# Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati
equations

Ta-Chu Kao@ Computational and Biological Learning Lab, Department. of
Engineering, University of Cambridge, Cambridge, UK  Guillaume Hennequin
Computational and Biological Learning Lab, Department. of Engineering,
University of Cambridge, Cambridge, UK

<EMAIL_ADDRESS>
)

###### Abstract

Sylvester, Lyapunov, and algebraic Riccati equations are the bread and butter
of control theorists. They are used to compute infinite-horizon Gramians,
solve optimal control problems in continuous or discrete time, and design
observers. While popular numerical computing frameworks (e.g.,  scipy )
provide efficient solvers for these equations, these solvers are still largely
missing from most automatic differentiation libraries. Here, we derive the
forward and reverse-mode derivatives of the solutions to all three types of
equations, and showcase their application on an inverse control problem.

##  1 Introduction

In recent years, automatic differentiation (AD) has become the backbone of
most modern machine learning applications and an important tool for scientific
enquiry (Baydin et al.,, 2017). The success of AD owes in part to the myriad
of forward and reverse-mode derivatives that have been derived for
matrix/tensor operations including most known linear operators and
factorizations (Giles,, 2008). These are commonly available in popular
automatic differentiation libraries such as PyTorch (Paszke et al.,, 2019),
Tensorflow (Paszke et al.,, 2019), Jax (Bettencourt et al.,, 2019), Diffsharp
(Baydin et al.,, 2017), and Zygote (Innes,, 2018).

In this technical note, we derive the forward and reverse-mode gradients for
solutions to three important types of matrix equations used extensively in
control theory: Sylvester, Lyapunov, and algebraic Riccati equations. Although
efficient solvers are readily available in popular scientific programming
libraries, to the best of our knowledge they remain missing from most
automatic differentiation packages (though CasADi seems to provide Lyapunov
solvers; Gillis,, 2015; Andersson et al.,, 2019). We have added these solvers
to Owl (Wang,, 2017), a numerical library written in OCaml with a full-
featured AD module, and we hope that this technical note will enable rapid
integration in other popular AD packages.

##  2 Preliminaries

We use bold upper-case letters to denote matrices and bold lower-case letters
to denote vectors. We use $\bm{X}^{T}$, $\bm{X}^{-1}$ and $\text{tr}(\bm{X})$
to denote the transpose, inverse, and trace of a matrix
$\bm{X}\in\mathbb{R}^{n\times n}$. Following Giles, (2008), we use
$\overline{\bm{X}}$ to denote the adjoint of matrix $\bm{X}$ w.r.t. a scalar
output $\ell$, i.e. $\partial\ell/\partial\bm{X}$. We also use $\dot{\bm{X}}$
to denote the tangent of a matrix $\bm{X}$ w.r.t. some scalar input $s$, i.e.
$\partial\bm{X}/\partial s$.

Consider a differentiable solver $\bm{P}=f(\bm{A},\bm{B},\ldots)$. In forward
mode, the input tangents $(\dot{\bm{A}},\dot{\bm{B}},\ldots)$ are known and
used to calculate $\dot{\bm{P}}$. In reverse mode, the solution’s adjoint
$\overline{\bm{P}}$ is known and used to update the input’s adjoints
$(\overline{\bm{A}},\overline{\bm{B}},\ldots)$. This note consists mainly in
deriving the rules by which these tangents and adjoints should be computed for
various types of matrix equation solvers. All our results are summarized in
Tables 1 and 2.

##  3 Sylvester equations

###  3.1 Continuous time Sylvester equation

The continuous time Sylvester equation takes the form

$\bm{A}\bm{P}+\bm{P}\bm{B}+\bm{C}=\bm{0},$ (1)

where $\bm{A}$, $\bm{P}$, $\bm{B}$, and $\bm{C}$ are all square matrices that
have the same dimensions. There is a unique solution $\bm{P}$ that satisfies
Equation 1 for a given $\bm{A}$, $\bm{B}$, and $\bm{C}$ if and only if
$\bm{A}$ and $\bm{B}$ have no common eigenvalue. As we shall see in the
following, the matrix derivatives for the solution to the Sylvester equation
is also unique under the same condition on the spectrum of $\bm{A}$ and
$\bm{B}$.

#### Forward mode

Taking the differential on both sides of Equation 1, we have

$\bm{A}\,\bm{dP}+\bm{dP}\,\bm{B}+\big{(}\bm{dA}\,\bm{P}+\bm{P}\,\bm{dB}+\bm{dC}\big{)}=\bm{0}.$
(2)

The forward mode derivative $\dot{\bm{P}}$ can thus be computed by solving
another continuous time Sylvester equation involving the tangents
$\bm{\dot{A}}$, $\bm{\dot{B}}$, and $\bm{\dot{C}}$:

$\bm{A}\,\bm{\dot{P}}+\bm{\dot{P}}\,\bm{B}+\big{(}\bm{\dot{A}}\,\bm{P}+\bm{P}\,\bm{\dot{B}}+\bm{\dot{C}}\big{)}=\bm{0}.$
(3)

#### Reverse mode

To derive the adjoints
$\overline{\bm{A}},\overline{\bm{B}},\overline{\bm{C}}$, we consider the
Lagrangian

$\mathcal{L}(\bm{A},\bm{B},\bm{C},\bm{P})=\ell(\bm{P})+\text{tr}\big{(}\bm{S}^{T}(\bm{A}\bm{P}+\bm{P}\bm{B}+\bm{C})\big{)},$
(4)

where $\ell(\bm{P})$ is the scalar output w.r.t. which we wish to compute
adjoints, and $\bm{S}$ is a matrix of Lagrange multipliers used to enforce
Equation 1. Taking partial derivatives on both sides of the Lagrangian and
setting them to zero, we obtain:

$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial\ell(\bm{P})}{\partial\bm{P}}+\bm{A}^{T}\bm{S}+\bm{S}\bm{B}^{T},$
(5) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}\bm{P}+\bm{P}\bm{B}+\bm{C},$
(6) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{S}\bm{P}^{T},$
(7) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{B}}=\bm{P}^{T}\bm{S},$
(8) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{C}}=\bm{S}.$ (9)

