The Role of Topological Photon Spheres in Constraining the Parameters of Black
Holes
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Ali S. Afshar ⋆222Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

In this paper, we investigate the topological photon sphere from two distinct
perspectives. In the first view, we examine the existence and characteristics
of topological photon(anti-photon)spheres for black holes with different
structures, such as Einstein-Young-Mills non-minimal, AdS black holes
surrounded by Chaplygin-like dark fluid, and Bardeen-like black holes in
Einstein-Gauss-Bonnet gravity. Furthermore, we delve into the deeper
perspective of the necessity of photon spheres for super-compact gravitational
structures such as black holes. By leveraging this necessity, we propose a
classification of the parameter space of black hole models based on the
existence and positioning of photon spheres. This approach enables the
determination of parameter ranges that delineate whether a solution represents
a black hole or a naked singularity.
In essence, the paper illustrates the utility of the photon sphere as a
notable test for establishing the permissible and non-permissible parameter
ranges within specific theories of black hole solutions.

Keywords: Black hole, Photon sphere, Topological classification, parameters of
black hole.

###### Contents

  1. 1 Introduction
  2. 2  Standard and Topological Methods for Investigating the Photon Sphere
  3. 3  Einstein-Young-Mills non-minimal regular black hole
    1. 3.1  photon sphere ranges
    2. 3.2 naked singularity ranges
  4. 4  AdS black holes surrounded by dark fluid with Chaplygin-like equation of state
  5. 5  Bardeen-like regular black holes in Einstein-Gauss-Bonnet gravity
    1. 5.1 $5D:\Lambda=0$
      1. 5.2 $4D:\Lambda\neq 0$

##  1 Introduction

The fundamental concept of a black hole, which initially emerged as a
theoretical construct and perhaps to some extent to lend practical credibility
to the complex concepts of general relativity, has significantly evolved to
this day. Observations today clearly indicate that they transcend mere
abstract models or purely mathematical structures. The researchers actively
expanded their understanding of general relativity instead of passively
waiting for experimental confirmation. They formulated various theoretical
amendments in the gravitational equations and integrated new concepts of
quantum gravity and string theory into this framework, which these efforts led
to the proposal of various models for the structures of black holes, that each
possessing unique structures and distinct parameters. One of our main
objectives in this article is to examine the impact of these parameters and
their permissible range, in several different models, on the topological
photon spheres. As we know, the allowed range for these parameters is affected
by the special preconditions that we impose on the model, which makes the
range of their changes in different black holes not the same. For instance,
one of these restrictive general preconditions is the Weak Cosmic Censorship
(WCC) conjecture, which states that singularities that appear when solving
Einstein’s equations must be hidden behind the event horizon and no
singularity should be observable from the infinite future. In other words, in
a singularity-free universe, as demonstrated by the theory of general
relativity, the evolution of the universe is deterministic and predictable. A
closer look at the above statement reveals that this situation is somewhat
more indicative of a computational limitation than a scientific reality. In
fact, our inability to access a stronger geometry and mechanics that can
explain the complexities behind the event horizon has led us to discard
anything that cannot be justified by our current understanding. However, over
time, facing various conditions and the growth of knowledge has somewhat
undermined the credibility of this condition because today, the potential
emergence of points that we call ”naked singularities” has become apparent in
specific scenarios [1, 2, 3]. In the past, it was always tried to avoid these
singularities as much as possible due to the loss of space-time geometry, but
today there are several studies on the continuous gravitational collapse of
the inhomogeneous matter cloud that illustrates that space-time singularities
developed during gravitational collapse can be seen by an outside observer and
the visibility of space-time singularity depends upon the initial conditions
of the collapsing matter[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19]. This multiplicity of studies shows that, unlike previous studies in which
only the permitted areas were considered, one should be sensitive to the
classification of other areas because these areas may also be involved in the
results under certain conditions, so their domain must be defined. Given these
interpretations, we must be able to categorize the range of our model
parameters, i.e., know in what range the metric background of our black hole
solution remains untouched, or it exhibits the behavior of a naked
singularity, or the model completely falls apart. Distinguishing these domains
and developing a relatively precise method to accomplish this task may be
challenging. Nevertheless, the clichéd and fractal nature of the universe to
some extent holds promise. The fact that a structure must adhere to an almost
constant pattern to have a black hole-like form has led us to the idea that
perhaps, by using the method of inverse motion, it may be possible to identify
experiments and approaches to determine the range of parameters. The method we
propose for this purpose and finding the permissible range for parameters is
the use of the concept of a topological photon sphere. In classical physics,
the emergence of rotational motion due to gravitational forces is a normal and
obvious occurrence. The analogy of this classical pattern in cosmological
models, as well as observational evidence, indicates that the existence of a
photon ring is a necessity for the existence of a black hole. For example, in
an observational experiment, researchers observed a dark shadow resulting from
the bending of light by the massive black hole at the center of the giant
elliptical galaxy M87, explicitly confirming the existence of photon rings
around the central compact body [20]. Of course, this is not the only example,
and more such cases can be cited [20, 21, 22, 23, 24, 25]. From a theoretical
perspective, the necessity of a photon ring has been examined in various
models of gravitational potential in numerous articles [26, 27, 28, 29, 30,
31], to the extent that today, the existence of photon rings for different
black hole models can be somewhat assured. So that today we can be sure of the
existence of photon rings for black holes with high accuracy.
To calculate the photon sphere, you can follow different paths, for example,
you can use the Hamiltonian, effective potential, and Kiling symmetries, or
you can go for the topological method. In this article, we used the
topological method. To achieve this method Cardoso et al. stated that” the
nonlinear instability of ultra- compact stars would provide a strong argument
in favor of the ”black hole hypothesis,” once electromagnetic or
gravitational-wave observations confirm the existence of light rings”, the
objects with light rings are black holes [32]. In addition, Cunha and his
colleagues showed that the standard black holes with spherical symmetry, have
planar circular photon orbits, so that the stable type causes instability in
spacetime and its unstable type can help us to determine the shadows of the
black hole [33]. Then, they proved a theorem that further strengthened this
concept, a theorem that stated: ”axisymmetric, stationary solutions of the
Einstein field equations formed from classical gravitational collapse of
matter obeying the null energy condition, that are everywhere smooth and
ultra-compact (i.e., they have a light ring) must have at least two light
rings, and one of them is stable. It has been argued that stable light rings
generally lead to nonlinear spacetime instabilities[34]. In this context, the
work of Ghosh and Sarkar can also be mentioned [35]. Expanding this hypothesis
to four-dimensional black holes and testing its validity was the next step of
this team, which showed: ”A stationary, axisymmetric and asymptotically flat
black hole space-time in 1+3 dimensions, with a non-extremal, topologically
spherical and Killing horizon admits, at least, one standard light ring
outside the horizon for each rotation sense” [36]. At last, Shao-Wen Wei
utilized this concept and Duan mapping, extending the discussion of the photon
ring to the photon sphere with a topological approach”It is worth noting that
there exists at least one standard photon sphere outside the black hole not
only in asymptotically flat space-time but also in asymptotically Ads and dS
space-time” [37]. Collectively, the comprehensive information presented above
unequivocally confirms the presence of photon rings and spheres within black
hole structures. This essential feature of black hole structures motivated us
to explore whether we can determine the permissible range of the specific
parameters of each black hole with respect to the photon sphere. To achieve
this goal and obtain the equations that define the permissible ranges for each
parameter, we employed the method suggested by Wei to classify black holes
based on their total topological charge. By examining the required conditions
for the existence of a photon sphere for different black holes, we derived
intervals for the specific parameter of each black hole, such that in these
intervals:
1\. The black hole possesses a standard unstable photon sphere, meaning that
it exhibits the behavior of a normal black hole.
2\. It approaches a naked singularity that has its own distinctive behaviors.
3\. The absence of any photon structures, and thus the violation of the
condition for the existence of a photon sphere, renders these intervals as
prohibited for the structure to retain its classification as a black hole.

In general, this article can be examined from two separate perspectives,
firstly, based on the topological method, we have examined the photon sphere
for three separate and structurally different black hole samples.Secondly, we
tried to obtain the allowed ranges for the parameters of each model by using
the necessary conditions for the formation of the photon sphere.
All above information give us motivation to arrange the paper as follows. In
section 2 we briefly review the mathematical and physical foundations of the
work in such a way that firstly, the effective potential and states the
circular null geodesic conditions is introduced and then goes to the Duan’s
topological mapping and at the end of this section, in an important
conclusion, we recount the general way of working. In section 3,4,5 we will
test the obtained technique on different black holes, and we will determine
the allowed ranges of parameters with respect to different black holes.
Finally, in section 6 we summarized our conclusions.

##  2  Standard and Topological Methods for Investigating the Photon Sphere

Let’s start with some commonly used definitions. The event horizon, a null
one-sided hypersurface that acts as a causal boundary, ensuring that the weak
cosmic censorship (WCC) holds. Just opposite is the naked singularity, as a
hypothetical gravitational singularity without an event horizon. The photon
sphere is a null ring that is the lower bound for any stable orbit, showing
the extreme bending of light rays in a very strong gravity. It has two types:
unstable, where small perturbations make the photons either escape or fall
into the black hole, and stable, where the opposite happens. The unstable type
is useful for determining the black hole shadows, while the stable type causes
spacetime instability.
It is necessary to note that in this article we only consider and examine
structures that have spherical symmetry, that is, models in the following
form,

$\mathit{ds}^{2}=-\mathit{dt}^{2}f\\!\left(r\right)+\frac{\mathit{dr}^{2}}{g\\!\left(r\right)}+\left(d\theta^{2}+d\varphi^{2}\sin\\!\left(\theta\right)^{2}\right)h\\!\left(r\right).$
(1)

In short, In the standard method to find the photon sphere, we go to the
integral of the action and based on its lagrangian, we extract the Hamiltonian
in the form of four-momentum, and then by applying the condition of having
zero geodesics, we will have a relation in the following form,

$\Pi=\frac{p^{\nu}p^{\mu}\textit{g}_{\mu,\nu}}{2}=0.$ (2)

Now, according to the obtained Hamiltonian form, we try to obtain the
effective potential, which usually has the following standard form,

$\begin{split}V_{eff}=g(r)\bigg{(}\frac{L^{2}}{h(r)}-\frac{E^{2}}{f(r)}\bigg{)},\end{split}$
(3)

where $E$ and $L$ represent the photon’s energy and the total angular
momentum, respectively. The radial component of the null geodesic equations
will be as [37],

$\begin{split}\dot{r}^{2}+V_{eff}=0.\end{split}$ (4)

A circular null geodesic occurs at an extremum of the effective potential
$V_{eff}$(r), which is given by,

$\begin{split}V_{eff}=0,\hskip 28.45274pt\partial_{r}V_{eff}=0.\end{split}$
(5)

These local extrema of the effective potential are equivalent to unstable and
stable circular null geodesics, will correspond to a photon sphere and an
anti-photon sphere. By Considering the equations(5) at the same time, we have
following expression,

$\begin{split}\bigg{(}\frac{f(r)}{h(r)}\bigg{)}^{{}^{\prime}}_{r=r_{ps}}=0,\end{split}$
(6)

where prime is the derivative with respect to $r.$ By rewriting(6), we will
have,

$\begin{split}f(r)h(r)^{\prime}-f(r)^{\prime}h(r)=0.\end{split}$ (7)

The first term at the horizon will be disappear, but this case the second term
usually remains non-zero, it means that $r_{ps}$ and $r_{h}$ do not
coincide[37].
In our assessment, the standard approach commonly employed to investigate the
photon sphere possesses both merits and drawbacks. On one hand, this method is
not contingent on the symmetry of the metric function, rendering it applicable
to any metric form. However, in highly intricate structural models, it may be
challenging to straightforwardly ascertain the Hamiltonian function, or even
if it is determined, extracting its effective potential may prove unfeasible.
Furthermore, the differentiation between stable and unstable photon spheres
necessitates extensive computations. This implies that relying on the roots of
the metric function can only delineate the permissible regions for a black
hole-like structure. Essentially, we are only able to discern the range of
parameters with which unstable photon spheres can be precisely characterized.
Perhaps this is the motivation that we needed to lead us to the topological
method.
Yishi Duane, in $1984$, in examining the inner structure of the conserved
topological currents in SU(2) theory, introduced the topological flow related
to a point system like particles, which a foundation to discuss other kinds of
topological currents[38]. In the first step, we consider a general vector
field as $\phi$ which can be decomposed into two components, $\phi^{r}$ and
$\phi^{\theta}$,

$\phi=(\phi^{r},\phi^{\theta}),$ (8)

also here, we can rewrite the vector as $\phi=||\phi||e^{i\Theta}$, where
$||\phi||=\sqrt{\phi^{a}\phi^{a}}$, or $\phi=\phi^{r}+i\phi^{\theta}$. Based
on this, the normalized vector is defined as,

$n^{a}=\frac{\phi^{a}}{||\phi||},$ (9)

where $a=1,2$ and $(\phi^{1}=\phi^{r})$ , $(\phi^{2}=\phi^{\theta})$. Now we
introduce our antisymmetric superpotential as follows[37, 38],

$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^{a}\partial_{\rho}n^{b},\hskip
8.5359pt\mu,\nu,\rho=0,1,2$,

and the topological current will be as,

$j^{\mu}=\partial_{\nu}V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b}$.

