum Gravity 17, 123 (2000).
  * (17) L. Amarilla, E. F. Eiroa and G. Giribet, Phys. Rev. D 81, 124045 (2010).
  * (18) L. Amarilla and E. F. Eiroa, Phys. Rev. D 87, 044057 (2013).
  * (19) A. Yumoto, D. Nitta, T. Chiba and N. Sugiyama, Phys. Rev. D 86, 103001 (2012).
  * (20) A. Abdujabbarov, M. Amir, B. Ahmedov and S. G. Ghosh, Phys. Rev. D 93, 104004 (2016).
  * (21) M. Amir and S. G. Ghosh, Phys. Rev. D 94, 024054 (2016).
  * (22) N. Tsukamoto, Z. Li and C. Bambi, J. Cosmol. Astropart. Phys. 1406, 043 (2014).
  * (23) K. Hioki and K. i. Maeda, Phys. Rev. D 80, 024042 (2009).
  * (24) C. Bambi and N. Yoshida, Class. Quant. Grav. 27, 205006 (2010).
  * (25) C. Goddi et al., Int. J. Mod. Phys. D 26, 1730001 (2016).
  * (26) R. Takahashi, Publ. Astron. Soc. Jap. 57, 273 (2005).
  * (27) S. W. Wei and Y. X. Liu, J. Cosmol. Astropart. Phys. 11, 063 (2013).
  * (28) A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov, and U. Camci, Astrophys. Space Sci. 344, 429 (2013).
  * (29) L. Amarilla and E. F. Eiroa, Phys. Rev. D 85, 064019 (2012).
  * (30) C. Bambi and K. Freese, Phys. Rev. D 79, 043002 (2009).
  * (31) F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88, 064004 (2013).
  * (32) M. Wang, S. Chen and J. Jing, J. Cosmol. Astropart. Phys. 1710, 051 (2017).
  * (33) J. Schee and Z. Stuchlik, Int. J. Mod. Phys. D 18, 983 (2009).
  * (34) A. Grenzebach, V. Perlick, and C. Lämmerzahl, Phys. Rev. D 89, 124004 (2014).
  * (35) B. P. Singh, Annals Phys. 441, 168892 (2022)
  * (36) R. Kumar, B. P. Singh and S. G. Ghosh, Annals Phys. 420, 168252 (2020).
  * (37) R. Kumar, B. P. Singh, M. S. Ali and S. G. Ghosh, Phys. Dark Univ. 34, 100881 (2021).
  * (38) R. Kumar and S. G. Ghosh, JCAP 07, 053 (2020).
  * (39) B. Narzilloev, I. Hussain, A. Abdujabbarov and B. Ahmedov, Eur. Phys. J. Plus 137, no.5, 645 (2022) doi:10.1140/epjp/s13360-022-02872-8 [arXiv:2205.11760 [gr-qc]].
  * (40) S. W. Wei, Y. C. Zou, Y. X. Liu and R. B. Mann, JCAP 08, 030 (2019)
  * (41) U. Papnoi and F. Atamurotov, Phys. Dark Univ. 35, 100916 (2022) doi:10.1016/j.dark.2021.100916 [arXiv:2111.15523 [gr-qc]].
  * (42) F. Atamurotov, U. Papnoi and K. Jusufi, Class. Quant. Grav. 39, no.2, 025014 (2022)
  * (43) V. Perlick, O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Phys. Rev. D 92, no. 10, 104031 (2015)
  * (44) F. Atamurotov and B. Ahmedov, Phys. Rev. D 92, 084005 (2015).
  * (45) A. Abdujabbarov, B. Toshmatov, Z. Stuchlik and B. Ahmedov, Int. J. Mod. Phys. D 26, 1750051 (2016).
  * (46) F. Atamurotov, K. Jusufi, M. Jamil, A. Abdujabbarov and M. Azreg-Aïnou, Phys. Rev. D 104, no.6, 064053 (2021)
  * (47) U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Phys. Rev. D 90, 024073 (2014).
  * (48) A. Abdujabbarov, F. Atamurotov, N. Dadhich, B. Ahmedov and Z. Stuchlik, Eur. Phys. J. C 75, 399 (2015).
  * (49) B. P. Singh and S. G. Ghosh, Annals Phys. 395, 127 (2018).
  * (50) M. Amir, B. P. Singh and S. G. Ghosh, Eur. Phys. J. C 78, 399 (2018).
  * (51) P.V. P. Cunha, C.A. R. Herdeiro, E. Radu, and H.F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015).
  * (52) P. V. P. Cunha, C. A. R. Herdeiro, E. Radu and H. F. Runarsson, Int. J. Mod. Phys. D 25, 1641021 (2016).
  * (53) E. F. Eiroa and C. M. Sendra, Eur. Phys. J. C 78, 91 (2018).
  * (54) A. Belhaj, H. Belmahi and M. Benali, Phys. Lett. B 821, 136619 (2021).
  * (55) A. Belhaj, H. Belmahi, M. Benali, H. El Moumni, M. A. Essebani and M. B. Sedra, [arXiv:2202.10892 [gr-qc]].
  * (56) A. Belhaj, M. Benali, A. E. Balali, W. E. Hadri and H. El Moumni, Int. J. Geom. Meth. Mod. Phys. 18, no.12, 2150188 (2021).
  * (57) M. Afrin and S. G. Ghosh, Universe 8, no.1, 52 (2022).
  * (58) J. M. Bardeen,  Black Holes, Edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973, p. 215).
  * (59) B. Toshmatov, Z. Stuchlík and B. Ahmedov, Phys. Rev. D 95, 084037 (2017).
  * (60) H. Erbin, Universe 3, 19 (2017).
  * (61) S. G. Ghosh, Eur. Phys. J. C 75, 532 (2015).
  * (62) J. F. Plebanski and M. Demianski, Ann. Phys. 98 (1976).
  * (63) Z. Xu and J. Wang, Phys. Rev. D 95, no.6, 064015 (2017)
  * (64) C. Bambi, JCAP 1308, 055 (2013).
  * (65) C. Bambi, Astron. Rev. 8, 4 (2013).
  * (66) C. Bambi, Mod. Phys. Lett. A 26, 2453 (2011).
  * (67) A. Cardenas-Avendano, J. Jiang and C. Bambi, Phys. Lett. B 760, 254 (2016).
  * (68) B. Carter, Phys. Rev. 174, 1559 (1968).
  * (69) S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992).
  * (70) O. Y. Tsupko, Phys. Rev. D 95, no. 10, 104058 (2017).
  * (71) H. Falcke, F. Melia, E. Agol, Astrophys. J. Lett. 528, L13 (2000).
  * (72) S. R. Das, G. W. Gibbons and S. D. Mathur, Phys. Rev. Lett. 78, 417 (1997).
  * (73) Y. Decanini, G. Esposito-Farese and A. Folacci, Phys. Rev. D 83, 044032 (2011).
  * (74) B. Mashhoon, Phys. Rev. D 7, 2807 (1973).