Note that Equation 6 does indeed enforce that $\bm{P}$ be the solution of
Equation 1. Identifying
$\displaystyle\overline{\bm{P}}=\partial\ell/\partial\bm{P}$, we find that the
Lagrange multiplier $\bm{S}$ satisfies another Sylvester equation

$\bm{A}^{T}\bm{S}+\bm{S}\bm{B}^{T}+\overline{\bm{P}}=\bm{0}.$ (10)

After computing $\bm{S}$, and noting that when $\bm{P}$ solves Equation 6 then
$\mathcal{L}=\ell$, it is straightforward to compute the rest of the
adjoints111One can formally show that
$\overline{\bm{X}}=\partial\mathcal{L}/\partial\bm{X}$ for
$\bm{X}\in\\{\bm{A},\bm{B},\bm{C}\\}$ by applying the implicit function
theorem (see Recht,, 2016 and references therein).  :

$\overline{\bm{A}}=\bm{S}\bm{P}^{T},\qquad\overline{\bm{B}}=\bm{P}^{T}\bm{S},\qquad\overline{\bm{C}}=\bm{S}.$
(11)

###  3.2 Discrete time Sylvester equation

We follow a similar approach as above to differentiate through the discrete
time Sylvester equation, which takes the form

$\bm{A}\bm{P}\bm{B}-\bm{P}+\bm{C}=\bm{0},$ (12)

where $\bm{A}$, $\bm{P}$, $\bm{B}$, and $\bm{C}$ are all square matrices that
have the same dimensions. There is a unique solution to Equation 12 if and
only if $\sigma_{i}(\bm{A})\sigma_{j}(\bm{B})\neq 1$ for all $i,j$, where
$\\{\sigma_{i}(\bm{X})\\}$ denotes the eigenvalue spectrum of a matrix
$\bm{X}$.

#### Forward mode

Taking the differential on both sides of Equation 12, we have

$\bm{A}\,\bm{dP}\,\bm{B}-\bm{dP}+\big{(}\bm{dA}\,\bm{P}\bm{B}+\bm{A}\bm{P}\,\bm{dB}+\bm{dC}\big{)}=\bm{0}.$
(13)

The tangent of $\bm{P}$ is thus computed by solving another discrete time
Sylvester equation:

$\bm{A}\,\bm{\dot{P}}\,\bm{B}-\bm{\dot{P}}+\big{(}\bm{\dot{A}}\,\bm{P}\bm{B}+\bm{A}\bm{P}\,\bm{\dot{B}}+\bm{\dot{C}}\big{)}=\bm{0}.$
(14)

#### Reverse mode

To derive the adjoints $\overline{\bm{A}}$, $\overline{\bm{B}}$ and
$\overline{\bm{C}}$, we consider the Lagrangian

$\mathcal{L}(\bm{A},\bm{B},\bm{C},\bm{P},\bm{S})=\ell(\bm{P})+\text{tr}\big{(}\bm{S}^{T}(\bm{A}\bm{P}\bm{B}-\bm{P}+\bm{C})\big{)}.$
(15)

Taking partial derivatives on both sides of the Lagrangian and setting them to
zero, we get:

$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial
f(\bm{P})}{\partial\bm{P}}+\bm{A}^{T}\bm{S}\bm{B}^{T}-\bm{S},$ (16)
$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}\bm{P}\bm{B}-\bm{P}+\bm{C},$
(17) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{S}\bm{B}^{T}\bm{P}^{T},$
(18) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{B}}=\bm{P}^{T}\bm{A}^{T}\bm{S},$
(19) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{C}}=\bm{S}.$ (20)

Identifying $\overline{\bm{P}}=\partial f(\bm{P})/\partial\bm{P}$, we find
that the Lagrange multiplier $\bm{S}$ satisfies another discrete time
Sylvester equation

$\bm{A}^{T}\bm{S}\bm{B}^{T}-\bm{S}+\overline{\bm{P}}=\bm{0}.$ (21)

The adjoints are given by

$\overline{\bm{A}}=\bm{S}\bm{B}^{T}\bm{P}^{T},\quad\overline{\bm{B}}=\bm{P}^{T}\bm{A}^{T}\bm{S},\quad\overline{\bm{C}}=\bm{S}.$
(22)

##  4 Lyapunov equations

###  4.1 Continuous time Lyapunov equation

The continuous time Lyapunov equation is given by

$\bm{A}\bm{P}+\bm{P}\bm{A}^{T}+\bm{Q}=\bm{0},$ (23)

where $\bm{A}$, $\bm{P}$, and $\bm{Q}$ are square matrices that have the same
dimensions. Note that the Lyapunov equation is a special case of the Sylvester
equation in Equation 1, with $\bm{B}=\bm{A}^{T}$ and $\bm{C}=\bm{Q}$. In
control theory, we are most commonly concerned with solutions to the Lyapunov
equation when $\bm{Q}$ is a symmetric matrix, which also results in a
symmetric $\bm{P}$. Below, we derive $\overline{\bm{P}}$ and $\dot{\bm{P}}$ in
the general case, and specialize to the case where $\bm{Q}$ is symmetric
afterwards.

#### Forward mode

Taking the differential on both sides of Equation 23, we get

$\displaystyle\bm{A}\,\bm{dP}+\bm{dP}\,\bm{A}^{T}+\big{(}\bm{dA}\,\bm{P}+\bm{P}\,\bm{dA}^{T}+\bm{dQ}\big{)}=\bm{0}.$
(24)

Thus, $\bm{\dot{P}}$ satisfies the continuous time Lyapunov equation:

$\displaystyle\bm{A}\bm{\dot{P}}+\bm{\dot{P}}\bm{A}^{T}+\big{(}\bm{\dot{A}}\bm{P}+\bm{P}\bm{\dot{A}}^{T}+\bm{\dot{Q}}\big{)}=\bm{0}.$
(25)

#### Reverse mode

In reverse-mode, we consider the Lagrangian

$\mathcal{L}(\bm{A},\bm{Q},\bm{P},\bm{S})=\ell(\bm{P})+\text{tr}\big{(}\bm{S}^{T}(\bm{A}\bm{P}+\bm{P}\bm{A}^{T}+\bm{Q})\big{)},$
(26)

where $\bm{S}$ is the Lagrange multiplier and $\ell(\bm{P})$ is some loss
downstream that is a function of $\bm{P}$. Taking partial derivatives and
setting them to zero, we get the following equations