Based on this, Noether’s current and charges at the given $\Omega$ will be,

$\partial_{\nu}j^{\nu}=0$,

and

$Q=\int_{\Omega}j^{0}d^{2}x,$ (10)

where $j^{0}$ is the charge density. By replacing $\phi$ instead of $n$ and
using the Jacobi tensor, we will arrive at,

$j^{\mu}=\frac{J^{\mu}\\!(\frac{\phi}{x})\ln\\!\left({||\phi||}\right)\Delta_{\phi^{a}}}{2\pi}.$
(11)

By using two-dimensional Laplacian Green function in $\phi-mapping$ space, we
have,

$\ln\\!\left({||\phi||}\right)\Delta_{\phi^{a}}=2\delta\\!\left(\phi\right)\pi,$
(12)

and the topological current will be,

$j^{\mu}=J^{\mu}\\!\left(X\right)\delta^{2}\\!\left(\phi\right).$ (13)

From the properties of $\delta$, it is clear that $j^{\mu}$ is non-zero only
at the zero points of $\phi^{a}$, and this is exactly what we need to continue
the discussion. Finally, by using the above relations and inserting them in
relation (10), the topological charge become,

$Q=\int_{\Omega}J^{0}\\!\left(X\right)\delta^{2}\\!\left(\phi\right)d^{2}x,$
(14)

once again and this time around the topological charge Q, according to the
characteristic of the $\delta$ function, it can be said that the charge is
non-zero only at the zero point of $\phi$.
Now, according to the above statements, we will go to the definition of the
potential for the topological model, according to [37] we will have

$\begin{split}H(r,\theta)=\sqrt{\frac{-g_{tt}}{g_{\varphi\varphi}}}=\frac{1}{\sin\theta}\bigg{(}\frac{f(r)}{h(r)}\bigg{)}^{1/2},\end{split}$
(15)

the discussion of potential will allow us to search for the radius of our
photon sphere at,

$\partial_{r}H=0$,

so, we can use a vector field $\phi=(\phi^{r},\phi^{\theta})$,

$\begin{split}&\phi^{r}=\frac{\partial_{r}H}{\sqrt{g_{rr}}}=\sqrt{g(r)}\partial_{r}H,\\\
&\phi^{\theta}=\frac{\partial_{\theta}H}{\sqrt{g_{\theta\theta}}}=\frac{\partial_{\theta}H}{\sqrt{h(r)}}.\end{split}$
(16)

With the above definition $\phi$, and recalling the relationships of the
Duan’s section, we now define the current and charge (10) for this new
potential.
Also, considering relation (13) and according to the characteristics of the
$\delta$ function,one can say that the charges will be non-zero only at the
points where $\phi$ is zero. That is, right where the photon sphere is placed,
so we can attribute a topological charge Q to each photon sphere. According to
equation (13), when we take $\Omega$ as a manifold which covers one zero
point, in that case the investigations show that the charge Q is exactly equal
to the number of winding[37]. Let $C_{i}$ be a closed curve that is smooth,
positive oriented, and encloses only the i-th zero point of $\phi$, while the
other zero points are outside of it. So the winding number can be calculated
by the following formula,

$\omega_{i}=\frac{1}{2\pi}\oint_{C_{i}}d\Lambda,$ (17)

where $\Lambda$ is

$\Lambda=\frac{\phi^{2}}{\phi^{1}},$ (18)

then the total charge will be,

$Q=\sum_{i}\omega_{i}.$ (19)

Finally, when the closed curve includes a zero point, the investigation of
corresponding note show that the charge Q is exactly equal to the winding
number. In that case, if it covers multiple zero points, Q will be sum of the
winding number at each zero point and if encloses no zero point then we must
have zero charge.
Before the end of this section it is better to explain, In a general
conclusion, how to calculate topological charges and total charges for
studying each case.
When the contour includes a winding:
1.If the field lines diverge around the zero point, its topological charge is
+1 (purple loop in Fig 3a). According to the previous definitions, this state
will correspond to the naked singularity.
2\. If the field lines converge around the zero point, its topological charge
is -1 (purple loop in Fig 2). This state will correspond to a black hole with
unstable photon sphere.
3.If the contour does not include a winding, its charge will be zero.Finally,
the total charges for each shape are the algebraic sum of the charges, which
can be displayed by drawing a contour that includes all the zero points.
Based on the aforementioned findings, it can be affirmed that each black hole
may harbor a photon sphere. In the same way, the topological current is only
non-zero at the zero point of the vector field, which determines the photon
sphere’s location. Consequently, a single topological charge can be attributed
to each photon sphere. Previous research indicates that in the complete
exterior region, when the structure takes the form of a black hole, the total
topological charge consistently amounts to -1[37]. Conversely, for a naked
singularity with a vanishing topological charge, the total topological charges
in this region will be either zero or +1[37, 39].
Therefore, in light of the above information, the region where the behavior of
the black hole results in a total topological charge of -1 aligns with the
permissible values and our preferred parameter distance in the metric,
corresponding to the inherent behavior of a black hole. Conversely, the region
where the equations’ behavior yields a total topological charge of zero or 1
will denote an interval for the parameters steering the system toward the
naked singularity. In conclusion, it can be inferred that beyond these two
ranges, and due to the contravention of the photon sphere condition in these
areas, these regions will likely be deemed illicit areas for the parameters.
$R_{PLPS}$: According to the allowed range of parameters to preserve the
structure of the black hole, the minimum or maximum possible radius for the
appearance of an unstable photon sphere is called the radius $R_{PLPS}$
(Possible Radius Limit for the Photon Sphere).

##  3  Einstein-Young-Mills non-minimal regular black hole

Regular non-minimal Einstein Yang-Mills (R-nm-EYM) black holes are a class of
black hole solutions in a modified gravity theory that couples the curvature
of spacetime to non-Abelian gauge fields. These black holes have two
distinctive features: they are regular, meaning that they do not have a
singularity at the center, and they have a magnetic charge, which is related
to the Yang-Mills field strength.The EYM theory is a generalization of the
Einstein-Maxwell theory, which couples the curvature of spacetime to the
electromagnetic field. This theory replaces the electromagnetic field with a
non-Abelian gauge field, which is a type of field that has internal symmetries
and self-interactions. The EYM theory also introduces a non-minimal coupling
term, which is a term that involves the product of the curvature tensor and
the gauge field strength tensor. This term modifies the gravitational and
gauge field equations, and leads to new types of solutions.The EYM theory and
its black hole solutions are interesting for several reasons. They provide a
way to study the effects of non-Abelian gauge fields and non-minimal couplings
on the geometry and physics of black holes. They also offer a possibility to
avoid the singularity problem, which is one of the major challenges in
classical and quantum gravity. Moreover, they have potential applications to
astrophysics and cosmology, as they could describe some of the observed
phenomena related to black holes, such as gravitational lensing, gravitational
waves, and black hole shadows[40, 41, 42, 43, 44, 45, 46].
the metric function for the 4D spherically symmetric R-nm-EYM black hole is
given by [40],

$f=1+\frac{r^{4}\left(-\frac{2m}{r}+\frac{q^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}\right)}{r^{4}+2q^{2}\xi},$
(20)

where $\xi$ is the nonminimal parameter of the theory, $l$ is related to the
cosmological constant( $l^{2}=-\frac{3}{\Lambda}$), $q$ is the magnetic charge
of the Wu-Yang gauge field, and $m$ is the asymptotic mass. Since our main
purpose is to study the allowed range for specific parameters of the theory,
for this reason we value the usual parameters such as mass, AdS radius and
charge, where these selected values are $m=1,l=1,q=0.5$.
Now, if we solve the metric function (20) based on the assumed values to find
the event horizon in terms of $\xi$, we will find that our metric function
always has a solution for $\xi\leq 0.41914$ , fig(1a), and in better words,
the structure can have the form of a black hole. But, for values greater than
0.41914, fig(1b), the metric function is always unanswered, which means that
the structure has no event horizon, and as a result, it will be in the form of
a naked singularity.

Figure 1: Plot 1(a) represents $f(r)$ vs. r with $\xi=0.41914$ , Plot 1(b)
shows the behavior $f(r)$ vs. r with $\xi=0.42$

Now, to deal with the topological view, according to the parametric values
mentioned above and since we have the following conditions for this model

$f\\!\left(r\right)=g\\!\left(r\right),$ (21) $h\\!\left(r\right)=r,$ (22)

from equations (15) and (16), we will have,

$H=\frac{\sqrt{1+\frac{r^{4}\left(-\frac{2}{r}+\frac{0.25}{r^{2}}+r^{2}\right)}{r^{4}+0.50\xi}}}{\sin\\!\left(\theta\right)r},$
(23)
$\phi^{r}=\frac{-r^{8}+3.0r^{7}+\left(\xi-0.5\right)r^{6}-1.0r^{4}\xi-0.5r^{3}\xi-0.25\xi^{2}}{r^{2}\sin\\!\left(\theta\right)\left(r^{4}+0.5\xi\right)^{2}},$
(24)
$\phi^{\theta}=-\frac{\sqrt{\frac{r^{4}+0.5\xi-2.0r^{3}+0.25r^{2}+r^{6}}{r^{4}+0.5\xi}}\,\cos\\!\left(\theta\right)}{\sin\\!\left(\theta\right)^{2}r^{2}}.$
(25)

###  3.1  photon sphere ranges

As we can see in fig(2), in the allowed areas for $\xi$,($0<\xi\leq 041914$)
the structure has a total topological charge of -1 and the black hole shows a
normal behavior by showing unstable photon sphere. What can be pointed out
here and is important is that, as shown in Table (1), to have a black hole
form with respect to the main pre-selected parameters, is the restriction that
we will have for the choice of $\xi$. In addition, with this method, the
radius $R_{PLPS}$ can also be calculated based on parametric constraints.

