# A Lyapunov-like Characterization of Predefined-Time Stability

Esteban Jiménez-Rodríguez  Aldo Jonathan Muñoz-Vázquez  Juan Diego Sánchez-
Torres  Michael Defoort and Alexander G. Loukianov Submitted on: 11/19/2018.E.
Jiménez-Rodríguez and A. G. Loukianov are with the Department of Electrical
Engineering, Cinvestav-Guadalajara, Zapopan, Jalisco, 45019 México (e-mail:
{ejimenezr, louk}@gdl.cinvestav.mx). A. J. Muñoz-Vázquez is with
CONACYT–México, and with the School of Engineering at the Autonomous
University of Chihuahua, Chihuahua, Chihuahua, 31100 México (e-mail:
aldo.munoz.vazquezz@gmail.com).J. D. Sánchez-Torres is with the Research
Laboratory on Optimal Design, Devices and Advanced Materials -OPTIMA-,
Department of Mathematics and Physics, ITESO, Tlaquepaque, Jalisco, 45604
México (e-mail: dsanchez@iteso.mx).M. Defoort is with the LAMIH, UMR CNRS
8201, Polytechnic University of Hauts-de-France, Valenciennes, 59313 France
(e-mail: michael.defoort@uphf.fr).

###### Abstract

This technical note studies Lyapunov-like conditions to ensure a class of
dynamical systems to exhibit predefined-time stability. The origin of a
dynamical system is predefined-time stable if it is fixed-time stable and an
upper bound of the settling-time function can be arbitrarily chosen a priori
through a suitable selection of the system parameters. We show that the
studied Lyapunov-like conditions allow to demonstrate equivalence between
previous Lyapunov theorems for predefined-time stability for autonomous
systems. Moreover, the obtained Lyapunov-like theorem is extended for
analyzing the property of predefined-time ultimate boundedness with predefined
bound, which is useful when analyzing uncertain dynamical systems. Therefore,
the proposed results constitute a general framework for analyzing predefined-
time stability, and they also unify a broad class of systems which present the
predefined-time stability property. On the other hand, the proposed framework
is used to design robust controllers for affine control systems, which induce
predefined-time stability (predefined-time ultimate boundedness of the
solutions) w.r.t. to some desired manifold. A simulation example is presented
to show the behavior of a developed controller, especially regarding the
settling time estimation.

Nonlinear control systems, Predefined-time stability, Sliding mode control,
Stability of nonlinear systems.

##  1 Introduction

The development of control, observation, and optimization algorithms for many
industrial applications requires the fulfillment of certain time-response
constraints in order to comply with a certain quality or safety standards. To
deal with these requirements, several developments concerning the finite-time
stability concept have been carried out in [1, 2, 3, 4, 5]. Nevertheless, the
finite settling time provided by finite-time convergent algorithms is usually
an unbounded function of the system’s initial conditions. This concern gives
rise to a stronger form of stability called fixed-time stability, where the
settling-time function is bounded. The notion of fixed-time stability has been
investigated in [6, 7, 8].

Although the concept of fixed-time stability represents a significant
advantage over the concept of finite-time stability because of the boundedness
of the settling time, it cannot be guaranteed in general that the convergence
time can be arbitrarily selected through the system tunable parameters. To
overcome this mentioned drawback, it is necessary to consider another class of
dynamical systems that exhibit the property of predefined-time stability,
which has been studied in [9, 10]. For these systems, an upper bound of the
settling-time function can be arbitrarily chosen through an appropriate
selection of the system parameters.

On the other hand, Lyapunov methods have proved to be a handy tool for
analyzing and designing nonlinear control systems [11, 12]. In the same
manner, they have been highly used for convergence rate estimation in systems
exhibiting finite- and fixed-time stability properties [4, 8, 13], and in
particular in systems with sliding modes [14]. For systems exhibiting the
predefined-time stability property, similar methods have been applied. For
instance, different Lyapunov-like theorems for predefined-time stability were
proposed in [15, 9, 16, 17] allowing the development of several control
applications [18, 19, 20].

This paper investigates Lyapunov-like sufficient conditions for predefined-
time stability of autonomous systems. The derived Lyapunov-like theorem allows
to establish equivalence with several previous Lyapunov-like theorems [9, 16,
17], unifying all the past contributions in predefined-time stability for
autonomous systems. Moreover, this framework is extended to the analysis of
predefined-time ultimate boundedness, which is specially usefull when
analyzing uncertain systems. To demonstrate the applicability of the proposed
framework, the developed results are used to design a family of continuous
(respectively, discontinuous) controllers, which ensure predefined-time
ultimate boundedness of the solutions to an arbitrarily small vicinity of a
desired manifold (respectively, ensure predefined-time stability to a desired
manifold). Finally, all the mentioned properties are validated through a
simulation example, in order to show the behavior of the proposed controller,
especially regarding the settling time estimation.

##  2 Preliminaries

###  2.1 Notation

Throughout the paper, the following notation is prevalent:

  * •

$\mathbb{R}$ is the set of real numbers,
$\mathbb{R}_{+}=\\{x\in\mathbb{R}\,:\,x>0\\}$, $\mathbb{R}_{\geq
0}=\\{x\in\mathbb{R}\,:\,x\geq 0\\}$ and
$\bar{\mathbb{R}}_{+}=\mathbb{R}_{+}\cup\\{\infty\\}$.

  * •

For $\bm{x}\in\mathbb{R}^{n}$, $\bm{x}^{T}$ denotes its transpose,
$\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert=\sqrt{\bm{x}^{T}\bm{x}}$
and, for $r\in\mathbb{R}_{+}$,
$B_{r}(\bm{x})=\\{\bm{y}\in\mathbb{R}^{n}\,:\,\left\lvert\left\lvert\bm{y}-\bm{x}\right\rvert\right\rvert<r\\}$.

  * •

For any real number $h$, the functions
$\left\lfloor\cdot\right\rceil^{h}:\mathbb{R}\to\mathbb{R}$ and
$\left\lvert\left\lfloor\bm{\cdot}\right\rceil\right\rvert^{h}:\mathbb{R}^{n}\to\mathbb{R}^{n}$
are defined as $\left\lfloor x\right\rceil^{h}=|x|^{h}\mathrm{sign}(x)$ for
any $x\in\mathbb{R}\setminus\\{0\\}$ and
$\left\lvert\left\lfloor\bm{x}\right\rceil\right\rvert^{h}=\frac{\bm{x}}{\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert^{1-h}}$
for any $\bm{x}\in\mathbb{R}^{n}\setminus\\{\bm{0}\\}$, respectively.
Moreover, if $h>0$, $\left\lfloor 0\right\rceil^{h}=0$ and
$\left\lvert\left\lfloor\bm{0}\right\rceil\right\rvert^{h}=\bm{0}$.

  * •

Whereas $\dot{\bm{x}}=\frac{d\bm{x}}{dt}$ denotes the first derivative of the
function $\bm{x}:\mathbb{R}\to\mathbb{R}^{n}$ with respect to time,
$\theta^{\prime}(z)=\frac{d\theta}{dz}$ denotes the first derivative of the
function $\theta:\mathbb{R}\to\mathbb{R}$ with respect to the variable $z$,
which may represent anything but the time variable $t$.

  * •

For $\alpha,\beta\in\mathbb{R}_{+}$,
$\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t}\text{d}t$ is the Gamma
Function and
$\mathcal{B}(\alpha,\beta)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\text{d}t$
is the Beta Function; additionally,
$\gamma(\alpha,r)=\int_{0}^{r}t^{\alpha-1}e^{-t}\text{d}t$ and
$P(\alpha,r)=\frac{\gamma(\alpha,r)}{\Gamma(\alpha)}$ are the Incomplete Gamma
Function and the regularized Incomplete Gamma Function, respectively, which
are defined for all $r\in\mathbb{R}_{\geq 0}$; finally
$b(\alpha,\beta,r)=\int_{0}^{r}t^{\alpha-1}(1-t)^{\beta-1}\text{d}t$ and
$I(\alpha,\beta,r)=\frac{b(\alpha,\beta,r)}{\mathcal{B}(\alpha,\beta)}$ are
the Incomplete Beta Function and the regularized Incomplete Beta Function,
respectively, which are defined for all $r\in[0,1]$ [21].

###  2.2 On predefined-time stability

Consider the following autonomous system:

$\dot{\bm{x}}=\bm{f}(\bm{x};\bm{\rho}),\ \ \bm{x}(0)=\bm{x}_{0},$ (1)

where $\bm{x}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}$ is the system state, the
vector $\bm{\rho}\in\mathbb{R}^{l}$ stands for the tunable parameters of (1).
The function $\bm{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ may be
discontinuous, and such that the solutions of (1) exist and are unique in the
sense of Filippov (see [22] and [23, Proposition 5]). Thus,
$\bm{\Phi}(t,\bm{x}_{0})$ denotes the solution of (1) starting from
$\bm{x}_{0}\in\mathbb{R}^{n}$ at $t=0$. Moreover, the origin $\bm{x}=\bm{0}$
is the unique equilibrium point of (1).

###### Remark 1

The parameter dependent system (1) is equivalent to the controlled system

$\dot{\bm{x}}=\bm{g}(\bm{x},\bm{u}),$ (2)

where $\bm{g}:\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}$, the
control $\bm{u}\in\mathbb{R}^{m}$ is a feedback function of $\bm{x}$ with
tunable parameters $\bm{\rho}$, i.e., $\bm{u}=\bm{\phi}(\bm{x};\bm{\rho})$,
with $\bm{\phi}:\mathbb{R}^{n}\to\mathbb{R}^{m}$. Substituting
$\bm{u}=\bm{\phi}(\bm{x};\bm{\rho})$ in (2) eliminates $\bm{u}$ and yields
$\bm{f}(\bm{x};\bm{\rho}):=\bm{g}(\bm{x},\bm{\phi}(\bm{x};\bm{\rho}))$.

All the notions defined and treated hereafter are global, so we will omit to
indicate it.