$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial\ell(\bm{P})}{\partial\bm{P}}+\bm{A}^{T}\bm{S}+\bm{S}\bm{A}$
(27) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}\bm{P}+\bm{P}\bm{A}^{T}+\bm{Q}$
(28) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{Q}}=\bm{S}$ (29)
$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{S}\bm{P}^{T}+\bm{S}^{T}\bm{P}.$
(30)

In reverse-mode, $\partial\ell(\bm{P})/\partial\bm{P}=\overline{\bm{P}}$,
which is known in the reverse pass. We can thus compute $\bm{S}$, by solving
the Lyapunov equation

$\overline{\bm{P}}+\bm{A}^{T}\bm{S}+\bm{S}\bm{A}=\bm{0}.$ (31)

This then gives:

$\overline{\bm{A}}=\bm{S}\bm{P}^{T}+\bm{S}^{T}\bm{P}\quad\text{ and
}\quad\overline{\bm{Q}}=\bm{S}$ (32)

using the equations above. If $\bm{Q}$ is a symmetric matrix, as is the case
in most control applications, then $\bm{P}$ and $\bm{S}$ are also symmetric.
In addition, if $\bm{Q}$ is positive semi-definite and all eigenvalues of $A$
have negative real parts, then $\bm{P}$ and $\bm{S}$ are positive semi-
definite.

###  4.2 Discrete time Lyapunov equation

The discrete time Lyapunov equation is given by

$\bm{A}\bm{P}\bm{A}^{T}-\bm{P}+\bm{Q}=\bm{0},$ (33)

#### Forward mode

Taking the differential on both sides of Equation 33, we get another discrete
time Lyapunov equation:

$\bm{A}\,\bm{dP}\,\bm{A}^{T}-\bm{dP}+\big{(}\bm{dA}\,\bm{P}\bm{A}^{T}+\bm{A}\bm{P}\,\bm{dA}^{T}+\bm{dQ}\big{)}=\bm{0}.$
(34)

Thus, we can compute $\bm{\dot{P}}$ by solving:

$\bm{A}\bm{\dot{P}}\bm{A}^{T}-\bm{\dot{P}}+\big{(}\bm{\dot{A}}\bm{P}\bm{A}^{T}+\bm{A}\bm{P}\bm{\dot{A}}^{T}+\bm{\dot{Q}}\big{)}=\bm{0}.$
(35)

#### Reverse mode

In reverse-mode, we consider the following Lagrangian:

$\mathcal{L}(\bm{A},\bm{Q},\bm{P},\bm{S})=\ell(\bm{P})+\text{tr}\big{(}\bm{S}^{T}(\bm{A}\bm{P}\bm{A}^{T}-\bm{P}+\bm{Q})\big{)}.$
(36)

Taking partial derivatives and setting them to zero, we get:

$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial\ell(\bm{P})}{\partial\bm{P}}+\bm{A}^{T}\bm{S}\bm{A}-\bm{S}$
(37) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}\bm{P}\bm{A}^{T}-\bm{P}+\bm{Q}$
(38) $\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{Q}}=\bm{S}$ (39)
$\displaystyle\bm{0}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{S}\bm{A}\bm{P}^{T}+\bm{S}^{T}\bm{A}\bm{P}.$
(40)

In reverse-mode, $\partial\ell(\bm{P})/\partial\bm{P}=\overline{\bm{P}}$. We
can thus compute $\bm{S}$, by solving the discrete time Lyapunov equation

$\overline{\bm{P}}+\bm{A}^{T}\bm{S}\bm{A}-\bm{S}=\bm{0}.$ (41)

This then gives:

$\overline{\bm{A}}=\bm{S}\bm{A}\bm{P}^{T}+\bm{S}^{T}\bm{A}\bm{P}\quad\text{
and }\quad\overline{\bm{Q}}=\bm{S}.$ (42)

If $\bm{Q}$ is symmetric, then so are $\bm{P}$ and $\bm{S}$, leading to
$\overline{\bm{A}}=2\bm{S}\bm{P}\bm{A}$.

##  5 Algebraic Riccati equations

###  5.1 Continuous time algebraic Riccati equation

The continuous time algebraic Riccati equation (CARE) is given by

$\bm{A}^{T}\bm{P}+\bm{P}\bm{A}-\bm{P}\bm{B}\bm{R}^{-1}\bm{B}^{T}\bm{P}+\bm{Q}=\bm{0},$
(43)

where $\bm{Q}$ and $\bm{R}$ are symmetric matrices and $\bm{R}$ is positive-
definite. In most applications, the parameters of this equation will satisfy
the conditions that guarantee the existence of a unique (symmetric) solution
$\bm{P}$. Here, we derive $\bm{\dot{P}}$ and $\bm{\overline{P}}$ assuming that
these conditions are met, so that $\bm{P}=\bm{P}^{T}$.

#### Forward mode

Taking the differential on both sides of Equation 43, we obtain a Lyapunov
equation:

$\bm{dP}\,\bm{\tilde{A}}+\bm{\tilde{A}}^{T}\,\bm{dP}+\left(\bm{dZ}+\bm{dZ}^{T}+\bm{dQ}+\bm{K}^{T}\,\bm{dR}\,\bm{K}\right)=\bm{0},$
(44)

where $\bm{\tilde{A}}=\bm{A}-\bm{B}\bm{K}$,
$\bm{K}=\bm{R}^{-1}\bm{B}^{T}\bm{P}$, and
$\bm{dZ}=\bm{P}(\bm{dA}-\bm{dB\,K})$. We can thus compute $\bm{\dot{\bm{P}}}$
by solving the continuous time Lyapunov equation:

$\bm{\dot{P}}\bm{\tilde{A}}+\bm{\tilde{A}}^{T}\bm{\dot{P}}+\left(\bm{\dot{Z}}+\bm{\dot{Z}}^{T}+\bm{\dot{Q}}+\bm{K}^{T}\bm{\dot{R}}\bm{K}\right)=\bm{0}$
(45)

where $\bm{\dot{Z}}=\bm{P}(\bm{\dot{A}}-\bm{\dot{B}\,K})$.