Figure 2: the blue arrows represent the vector field n on a portion of
the$(r,\theta)$ plane, The photon sphere is located at
$(r,\theta)=(2.938684304,1.57)$ with respect to $(q=0.5,m=1,\xi=0.37,l=1)$

###  3.2 naked singularity ranges

From a mathematical point of view, using equations (24) and (25) to find the
topological charges, we find that unlike the metric function which assumes
that for all values of $\xi\leq 0.41914$ the structure has an event
horizon(that is, it can still behave like a black hole), but Our calculations
for the $\phi$ function show that in negative $\xi$ regions, the structure
only responds up to $\xi=-4.3605$, although this response is in the form of a
naked singularity fig(3a) and after this range the response completely
disappears. This is to some extent a confirmation and maybe a complement to
the statement that was expressed in the [40] article. Also, the calculations
for the positive regions show that the structure for $\xi$’s greater than
0.41914 with a total topological charge of zero behaves in the form of a naked
singularity (Fig 3b), which means that there is a limit for choosing positive
z’s as well.

Figure 3: Plot 3(a) The photon spheres is located at
$(r,\theta)=(1.087510124,1.57)$ and $(r,\theta)=(1.358858788,1.57)$ with
respect to $(q=0.5,m=1,\xi=-4.1,l=1)$ , Plot 3(b) The photon spheres is
located at $(r,\theta)=(0.9525586040,1.57)$ and
$(r,\theta)=(3.203877596,1.57)$ with respect to $(q=0.5,m=1,\xi=1.3,l=1)$

The full results of this section are given in Table (1).

R-nm-EYM black holes |  Fix parameters |  Conditions |  *TTC |  $(R_{PLPS})$
---|---|---|---|---
*Unauthorized area |  $q=0.5,m=1,l=1$ |  $\xi<-4.3605$ |  $nothing$ |  $-$
naked singularity |  $q=0.5,m=1,l=1$ |  $-4.3605\leq\xi<0$ |  $0$ |  $-$
unstable photon sphere |  $q=0.5,m=1,l=1$ |  $0<\xi\leq 0.41914$ |  $-1$ |  $2.953579630$
naked singularity |  $q=0.5,m=1,l=1$ |  $0.41914<\xi$ |  $0$ |  $-$
Table 1: *Unauthorized region: is the region where the roots of $\phi$
equations become negative or imaginary in this region.
TTC: *Total Topological Charge

##  4  AdS black holes surrounded by dark fluid with Chaplygin-like equation
of state

AdS black holes surrounded by dark fluid with Chaplygin-like equation of state
are a class of black hole solutions in a modified gravity theory that includes
a dark fluid as a cosmic background. The dark fluid is a hypothetical form of
matter that could account for the accelerated expansion of the universe. The
dark fluid has a Chaplygin-like equation of state, which means that its
pressure is inversely proportional to its density. The Chaplygin-like equation
of state can interpolate between different regimes, such as dust, radiation,
cosmological constant, and phantom energy. The dark fluid can also mimic the
effects of dark matter and dark energy, which are the two main components of
the dark sector of the universe. The AdS black holes surrounded by dark fluid
with Chaplygin-like equation of state are interesting for several reasons.
They provide a way to study the effects of dark fluid and Chaplygin-like
equation of state on the geometry and physics of black holes. They also offer
a possibility to resolve the singularity problem and to describe a regular
black hole. Moreover, they have potential applications to astrophysics and
cosmology, as they could describe some of the observed phenomena related to
black holes, such as gravitational lensing, gravitational waves, and black
hole shadows.[47, 48, 49]
the metric function for the 4D spherically symmetric Chaplygin-like Dark Fluid
black holes is given by [47],

$f=1-\frac{2m}{r}+\frac{\left(\frac{q}{r^{3}\sqrt{b}}+\frac{q^{3}}{6r^{9}b^{\frac{3}{2}}}\right)q}{3r}-\frac{\sqrt{b+\frac{q^{2}}{r^{6}}}\,r^{2}}{3}+\frac{r^{2}}{l^{2}},$
(26)

where in this case, the Chaplygin-like Dark Fluid has a non-linear form as
$p=-b/\rho$ , that $\rho$ is energy density and ; b ; is a positive constant,
$l$ is related to the cosmological constant( $l^{2}=-\frac{3}{\Lambda}$),
$q>0$, is a normalization factor that indicates the intensity of the
Chaplygin-like Dark Fluid, and $m$ denotes the mass of the black hole. Since
our main purpose is to study the allowed range for specific parameters of the
theory, for this reason we value the usual parameters such as mass, AdS radius
, where these selected values are $m=1,l=1$.
Now, if we solve the metric function (20) based on the assumed values to find
the event horizon in terms of $b=0.76$, we will find that our metric function
always has a solution for$,0<q\leq 1.02709$, fig(4), and in better words, the
structure can have the form of a black hole.But, for values greater than
1.02709, the metric function is always unanswered, which means that the
structure has no event horizon, and as a result, it will be in the form of a
naked singularity.

Figure 4: this plot represents ($f(r)$ vs. r) with $b=0.76$ and $q=1.02709$
where the event horizon located at $r_{+}=0.9366148712$

Now, to deal with the topological view, according to the parametric values
mentioned above and with respect to (21) and (22), for such black hole, the
potential H and $\phi$ (15) , (16) after some calculations are,

$H=\frac{\sqrt{9-\frac{18}{r}+\frac{3\left(\frac{q}{r^{3}\sqrt{b}}+\frac{q^{3}}{6r^{9}b^{\frac{3}{2}}}\right)q}{r}-3\sqrt{b+\frac{q^{2}}{r^{6}}}\,r^{2}+9r^{2}}}{3\sin\\!\left(\theta\right)r},$
(27)
$\begin{split}&\mathcal{A}=((r^{9}(r-3)b^{\frac{3}{2}}+b\,q^{2}r^{6}+\frac{q^{4}}{3})\sqrt{\frac{b\,r^{6}+q^{2}}{r^{6}}}-\frac{q^{2}r^{6}b^{\frac{3}{2}}}{2})\sqrt{3}\\\
&\times\sqrt{9-\frac{18}{r}+\frac{3(\frac{q}{r^{3}\sqrt{b}}+\frac{q^{3}}{6r^{9}b^{\frac{3}{2}}})q}{r}-3\sqrt{b+\frac{q^{2}}{r^{6}}}\,r^{2}+9r^{2}}\\\
&\mathcal{B}=3\sqrt{\frac{-\sqrt{\frac{b\,r^{6}+q^{2}}{r^{6}}}\,r^{12}b^{\frac{3}{2}}+(3r^{12}+3r^{10}-6r^{9})b^{\frac{3}{2}}+b\,q^{2}r^{6}+\frac{q^{4}}{6}}{b^{\frac{3}{2}}r^{10}}}\,\sqrt{\frac{b\,r^{6}+q^{2}}{r^{6}}}\,b^{\frac{3}{2}}r^{12}\sin\\!(\theta)\\\
\phi^{r}=\frac{\mathcal{A}}{\mathcal{B}},\end{split}$ (28)
$\phi^{\theta}=-\frac{\sqrt{9-\frac{18}{r}+\frac{3\left(\frac{q}{r^{3}\sqrt{b}}+\frac{q^{3}}{6r^{9}b^{\frac{3}{2}}}\right)q}{r}-3\sqrt{b+\frac{q^{2}}{r^{6}}}\,r^{2}+9r^{2}}\,\cos\\!\left(\theta\right)}{3\sin\\!\left(\theta\right)^{2}r^{2}}.$
(29)

Calculations for this structure show that at a certain(optional) $b=0.76$, the
black hole in the allowed range of $0<q\leq 1.02709$ in fig (5a) shows a total
topological charge of -1 and has an unstable photon sphere. The important
point about this fig (5a) is that two topological charges are seen in it, and
therefore the total charge must be zero. What should be noted is that the
radius in which one of the charges appears, the charge enclosed in the green
ring, is smaller than the radius of the event horizon $r_{+}=0.9366148712$,
that is, this charge is located behind the horizon and practically according
to the concept of WCC it will not be included in the calculations. In the
interval $,1.02709<q\leq 3.6282$, it tends to naked singularity and having two
separate charges outside the horizon of fig (5b), it shows zero topological
charge. Finally, upon reaching the $q=3.6282$ limit, for $q\geq 3.6283$ the
structure lacks any solution. The results of this section are shown in Table
2.

Figure 5: Plot 5(a) the photon spheres is located at $(r,\theta)=(0.9364211602,1.57)$ and $(r,\theta)=(2.977014940,1.57)$ with respect to $(b=0.76,m=1,q=1.02709,l=1)$ , Plot 5(b) the photon spheres is located at $(r,\theta)=(1.806680305,1.57)$ and $(r,\theta)=(2.741356787,1.57)$ with respect to $(b=0.76,m=1,q=3,l=1)$ Chaplygin-like Dark Fluid black holes |  Fix parameters |  Conditions |  *TTC |  $(R_{PLPS})$
---|---|---|---|---
unstable photon sphere |  $b=0.76,m=1,l=1$ |  $0<q\leq 1.02709$ |  $-1$ |  $2.977014940$
naked singularity |  $b=0.76,m=1,l=1$ |  $1.02709<q\leq 3.6282$ |  $0$ |  $-$
*Unauthorized area |  $b=0.76,m=1,l=1$ |  $q\geq 3.6283$ |  $nothing$ |  $-$
Table 2: *Unauthorized region: is the region where the roots of $\phi$
equations become negative or imaginary in this region.
TTC: *Total Topological Charge

##  5  Bardeen-like regular black holes in Einstein-Gauss-Bonnet gravity

The regularized Einstein–Gauss–Bonnet gravity is a recent modification of the
Einstein’s general relativity that incorporates higher-order curvature terms
in the action, namely the Gauss–Bonnet (GB) term, which is usually relevant
only in higher dimensions. The regularization procedure, introduced by Glavan
and Lin in 2020, allows the GB term to have a nontrivial effect in four
dimensions by rescaling the GB coupling constant by the inverse of the space-
time dimension and taking the limit as the dimension approaches four. This
way, the field equations remain second-order and free of ghosts, but also
acquire some novel features that differ from the standard Einstein gravity.
Several authors have studied the properties of Bardeen black holes in the
regularized Einstein–Gauss–Bonnet gravity, such as their horizon structure,
thermodynamics, stability, phase transitions, and strong gravitational
lensing. Some of the main results are:
• The GB parameter affects the minimum cutoff values of mass and charge for
the existence of a black hole horizon, and also the size and shape of the
horizons. Depending on the values of the parameters, there can be zero, one,
or two horizons, corresponding to a naked singularity, an extremal black hole,
or a regular black hole, respectively [51, 52].
• The black hole undergoes a second-order phase transition at a critical value
of the horizon radius, where the temperature has a local maximum and the heat
capacity diverges. The phase transition is similar to the Hawking–Page
transition in AdS space, where the black hole becomes thermodynamically
preferred over the pure de Sitter space below the critical temperature [51].
• The black hole acts as a strong gravitational lens in the strong deflection
limit, where the light rays can loop around the black hole multiple times
before reaching the observer. The GB parameter and the NED parameter affect
the observable of the strong gravitational lensing, such as the deflection
angle, the position and magnification of the relativistic images, and the time
delay between them [53].
We studied this structure separately with and without the cosmological
constant in 4 and 5 dimensions.

###  5.1 $5D:\Lambda=0$

the metric function for the 5D spherically symmetric Bardeen-like regular
black holes in Einstein-Gauss-Bonnet gravity is given by [50],

$f=1+\frac{r^{2}\left(1-\sqrt{1+\frac{8\alpha
m}{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}}}\right)}{2\alpha},$ (30)

where m and g, are respectively mass and magnetic charge and $\alpha>0$ is the
GB coupling constant. Again we value the usual parameter, $m=1$. If we solve
the metric function (20) based on the assumed values to find the event horizon
in terms of $\alpha=0.4$(optional), we will find that our metric function has
a solution for $-0.7026\leq g\leq 0.70259$, fig(6), and for values outside
this range, the metric function is always unanswered, which means that it will
be in the form of a naked singularity.