######  Definition 1 (Stability notions [24])

The origin of (1) is said to be

  * •

Lyapunov stable if for any $\bm{x}_{0}\in\mathbb{R}^{n}$, the solution
$\bm{\Phi}(t,\bm{x}_{0})$ is defined for all $t\geq 0$, and for any
$\epsilon>0$, there is $\delta>0$ such that for any
$\bm{x}_{0}\in\mathbb{R}^{n}$, if $\bm{x}_{0}\in B_{\delta}(\bm{0})$ then
$\bm{\Phi}(t,\bm{x}_{0})\in B_{\epsilon}(\bm{0})$ for all $t\geq 0$;

  * •

asymptotically stable if it is Lyapunov stable and $\bm{\Phi}(t,\bm{x}_{0})\to
0$ as $t\to\infty$, for any $\bm{x}_{0}\in\mathbb{R}^{n}$;

  * •

finite-time stable if it is Lyapunov stable and for any
$x_{0}\in\mathbb{R}^{n}$ there exists $0\leq\tau<\infty$ such that
$\bm{\Phi}(t,\bm{x}_{0})=\bm{0}$ for all $t\geq\tau$. The function
$T(\bm{x}_{0})=\inf\left\\{\tau\geq 0:\bm{\Phi}(t,\bm{x}_{0})=\bm{0},\,\forall
t\geq\tau\right\\}$ is called the settling-time function of (1);

  * •

fixed-time stable if it is finite-time stable and the settling-time function
of (1), $T(\bm{x}_{0})$, is bounded on $\mathbb{R}^{n}$, i.e. there exists
$T_{\max}$ such that $\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T(\bm{x}_{0})\leq
T_{\max}<\infty$

###### Example 1

Consider system

$\displaystyle\begin{split}\dot{x}&=-\frac{1}{\rho_{1}}\left\lfloor
x\right\rceil^{\rho_{2}}-\rho_{1}\left\lfloor
x\right\rceil^{2-\rho_{2}},\end{split}$ (3)

where $x\in\mathbb{R}$ is the system state,
$\bm{\rho}=\left[\rho_{1},\,\rho_{2}\right]^{T}\in\mathbb{R}^{2}$ is the
vector of tunable parameters of (3), which comply to $\rho_{1}>0$ and
$0<\rho_{2}<1$. Using [8, Lemma 1], one can easily show that the origin of (3)
is fixed-time stable. Moreover, from [17, Theorem 1], the settling-time
function of (3) satisfies

$\sup_{x_{0}\in\mathbb{R}}T(x_{0})=\frac{\mathcal{B}\left(1/2,1/2\right)}{2\rho_{1}^{-1/2}\rho_{1}^{1/2}(1-\rho_{2})}=\frac{\pi}{2(1-\rho_{2})}>\frac{\pi}{2}.$

This example shows that the convergence time for system (3), whose origin is
fixed-time stable, cannot be reduced arbitrarily no matter how the parameters
$\bm{\rho}$ are tuned. The case when the convergence time can be arbitrarily
assigned through an appropriate tuning of the system parameters $\bm{\rho}$
corresponds to the notion of predefined-time stability, which is defined as
follows:

###### Definition 2

The origin of (1) is said to be predefined-time stable if it is fixed-time
stable and for any $T_{c}\in\mathbb{R}_{+}$, there exists some
$\bm{\rho}\in\mathbb{R}^{l}$ such that the settling-time function of (1)
satisfies

$\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T(\bm{x}_{0})\leq T_{c}.$

###### Example 2

Consider system (see [8, 6])

$\displaystyle\begin{split}\dot{x}&=-\left\lfloor\rho_{1}\left\lfloor
x\right\rceil^{\rho_{3}}+\rho_{2}\left\lfloor
x\right\rceil^{\rho_{4}}\right\rceil^{\rho_{5}},\end{split}$ (4)

where $x\in\mathbb{R}$ is the state of the system,
$\bm{\rho}=\left[\rho_{1},\,\rho_{2},\,\rho_{3},\,\rho_{4},\,\rho_{5}\right]^{T}\in\mathbb{R}^{5}$
is the vector of tunable parameters of (4), which comply to
$\rho_{1},\rho_{2},\rho_{5}>0$ and $0<\rho_{5}\rho_{3}<1<\rho_{5}\rho_{4}$.
The origin of (4) is fixed-time stable, by [8, Lemma 1]. Moreover, given
$T_{c}\in\mathbb{R}_{+}$, there exist
$\rho_{1}=\rho_{2}=\frac{\Gamma(1/4)^{4}}{4\pi T_{c}^{2}}$, $\rho_{3}=1$,
$\rho_{4}=3$ and $\rho_{5}=\frac{1}{2}$, such that the settling-time function
of (4) satisfies (see [17, Theorem 1])

$\sup_{x_{0}\in\mathbb{R}}T(x_{0})=\frac{\Gamma(1/4)^{2}}{\left(\frac{\Gamma(1/4)^{4}}{4\pi
T_{c}^{2}}\right)^{1/2}\Gamma(1/2)(3-1)}=T_{c}.$

Thus, the origin of system (4) is, in fact, predefined-time stable.

The following proposition is an immediate consequence of Definition 2, of
predefined-time stability.

###### Proposition 1

If a system does not have tunable parameters, then its origin is not
predefined-time stable.

From Proposition 1, every system with fixed-time stable origin whose
parameters are fixed numerical values (they are not tunable), cannot exhibit
the predefined-time stability property.

On the other hand, when dealing with systems subject to uncertainties or
external perturbations, it is difficult or even impossible to ensure exact
convergence to the origin. Instead, it is common to provide convergence not to
the origin but to a vicinity of it. In this sense, it would be useful to
ensure that, not only the convergence time can be arbitrarily assigned, but
also that the radius of the vicinity can be arbitrarily selected through an
appropriate tuning of the parameters of the system. This notion is formally
defined as follows:

###### Definition 3

A solution $\bm{\Phi}(t,\bm{x}_{0})$ of (1) is said to be predefined-time
ultimately bounded with predefined bound if for any
$T_{c},b\in\mathbb{R}_{+}$, there exist some $\bm{\rho}\in\mathbb{R}^{l}$ such
that for any $\bm{x}_{0}\in\mathbb{R}^{n}$,
$\left\lvert\left\lvert\bm{\Phi}(t,\bm{x}_{0})\right\rvert\right\rvert\leq b$
for all $t\geq T_{c}$.

###  2.3 Class $\mathcal{K}^{1}$ functions

Inspired in the class $\mathcal{K}$ functions in [25, Definition 1] and [12,
Definition 4.2], the class $\mathcal{K}^{1}$ functions are defined as follows:

######  Definition 4 ($\mathcal{K}^{1}$ functions)

A scalar continuous function $\kappa:\mathbb{R}_{\geq 0}\to\left[0,1\right)$
is said to belong to class $\mathcal{K}^{1}$, denoted as
$\kappa\in\mathcal{K}^{1}$, if it is strictly increasing, $\kappa(0)=0$ and
$\kappa(r)\to 1$ as $r\to\infty$.

If $\kappa\in\mathcal{K}^{1}$ is also differentiable, it is said to be a
differentiable class $\mathcal{K}^{1}$ function. In such a case, there exists
a continuous function $\Phi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{+}$ such that
$\frac{d\kappa}{dr}=\Phi(r)>0$.

The above can be equivalently written in an integral form as
$\kappa(r)=\int_{0}^{r}\Phi(z)\text{d}z$. Since $\kappa(r)\to 1$ as
$r\to\infty$, the function $\Phi$ is required to satisfy
$\int_{0}^{\infty}\Phi(z)\text{d}z=1$, i.e., functions $\Phi$ and $\kappa$ can
be viewed as probability density functions and cumulative distribution
functions, respectively, of positive random variables.

###### Proposition 2

Every class $\mathcal{K}^{1}$ function is bijective.

######  Proof 2.1.

Let $\kappa\in\mathcal{K}^{1}$. It is injective because $\kappa$ is continuous
and strictly increasing. Moreover, since $\kappa(0)=0$, $\kappa$ is continuous
and strictly increasing, and $\lim_{r\to\infty}\kappa(r)=1$, its image is
$\kappa(\mathbb{R}_{\geq 0})=[0,1)$. Thus, it is surjective. Hence, it is
concluded that $\kappa$ is bijective.

Since every class $\mathcal{K}^{1}$ function is bijective, their inverse
exist.

######  Proposition 2.2.

Let $\kappa\in\mathcal{K}^{1}$. Then:

  * (i)

$\kappa^{-1}(0)=0$;

  * (ii)

$\kappa^{-1}$ is continuous (i.e., $\kappa$ is a homeomorphism) and strictly
increasing.

  * (iii)

$\lim_{r\to 1^{-}}\kappa^{-1}(r)=\infty$.

######  Proof 2.3.

These properties follow directly from Definition 4, and the fact that the
inverse of a strictly increasing function is continuous and strictly
increasing.

The next lemma states some useful properties of class $\mathcal{K}_{\infty}$
and class $\mathcal{K}^{1}$ functions, which will be used in the next section.

######  Lemma 2.4.

Let $\alpha\in\mathcal{K}_{\infty}$ (see [25, Definition 1]) and
$\kappa_{1},\kappa_{2}\in\mathcal{K}^{1}$. Then,
$\kappa_{1}\circ\alpha\in\mathcal{K}^{1}$, and
$\kappa_{1}^{-1}\circ\kappa_{2}\in\mathcal{K}_{\infty}$.

######  Proof 2.5.

The composition of increasing functions is increasing. Moreover, note that
$(\kappa_{1}\circ\alpha)(0)=\kappa_{1}(\alpha(0))=\kappa_{1}(0)=0$ and
$(\kappa_{1}^{-1}\circ\kappa_{2})(0)=\kappa_{1}^{-1}(\kappa_{2}(0))=\kappa_{1}^{-1}(0)=0$.
Finally, since $\kappa_{1}$ is an homeomorphism,

$\lim_{r\to\infty}(\kappa_{1}\circ\alpha)(r)=\kappa_{1}\left(\lim_{r\to\infty}\alpha(r)\right)=1,$

and

$\lim_{r\to\infty}(\kappa_{1}^{-1}\circ\kappa_{2})(r)=\kappa_{1}^{-1}\left(\lim_{r\to\infty}\kappa_{2}(r)\right)=\infty.$

######  Example 2.6.

Some examples of $\mathcal{K}^{1}$ functions are:

  * (i)

$\kappa(r)=1-\exp(-r)$;

  * (ii)

$\kappa(r)=\frac{2}{\pi}\arctan(r)$;

  * (iii)

$\kappa(r)=\frac{r}{r+\alpha}$, with $\alpha>0$;

  * (iv)

$\kappa(r)=P(\alpha,r)$, with $\alpha>0$ (see Subsection 2.1).