#### Reverse mode

We consider the Lagrangian

$\mathcal{L}(\bm{A},\bm{B},\bm{Q},\bm{R},\bm{P},\bm{S})=\ell(\bm{P})+\text{tr}\big{(}\bm{S}^{T}(\bm{A}^{T}\bm{P}+\bm{PA}-\bm{PBR}^{-1}\bm{B}^{T}\bm{P}+\bm{Q})\big{)},$
(46)

where $\ell(P)$ is a scalar objective function that depends on $\bm{P}$ and
$\bm{S}$ is a Lagrange multiplier. We are interested in computing the adjoints
$\overline{\bm{A}}$, $\overline{\bm{B}}$, $\overline{\bm{Q}}$, and
$\overline{\bm{R}}$. Taking partial derivatives on both sides of the
Lagrangian and setting them to zero, we get

$\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial\ell(\bm{P})}{\partial\bm{P}}+\bm{\tilde{A}}\bm{S}+\bm{S}\bm{\tilde{A}}^{T}$
(47) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}^{T}\bm{P}+\bm{PA}-\bm{PBR}^{-1}\bm{B}^{T}\bm{P}+\bm{Q}$
(48) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{Q}}=\bm{S}$ (49)
$\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{R}}=\bm{K}\bm{S}\bm{K}^{T}$
(50) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{P}\bm{S}^{T}+\bm{P}\bm{S}$
(51) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{B}}=-\bm{PS}\bm{K}^{T}-\bm{P}\bm{S}^{T}\bm{K}^{T}.$
(52)

The Lagrange multiplier $\bm{S}$ satisfies the Lyapunov equation

$\bm{0}=\frac{1}{2}\left(\overline{\bm{P}}+\overline{\bm{P}}^{T}\right)+\bm{\tilde{A}}\bm{S}+\bm{S}\bm{\tilde{A}}^{T}\quad$
(53)

where we have enforced symmetry of $\overline{\bm{P}}$ because $\bm{P}$ itself
is symmetric – this ensures that $\bm{S}$ is also symmetric. The adjoints are
given by:

$\displaystyle\overline{\bm{A}}$ $\displaystyle=2\bm{P}\bm{S}$ (54)
$\displaystyle\overline{\bm{B}}$ $\displaystyle=-2\bm{PS}\bm{K}$ (55)
$\displaystyle\overline{\bm{Q}}$ $\displaystyle=\bm{S}$ (56)
$\displaystyle\overline{\bm{R}}$ $\displaystyle=\bm{K}\bm{S}\bm{K}^{T}.$ (57)

###  5.2 Discrete time algebraic Riccati equation

The discrete time algebraic Riccati equation (DARE) is given by

$\bm{A}^{T}\bm{P}\bm{A}-\bm{P}-(\bm{A}^{T}\bm{PB})(\bm{R}+\bm{B}^{T}\bm{P}\bm{B})^{-1}(\bm{A}^{T}\bm{P}\bm{B})^{T}+\bm{Q}=\bm{0}.$
(58)

Here, we derive $\bm{\dot{P}}$ and $\bm{\overline{P}}$ assuming that all
conditions are satisfied that ensure the existence of a unique, symmetric
solution $\bm{P}$.

#### Forward mode

Taking the differential on both sides of Equation 43 and simplifying, we get:

$\bm{\tilde{A}}^{T}\bm{dP}\,\bm{\tilde{A}}-\bm{dP}\\\
+\left(\bm{dZ}+\bm{dZ}^{T}+\bm{dQ}+\bm{K}^{T}\,\bm{dR}\,\bm{K}\right)=\bm{0}$
(59)

where $\bm{\tilde{A}}=\bm{A}-\bm{B}\bm{K}$ and
$\bm{K}=(\bm{R}+\bm{B}^{T}\bm{PB})^{-1}\bm{B}^{T}\bm{PA}$, and
$\bm{dZ}=\bm{\tilde{A}}^{T}\bm{P}\left(\bm{dA}-\bm{dB}\,\bm{K}\right)$. Thus,
in forward mode, we have the discrete time Lyapunov equation:

$\bm{\tilde{A}}^{T}\bm{\dot{P}}\bm{\tilde{A}}-\bm{\dot{P}}+\left(\bm{\dot{Z}}+\bm{\dot{Z}}^{T}+\bm{\dot{Q}}+\bm{K}^{T}\bm{\dot{R}}\bm{K}\right)=\bm{0}.$
(60)

where
$\dot{\bm{Z}}=\bm{\tilde{A}}^{T}\bm{P}(\dot{\bm{A}}-\dot{\bm{B}}\bm{K})$.

#### Reverse mode

We consider the Lagrangian

$\begin{split}\mathcal{L}(&\bm{A},\bm{B},\bm{Q},\bm{R},\bm{P},\bm{S})=\\\
&f(\bm{P})+\text{tr}\bigg{(}\bm{S}^{T}(\bm{A}^{T}\bm{P}\bm{A}-\bm{P}-(\bm{A}^{T}\bm{PB})(\bm{R}+\bm{B}^{T}\bm{P}\bm{B})^{-1}(\bm{A}^{T}\bm{PB})^{T}+\bm{Q})\bigg{)}.\end{split}$
(61)

Taking partial derivatives on both sides of the Lagrangian and setting them to
zero, we get

$\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{P}}=\frac{\partial
f(\bm{P})}{\partial\bm{P}}+\bm{\tilde{A}}\bm{S}\bm{\tilde{A}}^{T}-\bm{S}$ (62)
$\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{S}}=\bm{A}^{T}\bm{P}\bm{A}-\bm{P}-(\bm{A}^{T}\bm{PB})(\bm{R}+\bm{B}^{T}\bm{P}\bm{B})^{-1}(\bm{A}^{T}\bm{PB})^{T}+\bm{Q}$
(63) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{Q}}=\bm{S}$ (64)
$\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{R}}=\bm{K}\bm{S}\bm{K}^{T}$
(65) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{A}}=\bm{P}\bm{\tilde{A}}\bm{S}^{T}+\bm{P}\bm{\tilde{A}}\bm{S}$
(66) $\displaystyle 0$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\bm{B}}=-\bm{P\tilde{A}SK}^{T}-\bm{P\tilde{A}}\bm{S}^{T}\bm{K}^{T}$
(67)