Figure 6: this plot represents ($f(r)$ vs. r) with $\alpha=0.4$ and $g=0.7026$
where the event horizon located at $r_{+}=1.661555644$

According to the parametric values mentioned above and with respect to (21)
and (22), for such black hole, the potential H and $\phi$ (15),(16) are,

$H=\frac{\sqrt{4+\frac{2r^{2}\left(1-\sqrt{1+\frac{8\alpha}{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}}}\right)}{\alpha}}}{2\sin\\!\left(\theta\right)r},$
(31)
$\phi^{r}=-\frac{\left(g^{2}+r^{2}\right)^{\frac{7}{3}}\sqrt{\frac{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}+8\alpha}{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}}}-\frac{8r^{4}}{3}}{\sqrt{\frac{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}+8\alpha}{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}}}\,\left(g^{2}+r^{2}\right)^{\frac{7}{3}}r^{2}\sin\\!\left(\theta\right)},$
(32)
$\phi^{\theta}=-\frac{\sqrt{4+\frac{2r^{2}\left(1-\sqrt{1+\frac{8\alpha}{\left(g^{2}+r^{2}\right)^{\frac{4}{3}}}}\right)}{\alpha}}\,\cos\\!\left(\theta\right)}{2\sin\\!\left(\theta\right)^{2}r^{2}}.$
(33)

Calculations for this structure show that, the black hole in the allowed range
$-0.7026\leq g\leq 0.70259$ in fig (7a) shows a total topological charge of -1
and has an unstable photon sphere. In this fig again two topological charges
exist and therefore the total charge must be zero. but the event horizon
located at $r_{+}=1.661555644$, that is, one of the charge is located behind
the horizon and practically according to the concept of WCC and it will not be
included in the calculations. Calculations show that outside of the above
normal range, the structure will remain naked singularity form only up to
g=0.865, that is, the total topological charge will be zero, and after this
value, the solution will be practically lost. The full results are in Table 3.

Figure 7: Plot 7(a) the photon spheres is located at
$(r,\theta)=(1.648235913,1.57)$ and $(r,\theta)=(3.523343153,1.57)$ with
respect to $(g=0.70259,m=1,\alpha=0.4)$ , plot 7(b) the photon spheres is
located at $(r,\theta)=(2.554954982,1.57)$ and $(r,\theta)=(2.704224286,1.57)$
with respect to $(g=0.865,m=1,\alpha=0.4)$

5D Bardeen black holes in EGB |  Fix parameters |  Conditions |  *TTC |  $(R_{PLPS})$
---|---|---|---|---
unstable photon sphere |  $m=1,\alpha=0.4$ |  $-0.7026\leq g\leq 0.70259$ |  $-1$ |  $3.523343153$
naked singularity |  $m=1,\alpha=0.4$ |  $-0.865\leq g\leq-0.70259$ |  |
$0.70259\leq g\leq 0.865$ |  $0$ |  $-$ |  |
*Unauthorized area |  $m=1,\alpha=0.4$ |  $g<-0.865$ |  |
$g>0.865$ |  $nothing$ |  $-$ |  |

Table 3: *Unauthorized region: is the region where the roots of $\phi$
equations become negative or imaginary in this region. TTC: *Total Topological
Charge

###  5.2 $4D:\Lambda\neq 0$

Since the behavior is very similar to the 5D model, we only mention the main
formulas and show the results in the final table4. [54]

$f=1+\frac{r^{2}\left(1-\sqrt{1+4\left(\frac{2m}{\left(g^{2}+r^{2}\right)^{\frac{3}{2}}}-\frac{1}{l^{2}}\right)\alpha}\right)}{2\alpha},$
(34)
$H=\frac{\sqrt{4+\frac{2r^{2}\left(1-\sqrt{1+4\left(\frac{2}{\left(g^{2}+r^{2}\right)^{\frac{3}{2}}}-1\right)\alpha}\right)}{\alpha}}}{2\sin\\!\left(\theta\right)r},$
(35)
$\phi^{r}=-\frac{2\left(g^{2}+r^{2}\right)^{\frac{5}{2}}\sqrt{-\frac{\left(\alpha-\frac{1}{4}\right)\left(g^{2}+r^{2}\right)^{\frac{3}{2}}-2\alpha}{\left(g^{2}+r^{2}\right)^{\frac{3}{2}}}}-3r^{4}}{2\sqrt{-\frac{\left(\alpha-\frac{1}{4}\right)\left(g^{2}+r^{2}\right)^{\frac{3}{2}}-2\alpha}{\left(g^{2}+r^{2}\right)^{\frac{3}{2}}}}\,\left(g^{2}+r^{2}\right)^{\frac{5}{2}}r^{2}\sin\\!\left(\theta\right)},$
(36)
$\phi^{\theta}=-\frac{\sqrt{4+\frac{2r^{2}\left(1-\sqrt{1+4\left(\frac{2}{\left(g^{2}+r^{2}\right)^{\frac{3}{2}}}-1\right)\alpha}\right)}{\alpha}}\,\cos\\!\left(\theta\right)}{2\sin\\!\left(\theta\right)^{2}r^{2}}.$
(37)

From a deeper perspective, if we want to discuss our primary motivation for
these calculations, it is essential to acknowledge that the existence of a
photon sphere for ultra-compact gravitational structures like black holes is a
necessity which is largely confirmed by studies conducted on various black
hole models[20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68]. Leveraging this necessity can bring
several interesting implications. As the simplest outcome, the calculation of
unstable photon spheres could obviate the need for a simple computation of the
WCCC, as an unstable photon sphere will only dominate the space of the system
when an event horizon exists, and thus the WCCC condition is satisfied.
However, the most significant implication, which shapes the main goal of our
paper, is that we can simply classify the parameter space of the models under
study based on the existence and location of the topological photon spheres
and anti-photon spheres. That is, we can accurately demonstrate which
parameter ranges of the model correspond to a black hole structure, a naked
singularity, or even a model that, mathematically, lacks a physical and
scientific response for its surrounding space, which we have termed the
forbidden region. The other achievement of the necessity of the photon sphere
existing and its integration with a topological perspective is that we can
also use the aforementioned classification inversely. In a process akin to
reverse engineering, we can balance the parameters of our system in such a way
that the model under study is compelled to exhibit the behavior we desire.
This outcome is quite intriguing in its own right, as in certain gravitational
studies, we require a gravitational structure that adheres to specific
principles and behaviors. In such cases, this method could serve as a
significant contribution to these studies, which necessitate particular
parameter ranges while simultaneously requiring the structure to remain in the
form of a black hole or any other format. To elucidate this point further, let
us provide an example. In the study model of the Swampland Conjecture, there
are hypotheses such as the Weak Gravity Conjecture (WGC), among others, which
require that the black hole, while becoming overcharged, must still retain a
quasi-black hole form, or for instance, that the strength of terms related to
magnetic parameters be compared and influenced by electric and gravitational
counterparts. In this work, we suggest that for such scenarios, one can
utilize the topological behavior of the photon sphere and, after adjusting the
parameters to the desired form, examine the topological photon sphere to
determine whether the desired conditions can be achieved. In summary, and as a
synthesis of all the above discussions, it can be proposed that a
bidirectional relationship exists between the parameters and the topological
photon sphere. On one hand, based on the behavior of the topological photon
sphere, the permissible range for the parameters of the model under study can
be determined, and the space surrounding the model can be accurately
classified according to the behavior of the photon sphere. Conversely, by
choosing the parameters in the form required in the study, they can be used as
a tool to control the behavior of the black hole. Finally, we remark that in
this article we focus on the specific parameters of each model, such as the
non-minimal coupling, the Chaplygin gas parameter, and the Bardeen charge.
However, if these parameters are fixed, the method of range finding can be
extended to the common and main parameters of the black holes, such as the
mass, the charge, and the angular momentum. We leave this generalization for
future work.

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# Characterizing matrices with eigenvalues in an LMI region:
A dissipative-Hamiltonian approach

Neelam Choudhary Nicolas Gillis Punit Sharma  School of Engineering and
Applied Sciences, Department of Mathematics, Bennett University, Greater
Noida-201310, Uttar Pradesh, India<EMAIL_ADDRESS>Department of Mathematics and Operational Research, Faculté Polytechnique,
Université de Mons, Rue de Houdain 9, 7000 Mons, Belgium;
<EMAIL_ADDRESS>N. Gillis acknowledges the support by the Fonds de
la Recherche Scientifique - FNRS and the Fonds Wetenschappelijk Onderzoek -
Vlanderen (FWO) under EOS Project no O005318F-RG47, and by the Francqui
Foundation. Department of Mathematics, Indian Institute of Technology Delhi,
Hauz Khas, New Delhi-110016, India<EMAIL_ADDRESS>P. Sharma
acknowledges the support of the DST-Inspire Faculty Award (MI01807-G) by the
Government of India and Institute SEED Grant (NPN5R) by IIT Delhi.

###### Abstract

In this paper, we provide a dissipative Hamiltonian (DH) characterization for
the set of matrices whose eigenvalues belong to a given LMI region. This
characterization is a generalization of that of Choudhary et al. (Numer.
Linear Algebra Appl, 2020) to any LMI region. It can be used in various
contexts, which we illustrate on the nearest $\Omega$-stable matrix problem:
given an LMI region $\Omega\subseteq{\mathbb{C}}$ and a matrix
$A\in{\mathbb{C}}^{n,n}$, find the nearest matrix to $A$ whose eigenvalues
belong to $\Omega$. Finally, we generalize our characterization to more
general regions that can be expressed using LMIs involving complex matrices.

Keywords: $\Omega$-stability, linear matrix inequalities, dissipative-
Hamiltonian systems, nearest stable matrix

##  1 Introduction

In this paper, we study matrices $A\in{\mathbb{R}}^{n,n}$ whose eigenvalues
belong to a subset of the complex plane, $\Omega\subseteq{\mathbb{C}}$, such
matrices are called $\Omega$-stable.

######  Definition 1.

($\Omega$-stability) For $\Omega\subseteq\mathbb{C}$, the matrix
$A\in\mathbb{R}^{n,n}$ is said to be $\Omega$-stable if every eigenvalue of
$A$ lie inside the region $\Omega$.

The two most famous examples of $\Omega$-stable matrices are Hurwitz stable
matrices for which
$\Omega=\\{z\in{\mathbb{C}}:~{}\mathop{\mathrm{Re}}{z}<0\\}$, and Schur stable
matrices for which $\Omega=\\{z\in{\mathbb{C}}:~{}|z|<1\\}$. Hurwitz stable
matrices play a significant role in the study of linear time-invariant (LTI)
systems of the form

$\displaystyle\dot{x}(t)=Ax(t)+Bu(t),\quad y(t)=Cx(t),$

where, for all $t\in{\mathbb{R}}$, $x(t)\in{\mathbb{R}}^{n}$,
$u(t)\in{\mathbb{R}}^{m}$, $y(t)\in{\mathbb{R}}^{p}$,
$A\in{\mathbb{R}}^{n,n}$, $B\in{\mathbb{R}}^{n,m}$, and
$C\in{\mathbb{R}}^{p,n}$. In fact, such a system is stable if $A$ is Hurwitz
stable. Moreover, the transient response of such a system is directly related
to the location of its poles [3] in the complex plane. The poles in a specific
region in the complex plane can bound the maximum overshoot, the frequency of
oscillatory modes, the delay time, the rise time, and the settling time. The
problem of locating all the closed-loop poles of a controlled system inside a
specific region $\Omega\subseteq{\mathbb{C}}$ is known as the _$\Omega$ -pole
placement problem_ and has appeared in several applications [3, 13, 1, 23, 4,
24].