  * (v)

$\kappa(r)=I(\alpha,\beta,\frac{r}{r+1})$, with $\alpha,\beta>0$ (see
Subsection 2.1).

##  3 A Lyapunov characterization of predefined-time stability

This section states a Lyapunov-like theorem for predefined-time stability. The
importance of this theorem is twofold. On the one hand, it establishes
equivalence between previous Lyapunov theorems for predefined-time stability,
constituting a unifying result. On the other hand, it allows designing
predefined-time stable controllers, as shown in Section 4. Moreover, this
Lyapunov theorem is extended for analyzing the property of predefined-time
ultimate boundedness with predefined bound. Consequently, the results
presented in this section constitute the main contribution of this note.

######  Theorem 3.7.

Let $\kappa\in\mathcal{K}^{1}$ be differentiable in
$\mathbb{R}\setminus\\{0\\}$, and $V:\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}$ be
a continuous, positive definite and radially unbounded function. If for any
$T_{c}\in\mathbb{R}_{+}$, there exists some $\bm{\rho}\in\mathbb{R}^{l}$, such
that the time-derivative of $V$ along the trajectories of (1) satisfies

$\dot{V}(\bm{x})\leq-\frac{1}{(1-p)T_{c}}\frac{\kappa(V(\bm{x}))^{p}}{\kappa^{\prime}(V(\bm{x}))},\quad\text{for
}\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\},$ (5)

for some $0\leq p<1$, then the origin of (1) is predefined-time stable.
Moreover, if (5) is an equality, then
$\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T(\bm{x}_{0})=T_{c}$.

######  Proof 3.8.

Let $T_{c}\in\mathbb{R}_{+}$. Then, there exists some
$\bm{\rho}\in\mathbb{R}^{l}$ such that (5) holds. Moreover, since
$V:\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}$ is a continuous, positive definite
and radially unbounded function, and its time-derivative (5) is negative for
$\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\}$, the origin of
system (1) is asymptotically stable [12].

Now, let $\bm{\Phi}(t,\bm{x}_{0})$ be a solution of (1) and let
$y:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a function that satisfies

$\dot{y}=-\frac{1}{(1-p)T_{c}}\frac{\kappa(y)^{p}}{\kappa^{\prime}(y)},$

and $V(\bm{x}_{0})\leq y(0)$. Hence,

$\kappa(y(t))=\left\\{\begin{array}[]{cl}\left[\kappa(y(0))^{1-p}-\frac{t}{T_{c}}\right]^{\frac{1}{1-p}}&\text{if
}0\leq t\leq T_{c}\kappa(y(0))^{1-p}\\\ 0&\text{if
}t>T_{c}\kappa(y(0))^{1-p},\end{array}\right.$

and $V(\bm{\Phi}(t,\bm{x}_{0}))\leq y(t)$ (it is an equality only if (5), is
an equality) by the comparison lemma [12]. Thus,
$V(\bm{\Phi}(t,\bm{x}_{0}))=0$ for $t\geq T_{c}\kappa(V(\bm{x}_{0}))^{1-p}$,
implying that the trajectories of (1) reach the origin in finite time, and the
settling-time function satisfies

$\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T(\bm{x}_{0})\leq\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T_{c}\kappa(V(\bm{x}_{0}))^{1-p}=T_{c}.$

Hence, by Definition 2, the origin of system (1) is in fact predefined-time
stable. Moreover, if (5) is an equality, then
$\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T(\bm{x}_{0})=\sup_{\bm{x}_{0}\in\mathbb{R}^{n}}T_{c}\kappa(V(\bm{x}_{0}))^{1-p}=T_{c}$.

######  Remark 3.9.

Theorem 3.7 can be equivalently formulated in terms of a function
$W:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$, which satisfies
$W(\bm{x})=0$ if and only if $\bm{x}=\bm{0}$, $0\leq W(\bm{x})<1$, and
$\lim_{\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert\to\infty}W(\bm{x})=1$.
This equivalence is given by the transformation $W(\bm{x})=\kappa(V(\bm{x}))$.
Moreover, in this equivalent reformulation, inequality (5) is replaced by
inequality

$\dot{W}(\bm{x})\leq-\frac{1}{(1-p)T_{c}}W(\bm{x})^{p},\quad\text{for
}\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\}.$

In this sense, [15, Theorem 10] is a corollary of Theorem 3.7, which is
obtained fixing $p=0$.

######  Remark 3.10.

Although the result in Theorem 3.7 is independent of the choice of
$\kappa\in\mathcal{K}^{1}$, the form of the differential inequality (5)
strongly depends on the particular selection of this function. Indeed,
previous Lyapunov-like theorems for predefined-time stability reported in the
literature are, in fact, particular forms of Theorem 3.7. For instance:

  * (i)

[16, Theorem III.1] is obtained from Theorem 3.7 with the particular
selections of $\kappa(r)=P\left(\frac{1-\beta q}{s},\alpha
r^{s}\right)=\frac{\gamma\left(\frac{1-\beta q}{s},\alpha
r^{s}\right)}{\Gamma\left(\frac{1-\beta q}{s}\right)}$, with
$\alpha,\beta,s,q>0$ and $\beta q<1$, and $p=0$. Thus, inequality (5) then
becomes

$\dot{V}(\bm{x})\leq-\frac{\alpha^{\frac{\beta
q-1}{s}}\Gamma\left(\frac{1-\beta q}{s}\right)}{sT_{c}}\exp\left(\alpha
V(\bm{x})^{s}\right)V(\bm{x})^{\beta q}$ (6)

for $\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\}$.

  * (ii)

At the same time, with $\alpha=\beta=1$, $q=1-s$ and $0<s\leq 1$, inequality
(6) reduces to

$\dot{V}(\bm{x})\leq-\frac{1}{sT_{c}}\exp\left(V(\bm{x})^{s}\right)V(\bm{x})^{1-s},\quad\text{for
}\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\},$

which is equivalent to Theorem 3.7 with the particular selection of
$\kappa(r)=1-\exp(-r^{s})$. In this form, the result presented in [9, Theorem
2.1] is recovered.

  * (iii)

[17, Theorem 3] is retrieved from Theorem 3.7 with the particular selections
of $\kappa(r)=I\left(m_{s},m_{q},\frac{\beta r^{q-s}}{\beta
r^{q-s}+\alpha}\right),$ with $m_{s}=\frac{1-ks}{q-s}>0$,
$m_{q}=\frac{kq-1}{q-s}>0$, $\alpha,\beta,k>0$ and $0<ks<1<kq$, and $p=0$.
Replacing the above picks into inequality (5), it yields

$\dot{V}(\bm{x})\leq-\frac{\zeta}{T_{c}}\left(\alpha V(\bm{x})^{s}+\beta
V(\bm{x})^{q}\right)^{k},\quad\text{for
}\bm{x}\in\mathbb{R}^{n}\setminus\left\\{\bm{0}\right\\},$

where
$\zeta=\frac{\Gamma(m_{s})\Gamma(m_{q})}{\alpha^{k}\Gamma(k)(q-s)}\left(\frac{\alpha}{\beta}\right)^{m_{s}}$.

Thus, in this work, it is shown for the first time that all previous Lyapunov-
like theorems for predefined-time stability of autonomous systems are
equivalent.

Lyapunov analysis can also be extended to show predefined-time ultimate
boundedness with predefined bound of the solutions of (1) (see Definition 3),
even if there is no equilibrium point at the origin. Sufficient conditions are
stated in the following theorem:

######  Theorem 3.11.

Let $\kappa\in\mathcal{K}^{1}$ be differentiable in
$\mathbb{R}\setminus\\{0\\}$, and $V:\mathbb{R}^{n}\to\mathbb{R}_{\geq 0}$ be
a continuous, positive definite and radially unbounded function. If for any
$T_{c},\mu\in\mathbb{R}_{+}$, there exists some $\bm{\rho}\in\mathbb{R}^{l}$,
such that the time-derivative of $V$ along the trajectories of (1) satisfies

$\dot{V}(\bm{x})\leq-\frac{1}{(1-p)T_{c}}\frac{\kappa(V(\bm{x}))^{p}}{\kappa^{\prime}(V(\bm{x}))},\quad\text{for
}\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert\geq\mu,$ (7)

then, for any $\bm{x}_{0}\in\mathbb{R}^{n}$ the solution
$\bm{\Phi}(t,\bm{x}_{0})$ of (1) satisfies

$\left\lvert\left\lvert\bm{\Phi}(t,\bm{x}_{0})\right\rvert\right\rvert\leq
b=\alpha_{1}^{-1}(\alpha_{2}(\mu)),\quad\text{for all }t\geq T_{c},$

where $\alpha_{1},\alpha_{2}\in\mathcal{K}_{\infty}$.

Moreover, if
$V(\bm{x})=\alpha(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)$, with
$\alpha\in\mathcal{K}_{\infty}$, then $b=\mu$ in the above inequality. This
is, the solutions of (1) are predefined-time ultimately bounded with
predefined bound.

######  Proof 3.12.

Let $T_{c},\mu\in\mathbb{R}_{+}$. Then, there exists
$\bm{\rho}\in\mathbb{R}^{l}$ such that (7) holds. Since $V$ is a continuous,
positive definite and radially unbounded function, there exist
$\alpha_{1},\alpha_{2}\in\mathcal{K}_{\infty}$ such that
$\alpha_{1}(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)\leq
V(\bm{x})\leq\alpha_{2}(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)$
[12, Lemma 4.3].