The Lagrange multiplier $\bm{S}$ satisfies the Lyapunov equation

$\bm{0}=\frac{1}{2}(\overline{\bm{P}}+\overline{\bm{P}}^{T})+\bm{\tilde{A}}\bm{S}\bm{\tilde{A}}^{T}-\bm{S}$
(68)

where we have enforced the symmetry of $\overline{\bm{P}}$ because $\bm{P}$ is
symmetric. This ensures that $\bm{S}$ is also symmetric and the adjoints are
given by:

$\displaystyle\overline{\bm{A}}$ $\displaystyle=2\bm{P}\bm{\tilde{A}}\bm{S}$
(69) $\displaystyle\overline{\bm{B}}$ $\displaystyle=-2\bm{P\tilde{A}SK}^{T}$
(70) $\displaystyle\overline{\bm{Q}}$ $\displaystyle=\bm{S}$ (71)
$\displaystyle\overline{\bm{R}}$ $\displaystyle=\bm{K}\bm{S}\bm{K}^{T}.$ (72)

##  6 Example application: inverse LQR

To show a concrete application of automatic differentiation through these
algebraic matrix equations, we consider a discrete time linear system that
evolves according to the equations

$\bm{x}_{t+1}=\bm{A}\bm{x}_{t}+\bm{B}\bm{u}_{t},\quad\bm{x}_{0}=\bm{x}_{\text{init}}$
(73)

where $\bm{x}_{t}$ is the state of the system, $\bm{u}_{t}$ is some input that
enters the system through some matrix $\bm{B}$, and $\bm{A}$ is the state
transition matrix. The infinite-horizon, discrete-time LQR problem involves
finding the optimal inputs $\\{\bm{u}_{0},\bm{u}_{1},\ldots\\}$ that minimise
(Anderson and Moore,, 2007)

$J[\bm{u}_{0},\bm{u}_{1},\cdots]=\sum_{t=0}^{\infty}\bm{x}_{t}^{T}\bm{Q}\bm{x}_{t}+\bm{u}_{t}^{T}\bm{R}\bm{u}_{t}$
(74)

for some symmetric positive semi-definite matrices $\bm{Q}$ and $\bm{R}$. It
is well-known that the optimal solution is linear state-feedback:

$\bm{u}_{t}=-\bm{K}\bm{x}_{t},$ (75)

where $\bm{K}=(\bm{R}+\bm{B}^{T}\bm{PB})^{-1}\bm{B}^{T}\bm{PA}$ and $\bm{P}$
satisfies a DARE (c.f. Equation 58):

$\bm{A}^{T}\bm{P}+\bm{P}\bm{A}-\bm{PBR}^{-1}\bm{B}^{T}\bm{P}+\bm{Q}=\bm{0}.$
(76)

We consider the following inverse LQR problem: can we infer $\bm{Q}$, i.e. the
way state deviations from zero are penalized in various state space
directions, given sample state trajectories generated by the optimally
controlled system, and given knowledge of $\bm{A}$, $\bm{B}$, and $\bm{R}$?
More formally, given $K$ trajectories of the system under optimal LQR control
sampled at $T$ time points:

$\bm{x}^{(k)}_{0},\bm{x}^{(k)}_{1},\cdots\bm{x}^{(k)}_{T-1}\quad\text{ for
}\quad k=1,\cdots,K,$ (77)

can we determine the matrix $\bm{Q}$ used in the LQR algorithm’s objective
function (Equation 74)? Assuming that this problem is well-posed (i.e.,
$\bm{Q}$ is identifiable; Zhang et al.,, 2019), we can solve this problem by
differentiating through DARE (Section 5.2). More specifically, we start with
an initial guess of $\bm{Q}$, which we denote as $\bm{\hat{Q}}$. Next, we
solve the LQR problem and find the corresponding optimal inputs
$\bm{\hat{u}}_{t}$ and trajectories $\bm{\hat{x}}_{t}$. We then minimize the
objective

$\mathcal{\ell}(\bm{\hat{Q}})=\frac{1}{KT}\sum_{k=1}^{K}\sum_{t=0}^{T-1}(\bm{x}^{(k)}_{t}-\bm{\hat{x}}^{(k)}_{t})^{2}.$
(78)

with respect to $\bm{\hat{\bm{Q}}}$. To illustrate our solution, we consider
the following example system:

$\begin{split}\bm{A}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\quad&\quad\bm{B}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\\\
\bm{Q}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\quad&\quad\bm{R}=\begin{pmatrix}0.1&0\\\
0&0.3\end{pmatrix}.\end{split}$ (79)

We created a synthetic data set by sampling $\bm{K}=30$ initial states
$\bm{x}_{\text{init}}$ from $\mathcal{N}(\bm{0},\bm{I})$ and for each sample,
we simulated the activity trajectories under optimal control for $T=30$ steps.
We then optimized $\ell(\bm{\hat{Q}})$ for the synthetic data set using L-BFGS
(Liu and Nocedal,, 1989). After $7$ iterations, we were able to recover the
true $\bm{Q}$ to a good degree of accuracy (Figure 1). Code for this example
is available at https://github.com/tachukao/autodiff-inverse-lqr.

Figure 1:  Evolution of $\ell(\bm{\hat{Q}})$ (left) and $\|\bm{\hat{Q}}\|$
(right) duing optimization (see text).

##  7 Conclusion

In this note, we derived the forward and reverse mode derivatives for the
solutions to the discrete time and continuous time variants of the Sylvester,
Lyapunov, and Riccati equations (summarized in Tables 1 and 2). These
equations are widely-used in control theory and other branches of applied
mathematics. We demonstrate the usefulness of these derivatives on an inverse
LQR problem, where parameters of the optimal control problem are inferred from
observations of state trajectories by differentiating through the discrete
time Riccati equation (Section 5.2). These derivatives (with the exception of
the discrete time Sylvester equation, of more limited use), have been
implemented and numerically tested in Owl, a numerical library written in
OCaml with a full-featured automatic differentiation module.