For these reasons, characterizing $\Omega$-stable matrices is an important
topic in numerical linear algebra and control. In this paper, we focus on
regions of the complex plane that can be expressed by linear matrix
inequalities (LMIs). Such sets are referred to as _LMI regions_ [3] and
defined as follows.

######  Definition 2 (LMI Region, [3]).

A subset $\Omega\subseteq{\mathbb{C}}$ is called an LMI region if there exists
a symmetric matrix $B\in{\mathbb{R}}^{s,s}$ and a matrix
$C\in{\mathbb{R}}^{s,s}$ such that

$\Omega=\left\\{z\in{\mathbb{C}}:~{}f_{\Omega}(z)\prec 0\right\\},$ (1.1)

where

$f_{\Omega}\colon\mathbb{C}\mapsto{\mathbb{H}}^{s,s}\;\text{ is given by
}\;z\mapsto f_{\Omega}(z):=B+zC+\overline{z}C^{T},$ (1.2)

where ${\mathbb{H}}^{s,s}$ is the set of Hermitian matrices with real
eigenvalues, that is, ${\mathbb{H}}^{s,s}=\\{X\in{\mathbb{C}}^{s,s}\ :\
X=X^{*}\\}$ with $X^{*}$ the conjugate transpose of $X$, and for
$X\in{\mathbb{H}}^{s,s}$, $X\prec 0$ means that $X$ is negative definite, that
is, its eigenvalues are negative.

The function $f_{\Omega}(z)$ is called the characteristic function of the LMI
region $\Omega$, and $s$ is called the order of $f_{\Omega}(z)$. The
characteristic function of an LMI region is not unique [15]. Since
$f_{\Omega}(\overline{z})=(f_{\Omega}(z))^{T}$, any LMI region is symmetric
along the real axis. An LMI region is convex, and so is the intersection of
two or more LMI regions. Due to the strict inequality “$\prec$” in (1.1), the
LMI regions are open. Furthermore, the LMI regions are invariant under
congruence transformations. We refer to [15] for other topological and
geometrical properties of the LMI regions. A large number of subsets in the
complex plane can be expressed as LMI regions; for example, conic sectors,
vertical half-planes, vertical strips, discs, horizontal strips, ellipses,
parabolas, and hyperbolic sectors; see [2] and Section 3. The set of LMI
regions is dense in the set of convex regions symmetric to the real axis,
which are relevant for control systems [3, 2].

In [9], authors characterized $\Omega$-stable matrices using the so-called
dissipative Hamiltonian (DH) matrices.

######  Definition 3 (DH matrix).

A matrix $A\in{\mathbb{R}}^{n,n}$ is said to be a DH matrix if $A=(J-R)Q$ for
some $J,R,Q\in{\mathbb{R}}^{n,n}$ such that $J^{T}=-J$, $R\succeq 0$, and
$Q\succ 0$.

A DH matrix is always Hurwitz stable, that is, all its eigenvalues are in the
left half of the complex plane [9]. The term DH is inspired by the DH systems
in which the state matrix has the form $A=(J-R)Q$, where $J^{T}=-J$ is the
structure matrix describing the flux among energy storage elements, $R$ is a
positive semidefinite matrix describing the energy dissipation in the system,
and $Q$ is a positive definite matrix that describes the energy of the system
[21, 22]. By replacing the constraint on $R$, namely $R\succeq 0$, by other
LMI constraints on $(J,R,Q)$, DH matrices can represent different types of
$\Omega$-stable matrices. This was studied in [5] where $\Omega$-stable
matrices were written as DH matrices where $\Omega$ could be vertical strips,
disks, conic sectors, and their intersection; see the next section for more
details. An application of these characterizations is to solve the nearest
$\Omega$-stable matrix problem. For example, in system identification, one
needs to identify a $\Omega$-stable system from observations [19]. In fact,
sometimes numerical or modelling errors or approximation processes may produce
an unstable system in place of a stable one. The unstable system then has to
be approximated by a nearby stable system without perturbing its entries too
much. More precisely, for a region $\Omega\in{\mathbb{C}}$ and a matrix
$A\in{\mathbb{R}}^{n,n}$, this requires to solve the following optimization
problem

$\inf_{X\in\mathcal{S}_{\Omega}}{\|A-X\|}_{F}^{2},$ (1.3)

where $\|\cdot\|_{F}$ stands for the Frobenius norm and $\mathcal{S}_{\Omega}$
is the set of all $\Omega$-stable matrices. The DH characterization of stable
matrices has been proven very effective in solving several nearness problems
for LTI control systems. For example, distance to $\Omega$-stability [9, 5,
17], nearest admissible descriptor system problem [8], distance to passivity
[10], minimal-norm-static-state feedback problem [11], and learning data-
driven stable Koopman operators [16]. This DH characterization was also used
recently to design an optimization-based algorithm for parametric model order
reduction of LTI dynamical systems [20].

#### Contribution and outline of the paper

This paper is organized as follows. In Section 2, for a given LMI region
$\Omega\subseteq{\mathbb{C}}$, we characterize the set of all $\Omega$-stable
matrices as DH matrices of the form $A=(J-R)Q$ with LMI constraints on the
triplet $(J,R,Q)\in({\mathbb{R}}^{n,n})^{3}$. This characterization
generalizes the work on [5] that only considered three types of LMI regions,
namely conic sectors, vertical strips and disks. In Section 3, we provide
several examples of such LMI regions, including parabolas, ellipsoids,
hyperbolas and horizontal strips; this is the first time DH characterizations
of such regions are given via LMIs on the triplet $(J,R,Q)$. In Section 4, we
illustrate the use of these characterizations to solve the nearest
$\Omega$-stable matrix problem (1.3). Finally, in Section 5, we extend our
characterizations of LMI regions for complex matrices, where the set $\Omega$
is not necessarily symmetric with respect to the real line.

#### Notation

Throughout the paper, $X^{T}$ and $\|X\|$ stand for the transpose and the
spectral norm of a real matrix $X$, respectively. We write $X\succ 0$ $(X\prec
0)$ and $X\succeq 0$ $(X\preceq 0)$ if $X$ is symmetric and positive definite
(negative definite) or positive semidefinite (negative semidefinite),
respectively. By $I_{m}$ we denote the identity matrix of size $m\times m$.
The Kronecker product is represented by $\otimes$ and we refer to [14] for the
standard properties of the Kronecker product. The set of $n\times n$ Hermitian
matrices is denoted by ${\mathbb{H}}^{n,n}$.

##  2 DH characterization of matrices with eigenvalues in generic LMI regions

In this section, we consider matrices with eigenvalues in some generic LMI
regions, and for them, we provide a parametrization using the DH matrices.
This will allow us for example in Section 4 to use standard optimization tools
to find a nearby matrix to a given matrix with eigenvalues all in the given
LMI region.

The following result from [3] is crucial for our DH formulation of the set of
$\Omega$-stable matrices, where $\Omega$ is an LMI region.

######  Theorem 1.

[3, Theorem 2.2] Let $\Omega$ be an LMI region given by (1.1) and let
$A\in{\mathbb{R}}^{n,n}$. Then $A$ is $\Omega$-stable if and only if there
exists a symmetric matrix $X\in{\mathbb{R}}^{n,n}$ such that $X\succ 0$ and

$\mathcal{M}_{\Omega}(A,X):=B\otimes X+C\otimes(AX)+C^{T}\otimes(AX)^{T}\prec
0.$

Let us illustrate Theorem 1 on the two most well-known examples.

  1. 1)

For Hurwitz stability with
$\Omega=\\{z\in{\mathbb{C}}:~{}\mathop{\mathrm{Re}}{z}<0\\}$, we simply take
$B=0$ and $C=1$ to obtain $f_{\Omega}(z)=\mathop{\mathrm{Re}}{z}$. This choice
leads to the classical Lyapunov stability criterion, namely there exists
$X\succ 0$ such that $AX+XA^{T}\prec 0$.

  2. 2)

For Schur stability with $\Omega=\\{z\in{\mathbb{C}}:~{}|z|<1\\}$, we take
$B=-\binom{1\;0}{0\;1}$ and $C=\binom{0\;1}{0\;0}$, so that

$f_{\Omega}(z)=\left[\begin{array}[]{cc}-1&z\\\
\overline{z}&-1\end{array}\right]=\underbrace{\left[\begin{array}[]{cc}-1&0\\\
0&-1\end{array}\right]}_{=B}+z\underbrace{\left[\begin{array}[]{cc}0&1\\\
0&0\end{array}\right]}_{=C}+\bar{z}\underbrace{\left[\begin{array}[]{cc}0&0\\\
1&0\end{array}\right]}_{=C^{\top}}.$

In fact, $f_{\Omega}(z)\prec 0$ if and only if the trace of $f_{\Omega}(z)$ is
negative (the sum of the eigenvalues is negative) and the determinant is
positive (the product of the eigenvalues is positive). The trace is equal to
$-2$, while the determinant is given by $1-|z|^{2}$, which leads to the region
$|z|^{2}<1$, as desired. This choice leads to the well-known condition for
discrete-time stability: there exists $X\succ 0$ such that
$\left[\begin{array}[]{cc}-X&AX\\\ XA^{T}&-X\end{array}\right]\prec 0$ which
is equivalent, using the Schur complement, to $X-AXA^{T}\succ 0$.

To characterize matrices with eigenvalues in any LMI region, we rephrase the
definition of a DH matrix by relaxing the semidefinite constraint on $R$ and
obtaining the following characterization of $\Omega$-stable matrices.

######  Theorem 2.

Let $\Omega\subseteq C$ be an LMI region defined by (1.1) and let
$B,C\in{\mathbb{R}}^{n,n}$ be the corresponding matrices as given in (1.2).
Consider a matrix $A\in{\mathbb{R}}^{n,n}$. Then $A$ is $\Omega$-stable if and
only if $A=(J-R)Q$ for some $J,R,Q\in{\mathbb{R}}^{n,n}$ such that $J^{T}=-J$,
$R^{T}=R$, $Q\succ 0$, and

$\mathcal{M}_{\Omega}(J,R,Q):=B\otimes Q^{-1}+(C-C^{T})\otimes
J-(C+C^{T})\otimes R\quad\prec\quad 0.$ (2.1)

###### Proof.

First suppose that $A=(J-R)Q$ for some $J,R,Q\in{\mathbb{R}}^{n,n}$ such that
$J^{T}=-J$, $R^{T}=R$, $Q\succ 0$, and $\mathcal{M}_{\Omega}(J,R,Q)\prec 0$.
Let $\lambda\in{\mathbb{C}}$ be an eigenvalue of $A$ and
$v\in{\mathbb{C}}^{n}\setminus\\{0\\}$ be the corresponding eigenvector, that
is,

$Av=(J-R)Qv=\lambda v.$

Thus

$\displaystyle\left(B+\lambda C+\overline{\lambda}C^{T}\right)v^{*}Qv$
$\displaystyle=$ $\displaystyle B\otimes v^{*}Qv+\lambda C\otimes
v^{*}Qv+\overline{\lambda}C^{T}\otimes v^{*}Qv$ $\displaystyle=$
$\displaystyle(I\otimes v)^{*}\left(B\otimes Q^{-1}+(C-C^{T})\otimes
J-(C+C^{T})\otimes R\right)(I\otimes v)$ $\displaystyle=$
$\displaystyle(I\otimes v)^{*}\mathcal{M}_{\Omega}(J,R,Q)(I\otimes v)$
$\displaystyle\prec$ $\displaystyle 0,$

since $\mathcal{M}_{\Omega}(J,R,Q)\prec 0$ and $Q\succ 0$. This implies that
$(B+\lambda C+\overline{\lambda}C^{T})\prec 0$, and hence from (1.1)
$\lambda\in\Omega$. Conversely, let $A$ be $\Omega$-stable. Then by Theorem 1,
there exists $X\succ 0$ such that