Note that
$\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert<\mu\Longleftrightarrow\alpha_{2}(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)<\alpha_{2}(\mu)\Rightarrow
V(\bm{x})<\alpha_{2}(\mu)$, i.e. the set
$\left\\{\bm{x}\in\mathbb{R}^{n}:\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert<\mu\right\\}\subseteq\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})<\alpha_{2}(\mu)\right\\}$,
or equivalently
$\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})\geq\alpha_{2}(\mu)\right\\}\subseteq\left\\{\bm{x}\in\mathbb{R}^{n}:\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert\geq\mu\right\\}$.
Hence, inequality (7) holds for $V(\bm{x})\geq\alpha_{2}(\mu)$.

The above implies that the set
$\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})\leq\alpha_{2}(\mu)\right\\}$ is
positively invariant, since the derivative $\dot{V}(\bm{x})$ is negative in
its boundary
$\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})=\alpha_{2}(\mu)\right\\}$.

Now, we show that all trajectories starting in the set
$\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})\geq\alpha_{2}(\mu)\right\\}$, must
enter the set
$\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})\leq\alpha_{2}(\mu)\right\\}$ within
at most $T_{c}$ time units. Let $\bm{\Phi}(t,\bm{x}_{0})$ be a solution of
(1), with
$\bm{x}_{0}\in\left\\{\bm{x}\in\mathbb{R}^{n}:V(\bm{x})\geq\alpha_{2}(\mu)\right\\}$,
i.e. $V(\bm{x}_{0})\geq\alpha_{2}(\mu)$. From (7) and following similar
arguments as in Theorem 3.7,
$\kappa(V(\bm{\Phi}(t,\bm{x}_{0})))\leq\left[\kappa(V(\bm{x}_{0}))^{1-p}-\frac{t}{T_{c}}\right]^{\frac{1}{1-p}}$,
for
$t\in\left[0,T_{c}(\kappa(V(\bm{x}_{0}))^{1-p}-\kappa(\alpha_{2}(\mu))^{1-p})\right]$.
Hence,
$\kappa(V(\bm{\Phi}(t,\bm{x}_{0})))\leq\kappa(\alpha_{2}(\mu))\Longleftrightarrow
V(\bm{\Phi}(t,\bm{x}_{0}))\leq\alpha_{2}(\mu)$ for all $t\geq
T_{c}(\kappa(V(\bm{x}_{0}))^{1-p}-\kappa_{2}(\mu)^{1-p})$, and consequently
for all $t\geq T_{c}$.

Furthermore, note that
$V(\bm{x})\leq\alpha_{2}(\mu)\Rightarrow\alpha_{1}(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)\leq\alpha_{2}(\mu)\Longleftrightarrow\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert<\alpha_{1}^{-1}(\alpha_{2}(\mu))$.
Hence,
$\left\lvert\left\lvert\bm{\Phi}(t,\bm{x}_{0})\right\rvert\right\rvert\leq\alpha_{1}^{-1}(\alpha_{2}(\mu))$
for all $t\geq T_{c}$.

Moreover, if
$V(\bm{x})=\alpha(\left\lvert\left\lvert\bm{x}\right\rvert\right\rvert)$, one
can select $\alpha_{1}=\alpha_{2}=\alpha$, and the result is obtained.

##  4 Application: Lyapunov-based predefined-time controller design

###  4.1 Problem statement

Consider the following affine control system:

$\dot{\bm{x}}=\bm{f}(\bm{x})+\bm{B}(\bm{x})\bm{v}+\bm{\delta}(\bm{x},t)$ (8)

where $\bm{x}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{n}$ is the system state,
$\bm{v}\in\mathbb{R}^{m}$ is the control input,
$\bm{\delta}:\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{n}$
is a disturbance vector that includes plant parameter variations and external
unknown perturbations, and $\bm{B}:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}$ is
continuous and such that $\text{rank }\bm{B}(\bm{x})=m$ for all
$\bm{x}\in\mathbb{R}^{n}$.

The objective is to design a feedback control input $\bm{v}$ such that the
trajectories of (8) reach (a vicinity of) the manifold

$\bm{s}(\bm{x},t)=0,$ (9)

where $\bm{s}:\mathbb{R}^{n}\times\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}^{m}$ is a smooth mapping, in an arbitrarily selected
time $T_{c}\in\mathbb{R}_{+}$ and remain there for all $t\geq T_{c}$.

######  Remark 4.13.

There are several important control problems which take the form of system (8)
subject to (9). For instance,

  * (i)

An output tracking problem, where $\bm{s}(\bm{x},t)$ represents the output
tracking error vector; the relative degree of each output component with
respect to the control input is equal to one, and the system (8) with respect
to (9) is minimum phase [26]. The control objective is to ensure the tracking
error be predefined time stable.

  * (ii)

A sliding mode (SM) control design problem, where $\bm{s}(\bm{x},t)=0$ (9)
represents a sliding manofold with a desired SM motion [3]. In this case, the
objective is to induce the SM on the designed manifold in predefined-time
(predefined-time reaching phase).

  * (iii)

An optimization problem solved by dynamic networks, where $\bm{s}(\bm{x},t)$
is a variable which expresses the error in the satisfaction of equality
constraints [3].

The time derivative of $\bm{s}(\bm{x},t)$ is

$\dot{\bm{s}}=\bm{G}(\bm{x},t)\bm{f}(\bm{x})+\bm{G}(\bm{x},t)\bm{B}(\bm{x})\bm{v}+\bm{G}(\bm{x},t)\bm{\delta}(\bm{x},t)+\frac{\partial\bm{s}(\bm{x},t)}{\partial
t}.$ (10)

Then, assuming that $\text{rank}[\bm{G}(\bm{x},t)\bm{B}(\bm{x})]=m$, for all
$\bm{x}\in\mathbb{R}^{n}$ and $t\geq 0$, the control $\bm{v}$ is chosen as

$\bm{v}=-[\bm{G}(\bm{x},t)\bm{B}(\bm{x})]^{-1}\bigg{[}\bm{G}(\bm{x},t)\bm{f}(\bm{x})+\frac{\partial\bm{s}(\bm{x},t)}{\partial
t}+\bm{u}\bigg{]},$ (11)

where $\bm{u}\in\mathbb{R}^{m}$ is a virtual control input and
$\bm{G}(\bm{x},t)=\frac{\partial\bm{s}(\bm{x},t)}{\partial\bm{x}}$.

Substituting (11) in (10) results in

$\dot{\bm{s}}=\bm{u}+\bm{\Delta}(\bm{x},t),\ \
\bm{s}(\bm{x}_{0},0)=\bm{s}_{0},$ (12)

where $\bm{\Delta}(\bm{x},t)=\bm{G}(\bm{x},t)\bm{\delta}(\bm{x},t)$, which is
assumed to be globally bounded by
$\sup_{(\bm{x},t)\mathbb{R}^{n}\in\times\mathbb{R}_{\geq
0}}\left\lvert\left\lvert\bm{\Delta}(\bm{x},t)\right\rvert\right\rvert\leq\delta$
with $0\leq\delta<\infty$ a known constant.

From a control design point of view, the perturbation term
$\bm{\Delta}(\bm{x},t)$ can only be completely rejected by a discontinuous
control term (like the unit-vector controller
$\frac{\bm{s}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert}$), given
that it is only restricted to be bounded (no conditions of smoothness,
Lipschitz continuity neither continuity are assumed). However, such a
discontinuous control term might deteriorate the components of a real physical
system due to high-frequency oscillations, or might even be impossible to
implement due to limited actuator response.

A solution would be to sacrifice the exact convergence to the manifold
$\bm{s}(\bm{x},t)=0$ (9) in order to obtain a continuous controller (like the
continuous approximation of the unit-vector controller
$\frac{\bm{s}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\epsilon}$,
with $\epsilon>0$). In this case, as pointed out in Section 2, it can be
ensured that the trajectories converge to a vicinity of $\bm{s}(\bm{x},t)=0$
(see Eq. (9)).

Based on the above, the objective is to design the virtual control input
$\bm{u}$ as a feedback control law $\bm{u}=\bm{u}(\bm{s})$ that:

  * •

enforces predefined-time stability to the origin of (12), obtaining a
discontinuous controller; or

  * •

enforces the solutions of (12) to be predefined-time ultimately bounded with
predefined bound, obtaining a continuous controller.

###  4.2 Proposed solution

The proposed solution is a corollary of Theorems 3.7 and 3.11.

######  Corollary 4.14.

Consider system (12). Selecting $\bm{u}$ as

$\bm{u}=-\frac{1}{(1-\rho_{2})\rho_{1}}\frac{\kappa(\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert)^{\rho_{2}}}{\kappa^{\prime}(\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert)}\left\lvert\left\lfloor\bm{s}\right\rceil\right\rvert^{0}-\rho_{3}\frac{\bm{s}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}},$
(13)

where $\kappa\in\mathcal{K}^{1}$ is such that
$\kappa^{\prime}:\mathbb{R}_{\geq 0}\to\bar{\mathbb{R}}_{+}$, $\rho_{1}>0$,
$0\leq\rho_{2}<1$, $\rho_{3}>\delta$ and $\rho_{4}\geq 0$, the trajectories of
the closed-loop system (12)-(13) are predefined-time ultimately bounded with
predefined bound. In fact, for any $T_{c},b\in\mathbb{R}_{+}$ and
$\bm{s}_{0}\in\mathbb{R}^{m}$, the solution $\bm{\Phi}(t,\bm{s}_{0})$ of
(12)-(13) satisfies
$\left\lvert\left\lvert\bm{\Phi}(t,\bm{s}_{0})\right\rvert\right\rvert\leq
b=\frac{\delta\rho_{4}}{\rho_{3}-\delta}$ for all $t\geq T_{c}=\rho_{1}$.

Moreover, if $\rho_{4}=0$, then the origin $\bm{s}=\bm{0}$ of the closed-loop
system (12)-(13) is predefined-time stable.