## Acknowledgements

We are grateful for helpful comments on the manuscript by Kris Jensen.

## References

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##  Appendix A Forward-mode derivatives summary

Equation | Tangents
---|---
continuous time Sylvester  Equation 1
$\scriptstyle\bm{A}\bm{P}+\bm{P}\bm{B}+\bm{C}=\bm{0}$ | $\scriptstyle\bm{A}\,\bm{\dot{P}}+\bm{\dot{P}}\,\bm{B}+\big{(}\bm{\dot{A}}\,\bm{P}+\bm{P}\,\bm{\dot{B}}+\bm{\dot{C}}\big{)}=\bm{0}$
discrete time Sylvester  Equation 12
$\scriptstyle\bm{A}\bm{P}\bm{B}-\bm{P}+\bm{C}=\bm{0}$ | $\scriptstyle\bm{A}\,\bm{\dot{P}}\,\bm{B}-\bm{\dot{P}}+\big{(}\bm{\dot{A}}\,\bm{P}\bm{B}+\bm{A}\bm{P}\,\bm{\dot{B}}+\bm{\dot{C}}\big{)}=\bm{0}$
continuous time Lyapunov  Equation 23
$\scriptstyle\bm{A}\bm{P}+\bm{P}\bm{A}^{T}+\bm{C}=\bm{0}$ | $\scriptstyle\bm{A}\,\bm{\dot{P}}+\bm{\dot{P}}\,\bm{A}^{T}+\big{(}\bm{\dot{A}}\,\bm{P}+\bm{P}\,\bm{\dot{A}}^{T}+\bm{\dot{C}}\big{)}=\bm{0}$
discrete time Lyapunov  Equation 33
$\scriptstyle\bm{A}\bm{P}\bm{A}^{T}-\bm{P}+\bm{C}=\bm{0}$ | $\scriptstyle\bm{A}\,\bm{\dot{P}}\,\bm{A}^{T}-\bm{\dot{P}}+\big{(}\bm{\dot{A}}\,\bm{P}\bm{A}^{T}+\bm{A}\bm{P}\,\bm{\dot{A}}^{T}+\bm{\dot{C}}\big{)}=\bm{0}$
continuous time algebraic Riccati  Equation 43
$\scriptstyle\bm{A}^{T}\bm{P}+\bm{P}\bm{A}-\bm{P}\bm{B}\bm{R}^{-1}\bm{B}^{T}\bm{P}+\bm{Q}=\bm{0}$ | $\begin{array}[]{rl}\scriptstyle\bm{\dot{P}}\bm{\tilde{A}}+\bm{\tilde{A}}^{T}\bm{\dot{P}}+\big{(}\bm{\dot{A}}^{T}\bm{P}+\bm{P}\bm{\dot{A}}-\bm{P\dot{B}K}-\bm{K}^{T}\bm{\dot{B}}^{T}\bm{P}&\\\ \scriptstyle+\bm{\dot{Q}}+\bm{K}^{T}\bm{\dot{R}}\bm{K}\big{)}=\bm{0}&\end{array}$
discrete time algebriac Riccati  Equation 58
$\begin{array}[]{rr}\scriptstyle\bm{A}^{T}\bm{P}\bm{A}-\bm{P}&\\\ \scriptstyle-(\bm{A}^{T}\bm{PB})(\bm{R}+\bm{B}^{T}\bm{P}\bm{B})^{-1}(\bm{B}^{T}\bm{PA})+\bm{Q}=\scriptstyle\bm{0}\end{array}$ | $\begin{array}[]{rl}\scriptstyle\bm{\tilde{A}}^{T}\bm{\dot{P}}\bm{\tilde{A}}-\bm{\dot{P}}+\big{(}\bm{\dot{A}}^{T}\bm{P}\bm{\tilde{A}}+\bm{\tilde{A}}^{T}\bm{P}\bm{\dot{A}}&\\\ \scriptstyle-\bm{K}^{T}\bm{\dot{B}}^{T}\bm{P\tilde{A}}-\bm{\tilde{A}}^{T}\bm{P}\bm{\dot{B}}\bm{K}+\bm{\dot{Q}}+\bm{K}^{T}\bm{\dot{R}}\bm{K}\big{)}=\bm{0}\end{array}$
Table 1: Summary of forward-mode derivatives