$\mathcal{M}_{\Omega}(A,X)=B\otimes X+C\otimes(AX)+C^{T}\otimes(AX)^{T}\prec
0.$ (2.2)

By setting

$Q=X^{-1},\quad R=-\frac{(AX)+(AX)^{T}}{2},\quad\text{and}\quad
J=\frac{(AX)-(AX)^{T}}{2},$

we have $J^{T}=-J$, $R^{T}=R$, and $Q\succ 0$ so that $A=(J-R)Q$. Also, in
view of (2.2), we have that

$\displaystyle\mathcal{M}_{\Omega}(J,R,Q)$ $\displaystyle=$ $\displaystyle
B\otimes Q^{-1}+(C-C^{T})\otimes J-(C+C^{T})\otimes R$ $\displaystyle=$
$\displaystyle B\otimes
X+(C-C^{T})\otimes\frac{(AX)-(AX)^{T}}{2}+(C+C^{T})\otimes\frac{(AX)+(AX)^{T}}{2}$
$\displaystyle=$ $\displaystyle B\otimes X+C\otimes(AX)+C^{T}\otimes(AX)^{T}$
$\displaystyle\prec$ $\displaystyle 0.$

This completes the proof.

If we apply Theorem 2 to Hurwitz and Schur stable matrices, we obtain the
following DH characterizations:

  1. 1)

For Hurwitz stability, plugging $B=0$ and $C=1$ in (2.1) implies that
$A=(J-R)Q$ such that $J^{T}=-J$, $R^{T}=R$, $Q\succ 0$ is Hurwitz stable if
and only if $R\succ 0$, as shown in [9].

  2. 2)

For Schur stability, plugging $B=-\binom{1\;0}{0\;1}$ and
$C=\binom{0\;1}{0\;0}$ in (2.1) implies that $A=(J-R)Q$ such that $J^{T}=-J$,
$R^{T}=R$, $Q\succ 0$ is Schur stable if and only if

$\left[\begin{array}[]{cc}Q^{-1}&-J+R\\\ J+R&Q^{-1}\end{array}\right]\succ 0,$

as shown in [5].

In the next section, we apply Theorem 2 in the same way for other LMI regions.

##  3 Special LMI regions and DH parametrization

We have provided in Theorem 2 the DH parametrization of matrices with
eigenvalues inside a generic LMI region. As a corollary of which, in this
section, we provide LMI constraints on the matrix triplets $(J,R,Q)$ of a DH
matrix for some particular regions. In fact, as soon as an LMI region is
defined via the matrices $B$ and $C$, see Definition 2, we can provide its
corresponding DH characterization in terms of DH matrices. In this section, we
provide the following examples: left and right conic sectors, disks centered
on the real line, vertical left and right halfplanes, ellipsoid centred on the
real line, left and right parabolic regions centred on the real line, left and
right hyperbolas with vertices on the real line, and horizontal strip. By
following [2, Table III], we first define these regions as LMI regions for
$s=2$, and, in Table 3.1, we provide the DH characterization of matrices with
eigenvalues in these regions. Here are the considered LMI regions:

  * •

Left Conic sector: the left conic sector region of parameters
$a,\,\theta\in{\mathbb{R}}$ with $0\leq\theta\leq\pi/2$, denoted by
$\Omega_{C_{L}}(a,\theta)$, is defined as

$\Omega_{C_{L}}(a,\theta):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\
\sin(\theta)(x-a)<\cos(\theta)y<-\sin(\theta)(x-a),~{}x\leq a\right\\}.$

The conic sector $\Omega_{C_{L}}(a,\theta)$ can be characterized in form of
(1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}-a\sin(\theta)&0\\\
0&-a\sin(\theta)\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}\sin(\theta)&\cos(\theta)\\\
-\cos(\theta)&\sin(\theta)\end{array}\right].$

  * •

Right Conic sector: the right conic sector region of parameters
$a,\,\theta\in{\mathbb{R}}$ with $0\leq\theta\leq\pi/2$, denoted by
$\Omega_{C_{R}}(a,\theta)$, is defined as

$\Omega_{C_{R}}(a,\theta):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\
\sin(\theta)(a-x)<\cos(\theta)y<-\sin(\theta)(a-x),~{}x>a\right\\}.$

The conic sector $\Omega_{C_{R}}(a,\theta)$ is an LMI region in form of (1.1)
with matrices

$B=\left[\begin{array}[]{cc}a\sin(\theta)&0\\\
0&a\sin(\theta)\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}-\sin(\theta)&\cos(\theta)\\\
-\cos(\theta)&-\sin(\theta)\end{array}\right].$

  * •

Disks centred on the real line: the disk centred at $(q,0)$ with radius $r>0$,
denoted by $\Omega_{D}(q,r)$, is defined as

$\Omega_{D}(q,r):=\left\\{z\in{\mathbb{C}}\ \big{|}\ |z-q|<r\right\\}.$

The disk region $\Omega_{D}(q,r)$ can be characterized in form of (1.1) of an
LMI region with matrices

$B=\left[\begin{array}[]{cc}-r&q\\\ q&-r\end{array}\right]\quad\text{and}\quad
C=\left[\begin{array}[]{cc}0&0\\\ -1&0\end{array}\right].$

  * •

Vertical strip: the vertical strip region of parameters $h<k$, denoted by
$\Omega_{V}(h,k)$, is defined as

$\Omega_{V}(h,k):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\ h<x<k\right\\}.$

The vertical strip region $\Omega_{V}(h,k)$ can be characterized in form of
(1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}-k&0\\\ 0&h\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right].$

Note that $h$ (resp. $k$) can possibly be equal to $-\infty$ (resp. $+\infty$)
in which case $\Omega_{V}$ is a half space. In particular,
$\Omega_{V}(0,+\infty)$ is the open left half of the complex plane,
corresponding to stable matrices for continuous LTI systems.

More precisely, the vertical halfplane left of $k$, that is,
$\Omega_{V}(-\infty,k):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\ x<k\right\\}$ is
an LMI region with matrices $B=\left[\begin{array}[]{cc}-k&0\\\
0&-1\end{array}\right]$ and $C=\frac{1}{2}\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]$.

Similarly, the vertical halfplane right of $h$, that is,
$\Omega_{V}(h,\infty):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\ h<x\right\\}$ is
an LMI region with matrices $B=\left[\begin{array}[]{cc}h&0\\\
0&-1\end{array}\right]$ and $C=\frac{1}{2}\left[\begin{array}[]{cc}-1&0\\\
0&0\end{array}\right]$.

  * •

Ellipsoid: the ellipse centred at $(q_{e},0)$ with horizontal radius $a_{e}$
and vertical radius $b_{e}$, denoted by $\Omega_{E}(q_{e},a_{e},b_{e})$, is
defined as

$\Omega_{E}(q_{e},a_{e},b_{e}):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\
\frac{(x-q_{e})^{2}}{a_{e}^{2}}+\frac{y^{2}}{b_{e}^{2}}<1\right\\}.$

The ellipsoid $\Omega_{E}(q_{e},a_{e},b_{e})$ can be characterize in form of
(1.1) of an LMI region with matrices
$B=\left[\begin{array}[]{cc}-2a_{h}&-2q_{e}\\\
-2q_{e}&-2a_{e}\end{array}\right]$ and
$C=\left[\begin{array}[]{cc}0&(1+\frac{a_{e}}{b_{e}})\\\
(1-\frac{a_{e}}{b_{e}})&0\end{array}\right]$.

  * •

Left parabolic region: the parabolic region symmetric to $x$-axis with centred
at $(q_{p},0)$, and curvature $c_{p}>0$, denoted by
$\Omega_{P_{L}}(q_{p},c_{p})$, defined as

$\Omega_{P_{L}}(q_{p},c_{p}):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\
y^{2}<-\frac{2}{c_{p}}(x-q_{p})\right\\}.$

The parabolic region $\Omega_{P_{L}}(q_{p},c_{p})$ can be characterized in
form of (1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}-1&0\\\
0&-q_{p}\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}0&\sqrt{\frac{c_{p}}{2}}\\\
-\sqrt{\frac{c_{p}}{2}}&1\end{array}\right].$

  * •

Right parabolic region: the parabolic region symmetric to $x$-axis with
centred at $(q_{p},0)$, and curvature $c_{p}>0$, denoted by
$\Omega_{P_{R}}(q_{p},c_{p})$, defined as

$\Omega_{P_{R}}(q_{p},c_{p}):=\left\\{x+iy\in{\mathbb{C}}\ \big{|}\
y^{2}<\frac{2}{c_{p}}(x-q_{p})\right\\}.$

The parabolic region $\Omega_{P_{R}}(q_{p},c_{p})$ can be characterized in
form of (1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}-1&0\\\
0&q_{p}\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}0&\sqrt{\frac{c_{p}}{2}}\\\
-\sqrt{\frac{c_{p}}{2}}&-1\end{array}\right].$

  * •

Left hyperbolic region: the left hyperbolic region with the semi-major axis
$a_{h}$, $a_{h}>0$ and semi-minor axis $b_{h}$, $b_{h}>0$ and two vertices at
$(a_{h},0)$ and $(-a_{h},0)$, is denoted by $\Omega_{Hyp,L}(a_{h},b_{h})$,
defined as

$\Omega_{Hyp,L}(a_{h},b_{h}):=\left\\{x+iy\in{\mathbb{C}}\ :\
x<0,~{}\frac{x^{2}}{a_{h}^{2}}-\frac{y^{2}}{b_{h}^{2}}-1>0\right\\}.$

The hyperbolic region $\Omega_{Hyp,L}(a_{h},b_{h})$ can be characterized in
form of (1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]\quad\text{and}\quad
C=\left[\begin{array}[]{cc}\frac{1}{2a_{h}}&\frac{1}{2b_{h}}\\\
-\frac{1}{2b_{h}}&\frac{1}{2a_{h}}\end{array}\right].$

  * •

Right hyperbolic region: the right hyperbolic region with the semi-major axis
$a_{h}$ $a_{h}>0$ and semi-minor axis $b_{h}$, $b_{h}>0$ and two vertices at
$(a_{h},0)$ and $(-a_{h},0)$ is denoted by $\Omega_{Hyp,R}(a_{h},b_{h})$,
defined as

$\Omega_{Hyp,R}(a_{h},b_{h}):=\left\\{x+iy\in{\mathbb{C}}\ :\
x>0,~{}\frac{x^{2}}{a_{h}^{2}}-\frac{y^{2}}{b_{h}^{2}}-1>0\right\\}.$

The hyperbolic region $\Omega_{Hyp,R}(a_{h},b_{h})$ can be characterized in
form of (1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]\quad\text{and}\quad
C=\left[\begin{array}[]{cc}-\frac{1}{2a_{h}}&\frac{1}{2b_{h}}\\\
-\frac{1}{2b_{h}}&-\frac{1}{2a_{h}}\end{array}\right].$

  * •

Horizontal strip: the horizontal strip symmetric to the real axis with hight
$w$, denoted by $\Omega_{H}(w)$, defined as

$\Omega_{H}(w):=\left\\{x+iy\in{\mathbb{C}}\ :|y|<w\right\\}.$

The horizontal strip $\Omega_{Hyp,L}(a_{h},b_{h})$ can be characterized in
form of (1.1) of an LMI region with matrices

$B=\left[\begin{array}[]{cc}-w&0\\\ 0&-w\end{array}\right]\quad\text{and}\quad
C=\frac{1}{2}\left[\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right].$

We note that DH characterization for the sets of $\Omega$-stable matrices was
obtained in [5], where
$\Omega\in\\{\Omega_{C_{L}}(a,\theta),\Omega_{D}(q,r),\Omega_{V}(h,k)\\}$.
Because of Theorem 2, we can now extend these results for other LMI regions as
well, see Table 3.1.