######  Proof 4.15.

Consider the the continuous, positive definite and radially unbounded function
$V(\bm{s})=\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert$, and let
$T_{c},b\in\mathbb{R}_{+}$. The time-derivative of $V$ along the trajectories
of the closed-loop system (12)-(13) is
$\dot{V}(\bm{s})=\left\lvert\left\lfloor\bm{s}^{T}\right\rceil\right\rvert^{0}\left[-\frac{1}{(1-\rho_{2})\rho_{1}}\frac{\kappa(\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert)^{\rho_{2}}}{\kappa^{\prime}(\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert)}\left\lvert\left\lfloor\bm{s}\right\rceil\right\rvert^{0}\\\
-\rho_{3}\frac{\bm{s}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}}+\bm{\Delta}(\bm{x},t)\right]$.
Note that:

  * (a)

the continuous approximation of the unit-control term satisfies
$\frac{\bm{s}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}}=\left\lvert\left\lfloor\bm{s}\right\rceil\right\rvert^{0}-\frac{\rho_{4}\left\lvert\left\lfloor\bm{s}\right\rceil\right\rvert^{0}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}}$;

  * (b)

the product
$\left\lvert\left\lfloor\bm{s}^{T}\right\rceil\right\rvert^{0}\left\lvert\left\lfloor\bm{s}\right\rceil\right\rvert^{0}=1$;

  * (c)

by the Cauchy-Schwarz inequality
$\left\lvert\left\lfloor\bm{s}^{T}\right\rceil\right\rvert^{0}\bm{\Delta}\leq\delta$.

Therefore, considering (a), (b) and (c), the time derivative $\dot{V}(\bm{s})$
results in

$\dot{V}(\bm{s})\leq-\frac{1}{(1-\rho_{2})\rho_{1}}\frac{\kappa(V(\bm{s}))^{\rho_{2}}}{\kappa^{\prime}(V(\bm{s}))}-\left(\rho_{3}-\delta-\frac{\rho_{3}\rho_{4}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}}\right).$

Since
$\rho_{3}-\delta-\frac{\rho_{3}\rho_{4}}{\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert+\rho_{4}}\geq
0\iff\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert\geq\frac{\delta\rho_{4}}{\rho_{3}-\delta}$,
then

$\dot{V}(\bm{s})\leq-\frac{1}{(1-p)T_{c}}\frac{\kappa(V(\bm{s}))^{p}}{\kappa^{\prime}(V(\bm{s}))},\quad\text{for
}\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert\geq\mu,$ (14)

with $T_{c}=\rho_{1}$, $\mu=\frac{\delta\rho_{4}}{\rho_{3}-\delta}$ and
$p=\rho_{2}$.

From the above and using Theorem 3.11, the solutions of the closed-loop system
(12)-(13) satisfy
$\left\lvert\left\lvert\bm{\Phi}(t,\bm{s}_{0})\right\rvert\right\rvert\leq
b=\mu=\frac{\delta\rho_{4}}{\rho_{3}-\delta}$ for all $t\geq T_{c}=\rho_{1}$.

Moreover, if $\rho_{4}=0\Longleftrightarrow\mu=0$, inequality (14) holds for
all $\bm{s}\in\mathbb{R}^{m}$. Hence, by Theorem 3.7, the origin
$\bm{s}=\bm{0}$ of the closed-loop system (12)-(13) is predefined-time stable.

It is worth to notice that the predefined time $T_{c}=\rho_{1}$ and the
predefined bound $b=\frac{\delta\rho_{4}}{\rho_{3}-\delta}$ can be selected
independently since they depend on different parameters.

######  Example 4.16.

Let $\bm{x}=[x_{1}\quad x_{2}]^{T}\in\mathbb{R}^{2}$ be a point in the plane
whose dynamics is given by $\dot{\bm{x}}=\bm{u}$, where $\bm{u}=[u_{1}\quad
u_{2}]^{T}\in\mathbb{R}^{2}$ is the feedback control signal to be designed so
that the point $\bm{x}$ tracks a desired reference trajectory
$\bm{r}(t)=[r_{1}(t)\quad r_{2}(t)]^{T}:\mathbb{R}_{\geq 0}\to\mathbb{R}^{2}$.
Whereas the reference trajectory signal is assumed to be known, its derivative
$\dot{\bm{r}}(t)$ is assumed to be unknown but bounded of the form
$\sup_{t\in\mathbb{R}_{\geq
0}}\left\lvert\left\lvert\dot{{\bm{r}}}(t)\right\rvert\right\rvert\leq\delta$.
This model is a classic example in teleoperation tasks, such as haptic
interfaces, remote command of manipulators, land, aerial and underwater
robots, to name a few, where the position reference is given in real-time by
the user, but the desired velocity reference is unknown. The dynamics of the
error variable, $\bm{s}(\bm{x},t)=\bm{x}-\bm{r}(t)$, is then
$\dot{\bm{s}}=\bm{u}-\dot{\bm{r}}(t)$. Under all the above assumptions, the
feedback control signal $\bm{u}$ can be designed as (13) in Corollary 4.14.

For simulation purposes, the function $\kappa(r)$ in (13) is selected as
$\kappa(r)=I\left(m_{\rho_{7}},m_{\rho_{8}},\frac{\rho_{6}r^{\rho_{8}-\rho_{7}}}{\rho_{6}r^{\rho_{8}-\rho_{7}}+\rho_{5}}\right),$
with $m_{\rho_{7}}=\frac{1-\rho_{9}\rho_{7}}{\rho_{8}-\rho_{7}}>0$,
$m_{\rho_{8}}=\frac{\rho_{9}\rho_{8}-1}{\rho_{8}-\rho_{7}}>0$,
$\rho_{5},\rho_{6},\rho_{9}>0$ and $0<\rho_{9}\rho_{7}<1<\rho_{9}\rho_{8}$.
Furthermore, the reference signal is selected as $\bm{r}(t)=[\cos(2\pi
t)\quad\sin(2\pi t)]^{T}$ (i.e., the point $\bm{x}$ is required to follow a
circumference of radius $1$), whose derivative norm is
$\left\lvert\left\lvert\dot{\bm{r}}(t)\right\rvert\right\rvert=2\pi=\delta$.
Moreover, the following simulations are conducted using the Euler integration
method, with a fundamental step size of $1\times 10^{-5}$ s. Finally, the
initial conditions of the system are set as $\bm{x}_{0}=[x_{10}\quad
x_{20}]=x_{0}[1\quad 1]$, with $x_{0}$ taking the values of $10^{1}$, $10^{3}$
and $10^{21}$.

Part I: assume that the tracking error $\bm{s}$ is required to reach a
vicinity of the origin with a radius of measure $b=0.01$ in at most $T_{c}=1$
time units. To this end, the parameters of controller (13) are selected as
$\rho_{1}=1$, $\rho_{2}=0$, $\rho_{3}=4\pi$, $\rho_{4}=0.01$, $\rho_{5}=1$,
$\rho_{6}=1$, $\rho_{7}=0.9$, $\rho_{8}=1.1$, and $\rho_{9}=1$. Note that,
with this parameter selection, controller (13) is continuous.

Figure 1: Several trajectories of the state variables $x_{1},x_{2}$ vs. $t$,
and $\left\lvert\left\lvert\bm{e}\right\rvert\right\rvert$ vs. $t$. Continuous
controller. Figure 2: Several trajectories of the point $(x_{1},x_{2})$ in the
plane. Continuous controller.

Part II: assume that the tracking error $\bm{s}$ is required to reach the
origin in at most $T_{c}=1$ time units. To this end, the parameters of
controller (13) are selected as $\rho_{1}=1$, $\rho_{2}=0$, $\rho_{3}=4\pi$,
$\rho_{4}=0$, $\rho_{5}=1$, $\rho_{6}=1$, $\rho_{7}=0.9$, $\rho_{8}=1.1$, and
$\rho_{9}=1$. Note that, with this parameter selection, controller (13) is
discontinuous.

Figure 3: Several trajectories of the state variables $x_{1},x_{2}$ vs. $t$,
and $\left\lvert\left\lvert\bm{e}\right\rvert\right\rvert$ vs. $t$.
Discontinuous controller. Figure 4: Several trajectories of the point
$(x_{1},x_{2})$ in the plane. Discontinuous controller.

Figs. 1 and 3 show the state variables and the reference signal over time,
respectively, for the continuous ($\rho_{4}=0.01$) and the discontinuous
($\rho_{4}=0$) cases. They also show the tracking error norm
$\left\lvert\left\lvert\bm{s}\right\rvert\right\rvert$ for a better
appreciation of the convergence to the $0.01$-vicinity of the origin, in the
continuous case, or the convergence to the origin, in the discontinuous case.
Figs. 2 and 4 show the trajectories of the point $(x_{1},x_{2})$ in the plane,
respectively, for the continuous and the discontinuous cases.

##  5 Conclusion

This technical note was devoted to the study of sufficient Lyapunov-like
conditions to ensure a class of dynamic systems to exhibit a predefined time
stability property. The introduction of class $\mathcal{K}^{1}$ allowed to
establish equivalence with previous Lyapunov-like theorems for predefined-time
stability for autonomous systems [9, 16, 17, 15]. Moreover, the derived
Lyapunov theorem was extended for analyzing predefined-time ultimate
boundedness with predefined bound.

On the other hand, the developed framework was used to design a class of
robust controllers for uncertain affine control systems. This class of
controllers can be continuous, providing predefined-time ultimate boundedness
of the solutions, or discontinuous, providing predefined-time stability to
some desired manifold. Finally, the theoretical findings were validated
through a numerical simulation, which reveals the effectiveness of the
proposed control scheme.

As future work, the Lyapunov-like conditions introduced in Theorems 3.7-3.11,
require further research to exploit them for the controller design of
particular classes of nonlinear systems.

## Acknowledgment

Esteban Jiménez acknowledges to CONACYT–México for the D.Sc. scholarship
number 481467 and the project 252405. Aldo Jonathan acknowledges to
CONACYT–México for the Project Cátedras 1086 “Ambientes Inteligentes”.