##  Appendix B Reverse-mode derivatives summary

Equation | Adjoints
---|---
continuous time Sylvester  Equation 1
$\scriptstyle\bm{A}\bm{P}+\bm{P}\bm{B}+\bm{C}=\bm{0}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\bm{A}^{T}\bm{S}+\bm{S}\bm{B}^{T}+\overline{\bm{P}}\\\ &\scriptstyle\overline{\bm{A}}=\bm{S}\bm{P}^{T}\\\ &\scriptstyle\overline{\bm{B}}=\bm{P}^{T}\bm{S}\\\ &\scriptstyle\overline{\bm{C}}=\bm{S}\end{array}$
discrete time Sylvester  Equation 12
$\scriptstyle\bm{A}\bm{P}\bm{B}-\bm{P}+\bm{C}=\bm{0}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\bm{A}^{T}\bm{S}\bm{B}^{T}-\bm{S}+\overline{\bm{P}},\\\ &\scriptstyle\overline{\bm{A}}=\bm{S}\bm{B}^{T}\bm{P}^{T}\\\ &\scriptstyle\overline{\bm{B}}=\bm{P}^{T}\bm{A}^{T}\bm{S}\\\ &\scriptstyle\overline{\bm{C}}=\bm{S}\end{array}$
continuous time Lyapunov  Equation 23
$\scriptstyle\bm{A}\bm{P}+\bm{P}\bm{A}^{T}+\bm{C}=\bm{0}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\overline{\bm{P}}+\bm{A}^{T}\bm{S}+\bm{S}\bm{A}\\\ &\scriptstyle\overline{\bm{A}}=\bm{S}\bm{P}^{T}+\bm{S}^{T}\bm{P}\\\ &\scriptstyle\overline{\bm{Q}}=\bm{S}\end{array}$
discrete time Lyapunov  Equation 33
$\scriptstyle\bm{A}\bm{P}\bm{A}^{T}-\bm{P}+\bm{C}=\bm{0}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\overline{\bm{P}}+\bm{A}^{T}\bm{S}\bm{A}-\bm{S}\\\ &\scriptstyle\overline{\bm{A}}=\bm{S}\bm{A}\bm{P}^{T}+\bm{S}^{T}\bm{A}\bm{P}\\\ &\scriptstyle\overline{\bm{Q}}=\bm{S}.\end{array}$
continuous time algebraic Riccati  Equation 43
$\scriptstyle\bm{A}^{T}\bm{P}+\bm{P}\bm{A}-\bm{P}\bm{B}\bm{R}^{-1}\bm{B}^{T}\bm{P}+\bm{Q}=\bm{0}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\frac{1}{2}(\overline{\bm{P}}+\overline{\bm{P}}^{T})+\bm{\tilde{A}}\bm{S}+\bm{S}\bm{\tilde{A}}^{T}\\\ &\scriptstyle\overline{\bm{A}}=2\bm{P}\bm{S}\\\ &\scriptstyle\overline{\bm{B}}=-2\bm{PS}\bm{K}\\\ &\scriptstyle\overline{\bm{Q}}=\bm{S}\\\ &\scriptstyle\overline{\bm{R}}=\bm{K}\bm{S}\bm{K}^{T}.\end{array}$
discrete time algebriac Riccati  Equation 58
$\begin{array}[]{rr}\scriptstyle\bm{A}^{T}\bm{P}\bm{A}-\bm{P}&\\\ \scriptstyle-(\bm{A}^{T}\bm{PB})(\bm{R}+\bm{B}^{T}\bm{P}\bm{B})^{-1}(\bm{B}^{T}\bm{PA})+\bm{Q}=\scriptstyle\bm{0}\end{array}$ | $\begin{array}[]{rl}&\scriptstyle\bm{0}=\frac{1}{2}(\overline{\bm{P}}+\overline{\bm{P}}^{T})+\bm{\tilde{A}}\bm{S}\bm{\tilde{A}}^{T}-\bm{S}\\\ &\scriptstyle\overline{\bm{A}}=2\bm{P}\bm{\tilde{A}}\bm{S}\\\ &\scriptstyle\overline{\bm{B}}=-2\bm{P\tilde{A}SK}^{T}\\\ &\scriptstyle\overline{\bm{Q}}=\bm{S}\\\ &\scriptstyle\overline{\bm{R}}=\bm{K}\bm{S}\bm{K}^{T}\end{array}$
Table 2: Summary of reverse-mode derivatives


# Thermodynamics and phase transition of Bardeen-AdS-class black holes

Shan-Ping Wu, Shao-Wen Wei 111Corresponding author. E-mail<EMAIL_ADDRESS>1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Key Laboratory of Quantum Theory and Applications of
MoE, Lanzhou University, Lanzhou, Gansu 730000, China,
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, China 3 School of Physical Science and Technology,
Lanzhou University, Lanzhou 730000, China

###### Abstract

In a generalized parameter space, regular black holes can be regarded as non-
singular solutions under specific parameters in Einstein gravity theory
coupled with non-linear electromagnetic fields. Following this concept, we
investigate the thermodynamic states and phase transitions of Bardeen-AdS-
class black holes, revealing that the system can be classified into two
categories, Type I and Type II, based on whether it adopts a pure Bardeen-AdS
spacetime without event horizons or a Bardeen-AdS black hole as its phase
state, each exhibiting distinct thermodynamic properties. If one includes the
Bardeen-AdS black holes in the system (Type I), there will be three distinct
black hole states and the phase transitions between them are analogous to the
Reissner-Nordstrom-AdS black holes. On the other hand, if the pure Bardeen-AdS
spacetime is included (Type II), an additional tiny black hole state emerges.
A phase transition reminiscent of the Hawking-Page transition was found.
Significantly, in this scenario, thermodynamical characteristic curves exhibit
discontinuous behavior, which attributes to the multiple horizons. The
presence of a Bardeen-AdS black hole within the phase structure of the
Bardeen-AdS-class black hole profoundly modifies the thermodynamical
properties of the system, highlighting the novel aspects of regular black
holes.

Regular black hole, thermodynamics, phase transition.

###### pacs:

04.20.-q, 04.70.-s, 04.70.Dy

###### Contents

  1. I Introduction
  2. II General relativity coupled with nonlinear magnetic monopole
    1. II.1 Black hole solution
    2. II.2 Euclidean action and thermodynamics
  3. III Bardeen-AdS-class black hole and thermodynamics
  4. IV Phase States and Phase Transitions of Bardeen-AdS-class Black Holes
    1. IV.1 Type I
    2. IV.2 Type II
  5. V Conclusion and Discussion
  6. A The critical point for $r_{h}-T(r_{h})$ curve

##  I Introduction

Regular black holes are a distinct solutions characterized by the absence of
the center singularities. The first such model was implemented by Bardeen and
is known as the Bardeen black hole Bardeen1968 . Following it, the weak energy
condition was satisfied for this regular black hole Mars1996RBHweakcondition ;
Borde1996RBHtopologychange . Unlike many conventional black holes (such as
Kerr-Newman black holes), they do not violate the causality due to the absence
of singularities, nor do they challenge the cosmic censorship hypothesis
Penrose1969Gravitationalcollapse . Although the proposal of the Bardeen black
hole does not directly originate from the equations of field, it has been
shown that the Bardeen black hole can be accounted for within the framework of
general relativity coupled with nonlinear electromagnetic fields AyonBeato2000
. Nevertheless, this concept is general and convenient for further
understanding Hayward black hole Hayward2005Formation  and other regular black
hole solutions AyonBeato1998RegularBH ; Ayon-Beato:1999NonsingularChargedBH ;
Ayon-Beato:1999NewRegularBH ; Bronnikov:2000RegularMagneticBH ; Ayon-
Beato:2004FourParametricRegularBH ; Hassaine:2008HighRBH ;
Balart:2014RBHwithElectrodynamics ; FanConReBH2016 ; Canate2022Transforming ;
Li:2023RBHanalytic . Additionally, a range of fields can also be introduced as
sources for constructing regular black holes, such as phantom fields
Bronnikov:2005RegularPhantom , quasi-topological electromagnetic fields
Liu:2019QuasitopologicalElectromagnetism ;
Cisterna:2020QuasitopologicalElectromagnetism , and the Maxwell and scalar
fields in Maxwell-scalar theory Li:2024EMSGravity . Furthermore, various
gravitational theories offer different interpretations for the regular black
holes Nicolini:2005Noncommutative ; Eichhorn2022AsymptoticallySafeGravity ;
Ashtekar:2023LoopQuantumGravity ; Nicolini2023HowStringsRBHs ;
Bueno:2024RBHPureGravity ; Konoplya:2024DymnikovaBH .