LMI Region | Constraints on $J^{T}=-J$, $R^{T}=R$, and $Q\succ 0$
---|---
Left Conic Sector: $\Omega_{C_{L}}(a,\theta)$ | $\left[\begin{array}[]{cc}\sin(\theta)(aQ^{-1}+R)&-\cos(\theta)J\\\ \cos(\theta)J&\sin(\theta)(aQ^{-1}+R)\end{array}\right]\succ 0$
Right Conic Sector: $\Omega_{C_{R}}(a,\theta)$ | $\left[\begin{array}[]{cc}-\sin(\theta)(aQ^{-1}+R)&-\cos(\theta)J\\\ \cos(\theta)J&-\sin(\theta)(aQ^{-1}+R)\end{array}\right]\succ 0$
Disk: $\Omega_{D}(q,r)$ | $\left[\begin{array}[]{cc}rQ^{-1}&qQ^{-1}-J+R\\\ qQ^{-1}+J+R&rQ^{-1}\end{array}\right]\succ 0$
Vertical strip: $\Omega_{V}(h,k)$ | $\left[\begin{array}[]{cc}kQ^{-1}+R&0\\\ 0&-hQ^{-1}-R\end{array}\right]\succ 0$
Left halfplane: $\Omega_{V}(-\infty,k)$ | $kQ^{-1}+R\succ 0$
Right halfplane: $\Omega_{V}(h,\infty)$ | $-hQ^{-1}-R\succ 0$
Ellipsoid: $\Omega_{E}(q_{e},a_{e},b_{e})$ | $\left[\begin{array}[]{cc}a_{e}Q^{-1}&q_{e}Q^{-1}-\frac{a_{e}}{b_{e}}J+R\\\ q_{e}Q^{-1}+\frac{a_{e}}{b_{e}}J+R&a_{e}Q^{-1}\end{array}\right]\succ 0$
Left parabolic region: $\Omega_{P_{L}}(q_{p},c_{p})$ | $\left[\begin{array}[]{cc}Q^{-1}&-\sqrt{\frac{c_{p}}{2}}J\\\ \sqrt{\frac{c_{p}}{2}}J&q_{p}Q^{-1}+R\end{array}\right]\succ 0$
Right parabolic region: $\Omega_{P_{R}}(q_{p},c_{p})$ | $\left[\begin{array}[]{cc}Q^{-1}&-\sqrt{\frac{c_{p}}{2}}J\\\ \sqrt{\frac{c_{p}}{2}}J&-q_{p}Q^{-1}-R\end{array}\right]\succ 0$
Left hyperbola: $\Omega_{Hyp,L}(a_{h},b_{h})$ | $\left[\begin{array}[]{cc}\frac{R}{a_{h}}&-Q^{-1}-\frac{J}{b_{h}}\\\ -Q^{-1}+\frac{J}{b_{h}}&\frac{R}{a_{h}}\end{array}\right]\succ 0$
Right hyperbola: $\Omega_{Hyp,R}(a_{h},b_{h})$ | $\left[\begin{array}[]{cc}-\frac{R}{a_{h}}&-Q^{-1}-\frac{J}{b_{h}}\\\ -Q^{-1}+\frac{J}{b_{h}}&-\frac{R}{a_{h}}\end{array}\right]\succ 0$
Horizontal strip: $\Omega_{H}(w)$ | $\left[\begin{array}[]{cc}wQ^{-1}&-J\\\ J&wQ^{-1}\end{array}\right]\succ 0$
Table 3.1: DH parametrization of the LMI regions listed in Section 3.

We conclude this section with the following remark.

######  Remark 1 (Non-uniqueness of the DH characterization).

A remark similar to [9, Remark 6] also holds for the non-uniqueness of a DH
characterization for an $\Omega$-stable matrix. The non-uniqueness of the DH
characterization for $\Omega$-stable matrices can be partly characterized by
the non-uniqueness of the characteristic function of an LMI region. For
example, a different DH characterization involving two $n\times n$ LMIs is
obtained for the horizontal strip in Appendix A.

##  4 Application of the DH characterization of LMI regions: Nearest
$\Omega$-stable matrix problem

The DH characterization of matrices with eigenvalues in some LMI regions can
be used to solve various problems in systems and control. For example, it was
used to solve the nearest stable matrix problem (1.3) for Hurwitz stability in
[9], for Schur stability in [7, 17], and for some LMI regions (namely,
vertical strips, disks, and conic sectors) in [5]. It was also used for
static-state feedback (SSF) and static-output feedback (SOF) stabilization in
[11]. Our new general characterizations (see Theorem 2) can be used to
generalize these algorithms to any LMI region $\Omega$. In this section, we
illustrate this on the nearest $\Omega$-stable matrix problem.

###  4.1 Reformulation

Let $A\in{\mathbb{R}}^{n,n}$, $\Omega\subseteq{\mathbb{C}}$ be an LMI region,
and consider the nearest $\Omega$-stable matrix problem (1.3). Then in view of
Theorem 2, by setting $P=Q^{-1}$, we can parameterize the set $S_{\Omega}$ of
$\Omega$-stable matrices in terms of matrix triplets $(J,R,P^{-1})$ as follows

$S_{\Omega}=\left\\{(J-R)P^{-1}\in{\mathbb{R}}^{n,n}:~{}J^{T}=-J,R^{T}=R,P\succ
0,\mathcal{M}_{\Omega}(J,R,P^{-1})\prec 0\right\\}.$

This leads to the following equivalent formulation of the nearest
$\Omega$-stable matrix problem (1.3):

$\inf_{(J,R,P)\in\mathbb{D}H_{\Omega}}{\|A-(J-R)P^{-1}\|}_{F}^{2},$ (4.1)

where

$\mathbb{D}H_{\Omega}=\left\\{(J,R,P)\in({\mathbb{R}}^{n,n})^{3}:~{}J^{T}=-J,\,R^{T}=R,\,P\succ
0,\,M_{\Omega}(J,R,P^{-1})\prec 0\right\\}.$

Note that the feasible set in (4.1) is convex involving only convex LMI
constraints, but the objective function is nonconvex.

###  4.2 Algorithm

To solve the nonconvex optimization problem (4.1), we follow the same strategy
as in [5]. We have adapted the MATLAB code of [5] to handle all the LMI
regions summarized in Table 3.1. In a few words, the strategy in [5] uses a
simple gradient descent method, that is, it uses the update

$X\leftarrow\mathcal{P}\big{(}X-\gamma\nabla F(X)\big{)},$

where $X=(J,R,Q)$ is the variable, $F(X)={\|A-(J-R)P^{-1}\|}_{F}^{2}$,
$\mathcal{P}$ is the projection onto the feasible set $DH_{\Omega}$, which can
be computed via semidefinite programming (SDP), $\gamma$ is a step size
computed using backtracking line search ($\gamma$ is reduced until $F$
decreases), and $\nabla F(X)$ is the gradient of $F$ at $X$. After each such
update, the variables $(J,R)$, with $P$ being fixed, are updated using SDP,
which we solve using CVX [6, 12]. An effective and simple initialization for
the factors is to choose $P=I_{n}$ and set $(J,R)$ as the optimal solution of
(4.1) for $P$ fixed.

###  4.3 Numerical illustrations

As far as we know, only our approach can tackle the nearest $\Omega$-stable
matrix problem with LMI regions in full generality. Although Noferini and
Poloni [18] proposed a general framework for this problem using the Schur
decomposition of $A\approx UTU^{\top}$, they need to project $2\times 2$
diagonal blocks of $T$ onto the corresponding LMI region. It is unclear how to
perform this projection in full generality (in their paper, they work out the
details for the cases of Hurwitz and Schur stability).

To illustrate the use of our algorithm, we consider a similar setting as in
[5]: we generate $(J,R,P)$ randomly (namely using randn(n) in MATLAB), then
project it onto the desired LMI region so that $A=(J-R)P^{-1}$ is
$\Omega$-stable. Then we perturb $A$ using Gaussian noise: $A_{p}=A+N$ where
$N(i,j)\sim N(0,\sigma)$ where $\sigma=\frac{\epsilon{\|A\|}_{F}}{n}$ where
$\epsilon>0$ is a parameter so that the expected value of $\|N\|_{F}^{2}$ is
equal to $\epsilon^{2}{\|A\|}_{F}^{2}$ ($\epsilon$ is a measure of the noise-
to-signal ratio).

#### Example 1

Let us consider $\Omega$ as the intersection of

  * •

a vertical strip between -5 and 5,

  * •

a horizontal strip between -3 and 3,

  * •

a left parabolic region centered at $(6,0)$ and curvature 1, and

  * •

a right parabolic region centered at $(-6,0)$ and curvature 1.

We randomly generate a 10-by-10 matrix following the construction described
above with $\epsilon=1$ (the noise level is high to have several eigenvalues
of $A$ outside $\Omega$), and run our algorithm with the identity
initialization to approximate $A$ with an $\Omega$-stable matrix
$\tilde{A}=(J-R)Q$; see Figure 4.1 for the illustration. The approximation is
computed in approximately 1 minute on a standard laptop, and the relative
error is given by $\frac{\|A-\tilde{A}\|}{\|A\|_{F}}=18.1\%$.

Figure 4.1: Eigenvalues of $A$, and of its $\Omega$-stable approximation
$\tilde{A}=(J-R)Q$, where $\Omega$ is the intersection of a vertical strip, a
horizontal strip, and a left and a right parabolic region.

#### Example 2

Let us consider $\Omega$ as the intersection of

  * •

an ellipsoid centered at $(-1,0)$ with horizontal radius 3 and vertical radius
of 2,

  * •

a left hyperbolic region centered with semi-majos axis $a_{h}=b_{h}=0.5$, and

  * •

a right conic sector centered at $(-3.5,0)$ with angle $\frac{3}{8}\pi$.

We randomly generate a 10-by-10 matrix following the construction described
above with $\epsilon=1$, and run our algorithm with the identity
initialization to approximate $A$ with an $\Omega$-stable matrix
$\tilde{A}=(J-R)Q$; see Figure 4.2 for the illustration. The approximation is
computed in approximately 1 minute on a standard laptop, and the relative
error is given by $\frac{\|A-\tilde{A}\|}{\|A\|_{F}}=24.1\%$.

Figure 4.2: Eigenvalues of $A$, and of its $\Omega$-stable approximation
$\tilde{A}=(J-R)Q$, where $\Omega$ is the intersection of an ellipse, a right
conic sector and a left hyperbolic region.

######  Remark 2 (Code).

The MATLAB code is available from
https://sites.google.com/site/nicolasgillis/code. With the code, you can run
the examples presented above (we used the random seed 2017 to make the two
experiments above reproducible), and run any other example.

##  5 Extended LMI regions for complex matrices

In this section, we extend the concept of LMI regions to regions that are not
symmetric with respect to the real axis but can be represented by LMIs
involving complex matrices. We call such regions _extended LMI regions_ and
define them as follows.