## References

  * [1] E. Roxin, “On finite stability in control systems,” _Rendiconti del Circolo Matematico di Palermo_ , vol. 15, no. 3, pp. 273–282, 1966.
  * [2] V. Haimo, “Finite time controllers,” _SIAM Journal on Control and Optimization_ , vol. 24, no. 4, pp. 760–770, 1986.
  * [3] V. I. Utkin, _Sliding Modes in Control and Optimization_. Springer Verlag, 1992.
  * [4] S. Bhat and D. Bernstein, “Finite-time stability of continuous autonomous systems,” _SIAM Journal on Control and Optimization_ , vol. 38, no. 3, pp. 751–766, 2000.
  * [5] E. Moulay and W. Perruquetti, “Finite time stability conditions for non-autonomous continuous systems,” _International Journal of Control_ , vol. 81, no. 5, pp. 797–803, 2008.
  * [6] V. Andrieu, L. Praly, and A. Astolfi, “Homogeneous approximation, recursive observer design, and output feedback,” _SIAM Journal on Control and Optimization_ , vol. 47, no. 4, pp. 1814–1850, 2008.
  * [7] E. Cruz-Zavala, J. Moreno, and L. Fridman, “Uniform second-order sliding mode observer for mechanical systems,” in _Variable Structure Systems (VSS), 2010 11th International Workshop on_ , june 2010, pp. 14 –19.
  * [8] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 8, pp. 2106–2110, 2012.
  * [9] J. D. Sánchez-Torres, D. Gómez-Gutiérrez, E. López, and A. G. Loukianov, “A class of predefined-time stable dynamical systems,” _IMA Journal of Mathematical Control and Information_ , vol. 35, no. Suppl 1, pp. i1–i29, 2018.
  * [10] E. Jiménez-Rodríguez, J. D. Sánchez-Torres, and A. G. Loukianov, “On optimal predefined-time stabilization,” _International Journal of Robust and Nonlinear Control_ , vol. 27, no. 17, pp. 3620–3642, 2017.
  * [11] A. Bacciotti and L. Rosier, _Liapunov Functions and Stability in Control Theory_ , 2nd ed., S.-V. B. Heidelberg, Ed. Springer-Verlag Berlin Heidelberg, 2005.
  * [12] H. Khalil, _Nonlinear Systems_. Prentice Hall, 2000.
  * [13] M. Defoort, G. Demesure, Z. Zuo, A. Polyakov, and M. Djemai, “Fixed-time stabilisation and consensus of non-holonomic systems,” _IET Control Theory Applications_ , vol. 10, no. 18, pp. 2497–2505, 2016.
  * [14] A. Polyakov and L. Fridman, “Stability notions and Lyapunov functions for sliding mode control systems,” _Journal of the Franklin Institute_ , vol. 351, no. 4, pp. 1831 – 1865, 2014.
  * [15] E. Jiménez-Rodrıguez, J. D. Sánchez-Torres, A. J. Muñoz Vázquez, and A. G. Loukianov, “A note on predefined-time stability,” in _Second Conference on Modelling,Identification and Control of Nonlinear Systems (IFAC MICNON)_. Guadalajara, México: IFAC, June 2018.
  * [16] J. D. Sánchez-Torres, M. Defoort, and A. J. Munoz-Vázquez, “A second order sliding mode controller with predefined-time convergence,” in _2018 15th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)_ , Sept 2018, pp. 1–4.
  * [17] R. Aldana-López, D. Gómez-Gutiérrez, E. Jiménez-Rodríguez, J. D. Sánchez-Torres, and M. Defoort, “Enhancing the settling time estimation of a class of fixed-time stable systems,” _International Journal of Robust and Nonlinear Control_ , vol. 29, no. 12, pp. 4135–4148, Aug. 2019.
  * [18] E. Jiménez-Rodríguez, J. D. Sánchez-Torres, D. Gómez-Gutiérrez, and A. G. Loukianov, “Variable structure predefined-time stabilization of second-order systems,” _Asian Journal of Control_ , 2018.
  * [19] A. J. Muñoz-Vázquez, J. D. Sánchez-Torres, E. Jiménez-Rodríguez, and A. Loukianov, “Predefined-time robust stabilization of robotic manipulators,” _IEEE/ASME Transactions on Mechatronics_ , vol. 24, no. 3, pp. 1033–1040, June 2019.
  * [20] J. D. Sánchez-Torres, M. Defoort, and A. J. Muñoz-Vázquez, “Predefined-time stabilisation of a class of nonholonomic systems,” _International Journal of Control_ , pp. 1–8, 2019.
  * [21] M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables_ , 9th ed. Dover Publications, 1965.
  * [22] A. F. Filippov, _Differential equations with discontinuous righthand sides_ , Mathematics and . its Applications (Soviet Series), Eds. Kluwer Academic Publishers Group, Dordrecht, 1988\.
  * [23] J. Cortes, “Discontinuous dynamical systems,” _IEEE Control Systems Magazine_ , vol. 28, no. 3, pp. 36–73, June 2008.
  * [24] F. Lopez-Ramirez, D. Efimov, A. Polyakov, and W. Perruquetti, “On necessary and sufficient conditions for fixed-time stability of continuous autonomous system,” in _Proc. 17th European Control Conference (ECC)_ , Jun 2018\.
  * [25] C. M. Kellett, “A compendium of comparison function results,” _Mathematics of Control, Signals, and Systems_ , vol. 26, no. 3, pp. 339–374, Sep 2014.
  * [26] J. J. Ley-Rosas, L. E. González-Jiménez, A. G. Loukianov, and J. E. Ruiz-Duarte, “Robust observer-based sliding mode controller for vehicles with roll dynamics,” in _2016 IEEE 55th Conference on Decision and Control (CDC)_ , Dec 2016, pp. 740–745.


# Non-Minimally Coupled Einstein-Yang-Mills Field Equations and Wu-Yang
Monopoles in Bertotti-Robinson Spacetimes

Tekin<EMAIL_ADDRESS>Yorgo<EMAIL_ADDRESS>
Department of Physics, Koç University, 34450 Sarıyer-İstanbul, Turkey

Bertotti-Robinson spacetimes are topologically $AdS_{2}\times S^{2}$ and
described by a conformally flat metric. Together with the Coulomb electric
potential, they provide a class of static, geodetically complete Einstein-
Maxwell solutions. We show here that the Bertotti-Robinson metric together
with Wu-Yang magnetic pole potentials give a class of static solutions of a
system of non-minimally coupled Einstein-Yang-Mills equations that may be
relevant for investigating vacuum polarization effects in a first order
perturbative approach to quantum fields.

PACS numbers: o4.40.Nr, 04.40.Kd, 04.20.Jb

##  1 Introduction

Maxwell’s theory of electromagnetism is arguably one of the best established
classical field theories of Nature. Classically the coupling of
electromagnetic fields to gravity are described by the Einstein-Maxwell field
equations. This implies in the weak field approximation a quantum field theory
of massless spin-1 photons coupled to massless spin-2 gravitons. At the tree
level the photon propagation takes place along the null geodesics of the
Minkowski spacetime. In perturbative QED in curved spacetimes though many
unexpected new phenomena may occur. At the 1-loop level there exists for
instance vacuum polarization effects for which a photon produces a virtual
electron-positron pair that survives for a very short time interval before it
pair annihilates back into a photon. Such quantum vacuum processes confer a
length scale to the photon of the order of the Compton wavelength of an
electron. Thus a photon in motion in a curved spacetime will be affected by
the gravitational fields, even at the 1-loop order. Perturbative effects such
as these in curved spacetimes can be conveniently described by an effective
field theory at the classical level through certain non-minimal couplings of
the electromagnetic fields to gravity[1, 2, 3, 4].

In general, in an effective field theory defined at a mass scale $\mu$ , only
those fields with masses lighter than or of the of order $\mu$ are taken into
account. But the derivation of such an effective theory involves more than
integrating out heavy particles of masses $M>\mu$ [5, 6]. Suppose one
considers a renormalizable field theory at scales $\mu>>M$ with fields
included at all the mass scales. Then the theory is scaled down to $\mu\approx
M$ by the renormalization group flow. On the other hand, an effective field
theory at scales $\mu<M$ with only those fields with masses less than or equal
to $\mu$ is introduced which may include higher derivative couplings and
possibly many other types of non-renormalizable interactions. Then the short-
distance physics that is incorporated into the coefficients of the effective
Lagrangian is disentangled from the long-distance physics that remains
explicit in the actual low energy theory. Such a disentanglement can be
achieved order by order in perturbation theory by matching the physical
parameters of the high energy theory at $\mu>>M$ to the corresponding
parameters of the low energy theory at mass scales $\mu\approx M$.

In QED, it is found that when the electron field is integrated out, the
resulting effective Lagrangian at the 1-loop level involves certain
$F^{4}$-type self couplings of the electromagnetic fields. The corresponding
long-distance theory complements Maxwell electrodynamics by the higher
derivative Heisenberg-Euler terms [7]:

${\mathcal{L}}=\frac{1}{2}F\wedge*F+\frac{2\alpha^{2}}{45m^{4}}\left(X^{2}+\frac{7}{4}Y^{2}\right)*1.$

In a similar way, the effective action density 4-form at the 1-loop level in a
curved spacetime for QED had been calculated [8]:

$\displaystyle{\mathcal{L}}_{eff}$ $\displaystyle=$
$\displaystyle\frac{12\alpha}{45\pi
m^{2}}(d*F)\wedge*(d*F)+\frac{\alpha}{45\pi m^{2}}R_{ab}F^{ab}\wedge*F$
$\displaystyle-\frac{13\alpha}{45\pi
m^{2}}Ric_{a}\wedge\iota^{a}F\wedge*F+\frac{5\alpha}{90\pi m^{2}}RF\wedge*F.$

where $\alpha$ is the fine structure constant and $m$ is the free electron
mass. The 2-photon 2-graviton vertices are settled in perturbative QED by the
conventional $RF^{2}$ interactions in the effective action. These interaction
vertices were found to mean photon graviton oscillations at low energies
relative to the Planck scale and in weak gravitational fields [9]. This
process may be significant in the presence of strong magnetic fields near
heavy magnetars and pulsars [10, 11, 12]. In fact a curved spacetime acts as
an optically active medium for the photon with an index of refraction
determined by the higher-loop QED corrections [13, 14, 15]. Then dispersion
effects such as gravitational rainbows or gravitational bi-refringence may
occur.