######  Definition 4 (Extended LMI Regions).

A subset $\Omega\subseteq{\mathbb{C}}$ is called an extended LMI region if
there exists a Hermitian matrix $B\in{\mathbb{H}}^{m,m}$ and
$C\in{\mathbb{C}}^{m,m}$ such that

$\Omega_{{\mathbb{C}}}(B,C)=\left\\{z\in{\mathbb{C}}:~{}f_{\Omega}(z)\prec
0\right\\},$ (5.1)

where

$f_{\Omega}\colon\mathbb{C}\mapsto{\mathbb{H}}^{m,m}\;\text{ given by
}\;z\mapsto f_{\Omega}(z):=B+zC+\overline{z}C^{*}.$ (5.2)

The extended LMI set $\Omega_{{\mathbb{C}}}(B,C)$ is not in general symmetric
with respect to the real line. Further, the rotation, translation and scaling
of an LMI region $\Omega$ by nonzero scalar $\alpha\in C$ is in general not an
LMI region but an extended LMI region. For example, let
$\Omega=\\{z\in{\mathbb{C}}:~{}B+zC+\overline{z}C^{T}\prec 0\\}$ be an LMI
region as defined in (1.1) and $\alpha\in{\mathbb{C}}\setminus\\{0\\}$. If we
denote the scaling and rotation of $\Omega$ by $\alpha$ with
$\Omega_{\theta}:=\alpha\Omega=\\{\alpha z:~{}z\in\Omega\\}$ and the
translation of $\Omega$ by $\alpha$ with
$\Omega_{T}:=\Omega+\alpha=\\{z+\alpha\in{\mathbb{C}}:z\in\Omega\\}$, then it
is easy to verify that
$\Omega_{\theta}=\Omega_{\mathbb{C}}(B,\frac{1}{\alpha}C)$ and
$\Omega_{T}=\Omega_{\mathbb{C}}(B-\alpha C-\overline{\alpha}C^{*},C)$.

Next, we explain how the main results (Theorems 1 and 2) of Section 2 can be
generalized to extended LMI regions by using complex matrices $B$ and $C$ in
(1.2). We first define stability for a complex matrix
$A\in{\mathbb{C}}^{n,n}$.

######  Definition 5.

Let $\Omega(B,C)\subseteq{\mathbb{C}}$ be an extended LMI region (5.1). A
matrix $A\in{\mathbb{C}}^{n,n}$ is said to be complex $\Omega$-stable if all
eigenvalues of $A$ lie inside $\Omega(B,C)$.

The following result is a generalization of Theorem 1 for complex
$\Omega$-stable matrices, the proof of which is similar to the proof of
Theorem 1. We have included the proof for future reference.

######  Theorem 3.

Let $A\in{\mathbb{C}}^{n,n}$ and $\Omega(B,C)$ be an extended LMI region in
the form (5.1). Then $A$ is complex $\Omega$-stable if and only if there
exists a Hermitian matrix $X\in{\mathbb{C}}^{n,n}$ such that

$\displaystyle X\succ 0\quad\text{and}\quad\mathcal{M}_{\Omega}(A,X)=B\otimes
X+C\otimes(AX)+C^{*}\otimes(AX)^{*}\prec 0.$ (5.3)

###### Proof.

($\Leftarrow$) First suppose that there exists $X\in{\mathbb{C}}^{n,n}$
satisfying (5.3). Let $\lambda\in\Lambda(A)$ and
$x\in{\mathbb{C}}^{n}\setminus\\{0\\}$ such that $x^{*}A=\lambda x^{*}$. Then

$\displaystyle(I_{n}\otimes x^{*})\mathcal{M}_{\Omega}(A,X)(I_{n}\otimes
x)=f_{\Omega}(\lambda)x^{*}Xx.$ (5.4)

This implies that $f_{\Omega}(\lambda)\prec 0$, since
$\mathcal{M}_{\Omega}(A,X)\prec 0$ and $X$ is Hermitian positive definite.
Thus $\lambda\in\Omega(B,C)$ and hence $A$ is $\Omega$-stable.

($\Rightarrow$) Conversely, let $A$ be $\Omega$-stable. First assume that $A$
is a diagonal matrix (denote it by $\Delta$) with diagonal entries
$\lambda_{1},\ldots,\lambda_{n}\in\Omega(B,C)$. Then

$\displaystyle\mathcal{M}_{\Omega}(\Delta,I_{n})=\rm{diag}\left(f_{\Omega}(\lambda_{1}),\ldots,f_{\Omega}(\lambda_{n})\right)\prec
0,$ (5.5)

since $A$ is $\Omega$-stable implies that $f_{\Omega}(\lambda_{i})\prec 0$ for
all $i=1,\ldots,n$. Now suppose $A\in{\mathbb{C}}^{n,n}$ and let $\Delta$ be
the diagonal matrix containing the eigenvalues of $A$ (counting algebraic
multiplicities). Let $S$ be an invertible matrix such that $J=S^{-1}AS$ is the
Jordan matrix of $A$. Then there exists a squence of invertible matrices
${T_{k}}$ such that $\lim_{k\rightarrow\infty}T_{k}^{-1}JT_{k}=\Delta$. Since
$\mathcal{M}_{\Omega}(Y,I_{n})$ is a continuous function of $Y$, in view of
(5.5), we have that

$\lim_{k\rightarrow\infty}\mathcal{M}_{\Omega}(T_{k}^{-1}JT_{k},I_{n})=\mathcal{M}_{\Omega}(\Delta,I_{n})\prec
0.$

This implies that there exists a positive integer $k$ such that $T:=T_{k}$
satisfies $\mathcal{M}_{\Omega}(T^{-1}JT,I_{n})\prec 0$, or, equivalently
$\mathcal{M}_{\Omega}(T^{-1}S^{-1}AST,I_{n})\prec 0$. Thus, we have

$\displaystyle(I_{n}\otimes
ST)\mathcal{M}_{\Omega}(T^{-1}S^{-1}AST,I_{n})(I_{n}\otimes{(ST)}^{*})=\mathcal{M}_{\Omega}(A,X)\prec
0,$

where $X:=(ST)(ST)^{*}$ is a positive definite matrix, since $S$ and $T$ both
are invertible. This shows the existence of $X\succ 0$ such that
$M_{\Omega}(A,X)\prec 0$, and hence the proof.

The following result is a generalization of Theorem 2 for the complex matrices
that characterizes the set of all complex matrices with eigenvalues in an
extended LMI region in terms of complex matrix triplets
$(J,R,Q)\in({\mathbb{C}}^{n,n})^{3}$ with Hermitian and definite structures.

######  Theorem 4.

Let $\Omega(B,C)\subseteq{\mathbb{C}}$ be an extended LMI region defined by
(5.1) and let $B,C\in{\mathbb{C}}^{n,n}$ be the corresponding matrices defined
in (5.2). Consider a matrix $A\in{\mathbb{C}}^{n,n}$. Then $A$ is complex
$\Omega$-stable if and only if $A=(J-R)Q$ for some
$J,R,Q\in{\mathbb{C}}^{n,n}$ such that $J^{*}=-J$, $R^{*}=R$, $Q\succ 0$, and
$\mathcal{M}_{\Omega}(J,R,Q)\prec 0$, where

$\mathcal{M}_{\Omega}(J,R,Q):=B\otimes Q^{-1}+(C-C^{*})\otimes
J-(C+C^{*})\otimes R.$

###### Proof.

The proof follows on the lines of the proof of Theorem 2 by using Theorem 3 in
place of Theorem 1.

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##  Appendix A DH characterization of the horizontal strip with two $n\times
n$ LMIs

The following theorem provides another DH characterization for the horizontal
strip compared to the one provided in Table 3.1.

######  Theorem 5.

Let $A=(J-R)Q$ be a DH matrix with $J,R,Q\in{\mathbb{R}}^{n,n}$ such that
$J^{T}=-J$, $R^{T}=R$, and $Q\succ 0$. If

$-wQ^{-1}\prec iJ\prec wQ^{-1},$ (A.1)

then $A$ is $\Omega_{H}(w)$-stable.

###### Proof.

Let $\lambda=\lambda_{1}+i\lambda_{2},\lambda_{1},\lambda_{2}\in{\mathbb{R}}$
be an eigenvalue of $A$ and let $x\in{\mathbb{C}}^{n}\setminus\\{0\\}$ such
that $(J-R)Qx=\lambda x$. This implies that
$x^{*}QJQx-x^{*}QRQx=(\lambda_{1}+i\lambda_{2})x^{*}Qx$ and thus $ix^{*}QJQx-
ix^{*}QRQx=(i\lambda_{1}-\lambda_{2})x^{*}Qx$. By comparing the real and
imaginary parts, we have

$\lambda_{2}=\frac{x^{*}Q(-iJ)Qx}{x^{*}Qx},$ (A.2)

since $Q\succ 0$. As $J$ satisfies (A.1), we have
$-wx^{*}Qx<x^{*}Q(iJ)Qx<wx^{*}Qx$ and thus from (A.2) we have that

$-w<\frac{x^{*}Q(-iJ)Qx}{x^{*}Qx}<w\quad\implies\quad-w<\lambda_{2}<w.$ (A.3)

This implies $\lambda\in\Omega_{H}(w)$.

The converse of Theorem 5 is true when $A$ is semisimple, that is, algebraic
mulitiplicity is equal to the geometric multiplicity for every eigenvalue
$\lambda$ of $A$. More precisely, we have the following result.

######  Theorem 6.

Let $A\in{\mathbb{R}}^{n,n}$ be semisimple $\Omega_{H}(w)$-stable matrix. Then
$A=(J-R)Q$ for some $J,R,Q\in{\mathbb{R}}^{n,n}$ such that $J^{T}=-J$,
$R^{T}=R$, $Q\succ 0$, and

$-wQ^{-1}\prec iJ\prec wQ^{-1}.$ (A.4)

###### Proof.

In view of Theorem 2, A=$(J-R)Q$ for some $J,R,Q\in{\mathbb{R}}^{n,n}$ such
that $J^{T}=-J$, $R^{T}=R$, and $Q\succ 0$, since $A$ is
$\Omega_{H}(w)$-stable. Next, we show that $J$ satisfies (A.4), or
equivalently, $-wQ\prec Q(iJ)Q\prec wQ$, since $Q\succ 0$. We will prove this
by showing that

$x^{*}(Q(iJ)Q-wQ)x<0\quad\text{and}\quad x^{*}(Q(iJ)Q+wQ)x>0\quad\text{for
all}~{}x\in{\mathbb{C}}^{n}\setminus\\{0\\}.$ (A.5)

As $A$ is semisimple, there exists an orthonormal basis of ${\mathbb{C}}^{n}$
consisting of eigenvectors of $A$, say $y_{1},\ldots,y_{n}$. Thus, to prove
(A.5) for every $x$, it is sufficient to show it for vectors $y_{j}$,
$j=1,\ldots,n$. Since $y_{j}$ is an eigenvector of $A$, we have
$Ay_{j}=\lambda_{j}y_{j}$ for some eigenvalue $\lambda_{j}$ of $A$. This
implies that $(J-R)Qy_{j}=\lambda_{j}y_{j}$ and thus
$y_{j}^{*}Q(J-R)Qy_{j}=\lambda_{j}y_{j}^{*}Qy_{j}$. This implies that
$\mathop{\mathrm{Im}}(\lambda_{j})=\frac{y_{j}^{*}Q(-iJ)Qy_{j}}{y_{j}^{*}Qy_{j}}$
and thus we have $-w<\frac{y_{j}^{*}Q(-iJ)Qy_{j}}{y_{j}^{*}Qy_{j}}<w$, since
$A$ is $\Omega_{H}(w)$-stable.