The distinctive feature of a non-Abelian gauge field theory is the self -
interactions among its gauge bosons. In QCD, the exchange potentials between
color charges give rise to asymptotically free forces. Then in the UV limit
constituent quarks behave as if they are free particles inside the hadrons.
Perturbative quantization is viable in this case and the high-energy QCD tests
agree with observations up to logarithmic corrections. On the other hand the
perturbation theory fails altogether in the IR limit where the interaction
potential becomes very complicated. The corresponding low-energy limit of QCD
is not easy to define. A possible approach to deal with such long distance
effects of QCD is to work with an effective field theory that would be valid
at lower energy scales. A perturbative 1-loop effective field theory for QCD
includes the generic types of $F^{3}$ and $(\nabla_{A}F)^{2}$ Yang-Mills self-
interaction terms that show up as exotic strong-interaction contributions in
colour condensate models [16]. Then it should be expected in curved
spacetimes, the QCD vacuum polarization effects shall also contribute, as it
is in the case of perturbative QED, non-minimal coupling terms of the generic
type $RF^{2}$ at the 1-loop level to an effective Einstein-Yang-Mills theory
[17].

We consider in Section:2 a simple non-minimally coupled Einstein-$SU(2)$ Yang-
Mills effective field theory model. Variational field equations will be
derived in the coordinate-independent language of exterior differential forms.
We refer to our previous papers for the notation and conventions [18, 19, 20].
We provide in Section:3 a brief discussion of the unique conformally flat,
static, spherically symmetric Bertotti-Robinson solution of the source-free
Einstein-Maxwell equations. The celebrated Wu-Yang magnetic pole potentials
are introduced in Section:4. In particular we give a non-generic class of
regular static, spherically symmetric solutions of the coupled Einstein-Yang-
Mills field equations.

##  2 Non-minimally Coupled Einstein-Yang-Mills
Fields

We first consider the Yang-Mills potential 1-form333We set the gauge coupling
constant $e=1$

$A=A^{j}T_{j}=A^{j}_{\;\;\mu}dx^{\mu}\otimes T_{j}$ (1)

where $\\{T_{j}:j=1,2,3\\}$ are the anti-hermitian generators of the Lie
algebra $su(2)$ that satisfy the structure equations

$\left[T_{j},T_{k}\right]=\epsilon_{jkl}T_{l}.$ (2)

One may also introduce a local coordinate chart $\\{x^{\mu}:\mu=0,1,2,3\\}$,
even though we will be using extensively the coordinate independent formalism
of exterior differential forms over the spacetime. Then the Yang-Mills field
2-form is determined by

$F=dA+A\wedge A=F^{j}T_{j}=\frac{1}{2}F^{j}_{\;\;\mu\nu}dx^{\mu}\wedge
dx^{\nu}\otimes T_{j}.$ (3)

We work out

$F^{j}=dA^{j}+\epsilon_{jkl}A^{k}\wedge
A^{l}\;\;,\;\;F^{j}_{\;\;\mu\nu}=\partial_{\mu}A^{j}_{\;\;\nu}-\partial_{\nu}A^{j}_{\;\;\mu}+\epsilon_{jkl}A^{k}_{\;\;\mu}A^{l}_{\;\;\nu}.$

Under a local gauge transformation $U=e^{\theta^{j}T_{j}}\in SU(2)$, the Yang-
Mills fields transform as

$A\rightarrow UAU^{-1}+UdU^{=1}\quad,\quad F\rightarrow UFU^{-1}$

The integrability of $F$ implies the Bianchi identity

$\nabla_{A}F\equiv dF+A\wedge F-F\wedge A=0.$ (4)

We label the quadratic Yang-Mills invariants by

$X=*\hskip 1.42262ptTr(F\wedge*F),\quad Y=*\hskip 1.42262ptTr(F\wedge F).$ (5)

The Lagrangian density 4-form of Einstein-Yang-Mills theory with a
cosmological constant is given by

${\mathcal{L}}_{0}=\left(\frac{1}{2\kappa^{2}}R+\frac{1}{2}X+\Lambda\right)*1$
(6)

where

$R=-*(R_{ab}\wedge*e^{ab}).$ (7)

denotes the scalar curvature of space-time. $\kappa^{2}=4\pi G$ with $G$ being
the Newton’s universal constant and $\Lambda$ is a cosmological constant.

We consider, for the sake of simplicity, only the following non-minimal
coupling terms:

${\mathcal{L}}_{1}=\frac{\gamma}{2}XR*1+\frac{\gamma^{\prime}}{2}YR*1.$ (8)

$\gamma$ and $\gamma^{\prime}$ are two arbitrary coupling constants.
Furthermore we assume that the connection is the unique, torsion-free Levi-
Civita connection. This constraint will be imposed by the method of Lagrange
multipliers

${\mathcal{L}}_{C}=\left(de^{a}+\omega^{a}_{\;\;b}\wedge
e^{b}\right)\wedge\lambda_{a}.$ (9)

Therefore the total action that is to be varied becomes

$I[e^{a},\omega^{a}_{\;\;b},A,\lambda_{a}]=\int_{M}\left({\mathcal{L}}_{0}+{\mathcal{L}}_{1}+{\mathcal{L}}_{C}\right)$
(10)

After a long computation, the final form of the gravitational field equations
turn out to be

$\displaystyle(\frac{1}{\kappa^{2}}+\gamma X+\gamma^{\prime}Y)G_{a}$
$\displaystyle-$
$\displaystyle(\Lambda-\frac{\gamma}{2}XR-\frac{\gamma^{\prime}}{2}YR)*e_{a}$
$\displaystyle=$ $\displaystyle\left(1+\gamma R\right)\tau_{a}[F]+\gamma
D(\iota_{a}*dX)+\gamma^{\prime}D(\iota_{a}*dY),$

where the Einstein 3-forms

$G_{a}=-\frac{1}{2}R^{bc}\wedge*e_{abc}=G_{ab}*e^{b}$

and the Yang-Mills stress-energy-momentum 3-forms

$\tau_{a}[F]=\frac{1}{2}Tr\left(\iota_{a}F\wedge*F-F\wedge\iota_{a}*F\right)=T_{ab}[F]*e^{b}.$
(12)

The Yang-Mills field equations are also modified according to

$\nabla_{A}F=0,\quad\nabla_{A}*\left(F+\gamma RF+\gamma^{\prime}R*F\right)=0.$
(13)

In what follows we will be looking for the solutions of the coupled field
equations (11) and (14).

##  3 Bertotti-Robinson Spacetimes

The Bertotti-Robinson spacetimes [21, 22, 23, 24, 25, 26, 27] admit a product
topology $M^{1,3}=AdS_{2}\times S^{2}.$ They were first discussed as the
unique conformally flat, static spherically symmetric solution of the source-
free Einstein-Maxwell equations with a non-null electromagnetic potential. In
particular, we consider in spherical polar coordinates $(t,r,\theta,\varphi)$
a static, spherically symmetric metric

$g=-f_{1}^{2}(r)dt^{2}+f_{2}^{2}(r)dr^{2}+r_{0}^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (14)

together with a static, spherically symmetric electric potential

$A=q(r)dt.$ (15)

Then the source-free Einstein-Maxwell field equations

$-\frac{1}{2}R^{bc}\wedge*e_{abc}=\frac{\kappa^{2}}{2}\left(\iota_{a}F\wedge*F-F\wedge\iota_{a}*F\right),$
(16) $dF=0\quad,\quad d*F=0$ (17)

are satisfied for the choice

$f_{1}(r)=f_{2}(r)=\frac{r_{0}}{r}\quad,\quad q(r)=-\frac{Q}{r}.$

Hence the metric

$g=\frac{r_{0}^{2}}{r^{2}}\left[-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right]$ (18)

is conformally flat and the Weyl curvature 2-forms identically vanish:

$C_{ab}=R_{ab}-\frac{1}{2}(e_{a}\wedge Ric_{b}-e_{b}\wedge
Ric_{a})+\frac{R}{6}e_{a}\wedge e_{b}=0.$ (19)

The curvature scalar $R=0$ vanishes as well. $A$ turns out to be the Coulomb
electric potential so that the net electric charge

$\frac{1}{4\pi}\oint_{S^{2}}*F=Q.$ (20)

We also make note of an alternative way of writing down the Bertotti-Robinson
solution with a re-definition of the time coordinate as

$dt=(\frac{r}{r_{0}})^{2}d\tilde{t}.$ (21)

Then the conformally flat metric reads

$g=-\left(\frac{r^{2}}{r_{0}^{2}}\right)^{2}d{\tilde{t}}^{2}+\left(\frac{r_{0}^{2}}{r^{2}}\right)^{2}\left[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right],$ (22)

and the electric potential 1-form becomes

$A=-\frac{Q}{r_{0}^{2}}rd{\tilde{t}}.$ (23)

##  4 Wu-Yang Magnetic Pole Solutions

We now look for static, spherically symmetric solutions of our Einstein-Yang-
Mills system of equations for which the space-time metric in terms of
isotropic coordinates $\\{t,x,y,z\\}$ is given by

$g=-\left(\frac{r}{r_{0}}\right)^{2k}dt^{2}+\left(\frac{r_{0}}{r}\right)^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(24)

where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. We introduced a real parameter $k$ so that
our Bertotti-Robinson metric is no longer conformally flat unless $k^{2}=1$.
Yet the spacetime still admits a product topology $AdS_{2}\times S^{2}$. We
calculate the curvature scalar which turns out to be

$R=-\frac{2}{r_{0}^{2}}(1-k^{2}).$ (25)

Let us take the Wu-Yang potential 1-forms [28, 29, 30]

$\displaystyle A^{1}=\left(\frac{K(r)-1}{r^{2}}\right)(ydz-zdy),$
$\displaystyle A^{2}=\left(\frac{K(r)-1}{r^{2}}\right)(zdx-xdz),$
$\displaystyle A^{3}=\left(\frac{K(r)-1}{r^{2}}\right)(xdy-ydx).$ (26)

Then we calculate

$\displaystyle F^{1}$ $\displaystyle=$
$\displaystyle\big{[}(y^{2}+z^{2})\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}+2\left(\frac{K(r)-1}{r^{2}}\right)+2e^{2}x^{2}\left(\frac{K(r)-1}{r^{2}}\right)^{2}\big{]}dy\wedge
dz$ (27)
$\displaystyle+\left[\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}-2\left(\frac{K(r)-1}{r^{2}}\right)^{2}\right]xdx\wedge(ydz-
zdy),$

and by the cyclic permutations of $(xyz)$, we also write down

$\displaystyle F^{2}$ $\displaystyle=$
$\displaystyle\big{[}(z^{2}+x^{2})\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}+2\left(\frac{K(r)-1}{r^{2}}\right)+2e^{2}y^{2}\left(\frac{K(r)-1}{r^{2}}\right)^{2}\big{]}dz\wedge
dx$ (28)
$\displaystyle+\left[\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}-2\left(\frac{K(r)-1}{r^{2}}\right)^{2}\right]ydy\wedge(zdx-
xdz),$

and

$\displaystyle F^{3}$ $\displaystyle=$
$\displaystyle\big{[}(x^{2}+y^{2})\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}+2\left(\frac{K(r)-1}{r^{2}}\right)+2e^{2}z^{2}\left(\frac{K(r)-1}{r^{2}}\right)^{2}\big{]}dx\wedge
dy$ (29)
$\displaystyle+\left[\frac{1}{r}\left(\frac{K(r)-1}{r^{2}}\right)^{\prime}-2\left(\frac{K(r)-1}{r^{2}}\right)^{2}\right]zdz\wedge(xdy-
ydx).$

Suppose special solutions of the Yang-Mills equations for which $K(r)=K$ is a
constant are sought. In this case we calculate

$d*F^{1}+A^{2}\wedge*F^{3}-A^{3}\wedge*F^{2}=\left(\frac{r}{r_{0}}\right)^{k+1}\frac{K(K^{2}-1)}{r^{3}}dt\wedge
dr\wedge dx=0$ (30)

and the other two field equations are similarly determined by cyclic
permutations of $(123)$ and $(xyz)$. There exists three constant solutions:
$K=\pm 1$ and $K=0$. In the first two cases the Yang-Mills fields vanish,
$F=0$, so that both of these cases correspond to the Yang-Mills vacuum. The
non-trivial Wu-Yang magnetic pole solution corresponds to the choice $K=0$. On
the other hand, substituting in the Yang-Mills field 2-forms above and tracing
over the Lie algebra, we get

$X=-\left(\frac{r}{r_{0}}\right)^{4}\Big{(}\frac{{2K^{\prime}}^{2}}{r^{2}}+\frac{(K+1)^{2}(K-1)^{2}}{r^{4}}\Big{)},\quad
Y=0.$ (31)

Therefore it can easily be verified for the Wu-Yang solution that

$X=-\frac{1}{r_{0}^{4}},\quad R=-\frac{2}{r_{0}^{2}}(1-k^{2}).$ (32)

Finally the complete set of non-minimally coupled Einstein-Yang-Mills field
equations with a cosmological constant are solved provided

$\Lambda=\frac{1}{2r_{0}^{4}}\quad,\quad\gamma=\frac{r_{0}^{4}}{\kappa^{2}}.$
(33)

Then the metric parameter should be bound according to

$0<k^{2}=1-\frac{\kappa^{2}}{2r_{0}^{2}}<1.$ (34)

It is interesting to note that under the assumption that the coupling
constants are fixed by

$\kappa^{2}\Lambda=\frac{1}{2\gamma}=\frac{1}{2\gamma^{\prime}},$ (35)

the total Lagrangian density 4-form factorizes as follows:

${\mathcal{L}}_{0}+{\mathcal{L}}_{1}=\left(\frac{1}{2\kappa^{2}}R+\Lambda\right)\left(1+\gamma\kappa^{2}X+\gamma^{\prime}\kappa^{2}Y\right)*1.$
(36)

For the solution above, both of these factors vanish. Hence we get a very
special type of an exact solution of the Einstein-Yang-Mills field equations
derived from a Lagrangian density that admits a product structure. Then under
independent field variations with respect to the metric and the gauge
potential, the variational field equations will always contain either the
first factor or the second factor. For our solution, it is evident that the
coupled field equations are satisfied since both the gravity and gauge factors
vanish on their own.

##  5 Concluding Remark

Here a simple model of non-minimally coupled Einstein-Yang-Mills field
equations are considered. A very special class of exact static, spherically
symmetric solutions which are determined by the Wu-Yang magnetic pole
potentials in a non-conformally flat Bertotti-Robinson spacetime are found.
Such solutions may be related physically to regular black holes with colour
[31, 32, 33]. However, they are in no way generic. We regard the existence of
these non-generic solutions in a simple case of non-minimal couplings induced
by 1-loop quantum effects in curved spacetimes as an indication that Yang-
Mills vacuum polarization effects might in general imply unexpected new
results.

##  6 Acknowledgement

Y.Ş. is grateful to Koç University for its hospitality and partial support.

## References

  * [1] A.R.Prassana A new invariant for electromagnetic fields in curved space-time , Phys.Lett.A37 (1971) 337
  * [2] H.A.Buchdahl,On a Lagrangian for non-minimally coupled gravitational and electromagnetic fields, J. Phys. A12(1979)1037
  * [3] G.M.Shore,A local effective action for photon-gravity interactions,Nucl.Phys.
B646(2002)281

  * [4] A.B. Balakin,J.P.S.Lemos Non-minimal coupling for the electromagnetic fields: a general system of equations, Class.Q.Grav. 22 (2005) 1867
  * [5] H.Georgi,An effective field theory for heavy quarks at low energies,Phys.Lett.B240(1990)447
  * [6] M.Neubert, Heavy-quark effective theory, Lectures presented at the 34th International School of Subnuclear Physics: Effective Theories and Fundamental Theories, Erice,Italy,3-12 July 1996
  * [7] W.Heisenberg,H.Euler,Folgerungen aus der Diracschen Theorie des Electrons,Z.Phys.98(1936)714
  * [8] I.T.Dummond,S.J.Hathrell QED vacuum polarisation in a background gravitational field and its effects on the velocity of photons, Phys.Rev.D22 (1980) 343
  * [9] G.Raffelt,L.Stodolsky,Mixing of photons with low-mass particles,Phys.Rev.
D37(1988)1237

  * [10] G.Lambiase,A.R.Prassana,Gauge invariant wave equations in curved space-times and primordial magnetic fields, Phys.Rev.D70(2004)063502
  * [11] A.D.Dolgov,Breaking of conformal invariance and electromagnetic field generation in the Universe,Phys.Rev,D48(1993) 2499
  * [12] A.D.Dolgov,Electromagnetic phenomena around black holes.Talk presented at 8th International Conference on New Frontiers in Physics,21-30 August 2019,Crete,Greece. arXiv:2002.10262[gr-qc]
  * [13] G.M.Shore,Quantum gravitational optics,Contemp.Phys.44(2003)503
  * [14] T.Dereli,J.Gratus,R.W.Tucker,New perspectives on the relevance of gravitation for the covariant description of electromagnetically polarisable media, J.Phys.A40(2007) 5695
  * [15] T.Hollowood,G.M.Shore,The refractive index of curved spacetime: The fate of causality in QED,Nucl.Phys.B795(2008)138
  * [16] S.Başkal,T.Dereli,Higher derivative effective Yang-Mills theory and static,
spherically symmetric field configuration, J.Phys.G19(1993)477

  * [17] T.Dereli,G.Üçoluk,Direct-curvature Yang-Mills field couplings induced by the Kaluza-Klein reduction of Euler form actions in seven dimensions, Class.Q.Grav.7(1990)533
  * [18] T.Dereli,Ö.Sert, Nonminimal $ln(R)F^{2}$ couplings of electromagnetic fields to gravity: static,spherically symmetric solutions, Eur.Phys.J. C71 (2011) 1589
  * [19] A.Baykal, T.Dereli,Non-minimally coupled Einstein-Maxwell model in a non-Riemannian spacetime with torsion,Phys.Rev.D92(2015)065018
  * [20] T.Dereli,Y.Şenikoğlu,Electric charge screening by non-minimal coupling of electromagnetic fields to gravity, Europhys.Lett.131(2020)10006
  * [21] B.Bertotti,Uniform electromagnetic field in the theory of general relativity,Phys.Rev.116(1959)1331
  * [22] I.Robinson,A solution of the Einstein-Maxwell equations, Bull.Acad.Scie.Polon.
7(1959)351

  * [23] H.Stephani,Konform flache Gravitationsfeldser, Comm.Math.Phys.5(1967)337
  * [24] P.Dolan,A singularity free solution of the Einstein-Maxwell equations,
Commun.Math.Phys.9(1968)161

  * [25] N.Tariq,B.O.J.Tupper,The uniqueness of the Bertotti-Robinson electromagnetic universe,J.Math.Phys.15(1974)2232
  * [26] N.Dadhich,On product spacetime with 2-sphere of constant curvature,
arXiv:gr-qc/0003026

  * [27] D.Garfinkle,E.N.Glass,Bertotti-Robinson and Melvin spacetimes, Class.Q.Grav.
28(2011)215012

  * [28] T.T.Wu,C.N.Yang, Some solutions of the classical isotopic gauge field equations in The Properties of Matter under Unusual Conditions, Edited by H.Mark and S.Fernbach (Wiley,1969) p.349
  * [29] R.Bartnik,J.McKinnon,Particlelike solutions of the Einstein-Yang-Mills equations,Phys.Rev.Lett.61(1988)141
  * [30] A.B.Balakin, A.E.Zayats, Non-minimal Wu-Yang monopole, Phys.Lett.
B644(2007)294

  * [31] A.B.Balakin,S.V.Sushkov,A.E.Zayats,Magnetic black holes and monopoles in a non-minimal Einstein-Yang-Mills theory with a cosmological constant:exact solutions,Phys.Rev.D93(2016)084004
  * [32] F.-Y.Liu,Y.-F.Mai,W.-Y.Wu,Y.Xie,Probing a regular non-minimal Einstein-Yang-Mills black hole with gravitational lensing,Phys.Lett.B795(2019)475
  * [33] C.Zhang,R.B.Mann,Unified interacting quark matter and its astrophysical implications,arXiv:2009.07182[astro-ph.HE]