After the establishment of the four laws of black hole thermodynamics, it is
widely accepted that black holes are thermodynamical systems with Hawking
temperature and entropy Hawking1975ParticleCreation ; Bardeen1973FourLaws ;
Gibbons1976ActionIntegrals . Moreover, it seems feasible to consider black
holes with nonlinear electromagnetic fields within the context of
thermodynamics Rasheed1997Zerothandfirstlaws . The first law and Smarr
relation of thermodynamics for the Einstein-Born-Infeld (-AdS, -dS)
electrodynamics black hole are reasonable Born1934Foundations ;
Rasheed1997Zerothandfirstlaws ;
Gunasekaran2012ExtendedPhaseApaceThermodynamics . However, when one considers
other regular black holes with non-linear electromagnetic fields, their
thermodynamics gets some controversial issues. The requirements of the first
law may lead to a conflict between the entropy-area law and the Hawking
temperature Ma2014CorrectedForm ; Lan:2021Entropy ; Lan:2023Entropy , and
necessitate a redefinition of the electromagnetic potential. The main reason
for this situation is that the energy and magnetic charge, serving as coupling
constants, appear in the action of the nonlinear electromagnetic fields.
Furthermore, to address this “coupling constant issue”, numerous new forms of
the first law of thermodynamics and Smarr relations have been proposed
Ma2014CorrectedForm ; Azreg-Ainou:2014NoInconsistency ;
Zhang2016FirstLawandSmarrformula ; Fan2016CriticalPhenomena ;
Lan2021GlinerVacuum ; Guo:2023RecoverThermodynamicsyRegularBlackHoles ;
Simovic:2023EuclideanandHamiltonianRBH ; Lan:2023RBHTopicReview ;
Huang:2023IntermediateStates .

To clarify the reasons behind “coupling constant issue”, let’s revisit the
non-linear electromagnetic part of the action

$\mathcal{L}_{m}=\frac{12m}{\left|g\right|^{3}}\left(\frac{\sqrt{g^{2}F_{\mu\nu}F^{\mu\nu}/2}}{1+\sqrt{g^{2}F_{\mu\nu}F^{\mu\nu}/2}}\right)^{5/2},$
(1)

used in constructing the Bardeen black hole AyonBeato2000 . Here, the energy
and magnetic charge are denoted as $m$ and $g$, respectively. Within the
framework of the action, there is some unnaturalness because energy and
magnetic charge are integration constants, not necessarily equivalent to the
coupling constants. It implies that for the Bardeen-AdS black hole, the energy
and charge cannot deviate from the coupling constants. Ultimately, in the
thermodynamical discussion, we must simultaneously consider variations in
energy and its associated coupling constants, which could potentially lead to
inconsistencies between thermodynamical quantities and the first law of
thermodynamics.

To address this issue, a straightforward approach is to consider that the
energy and charge are not function of the coupling constants in the
Lagrangian, but rather treat $m$ and $g$ as “real” constants. Therefore, it
becomes necessary to explore more general black hole solutions. According to
the conclusions drawn in Refs. Rasheed1997Zerothandfirstlaws ;
Ma2014CorrectedForm ; Zhang2016FirstLawandSmarrformula ;
Guo:2023RecoverThermodynamicsyRegularBlackHoles , this approach can indeed
help reconcile the construction of thermodynamical quantities with the first
law of thermodynamics. However, within these generalized black hole solutions,
singularity avoidance occurs only under specific values of the black hole’s
energy and charge. This suggests that in the context of black hole
thermodynamics, regular black holes may correspond to particular intermediate
states Huang:2023IntermediateStates . Building on this premise, we intend to
undertake a detailed exploration of the thermodynamics and phase transition of
Bardeen black holes in the AdS background.

The first crucial step is to redefine the coupling constant within the
framework of nonlinear electromagnetic fields as a “fundamental” constant,
distinct from the energy or charge, thereby ensuring the internal consistency
of thermodynamics. Subsequently, a class of black hole solutions with a
negative cosmological constant, known as Bardeen-AdS-class black holes, is
derived. In particular, we present two types of Bardeen-AdS-class black holes:
one incorporating the Bardeen-AdS black hole and the other showcasing the pure
Bardeen-AdS spacetime. These configurations exhibit significantly different
thermodynamical properties in the issues of the black hole horizon,
thermodynamical states, and phase transitions.

The paper is organized as follows. In Sec. II, we provide a brief review of
black hole thermodynamics within the framework of the general spherically
symmetric nonlinear electromagnetic fields. In Sec. III the Bardeen-AdS-class
metric is obtained. The first law of thermodynamics and Smarr relation are
also considered. In Sec. IV, we discuss the black hole states and phase
transitions for Bardeen-AdS-class black holes featuring either the Bardeen-AdS
black hole or pure Bardeen-AdS spacetime. Finally, we summarize and discuss
our results in Sec. V.

##  II General relativity coupled with nonlinear magnetic monopole

As previously mentioned, the Bardeen black hole represents a specific class of
regular black hole solutions within general relativity, incorporating the
nonlinear magnetic monopoles. In this section, we review the general theory of
gravity and examine its thermodynamical implications.

###  II.1 Black hole solution

In Lagrangian formalism, the action for Einstein gravity minimally coupled to
nonlinear electromagnetic fields is given by

$I=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\left(R+\frac{6}{L^{2}}-\mathcal{L}_{m}(s,a)\right),$ (2)

where $s=F_{\mu\nu}F^{\mu\nu}$, and $a$ is the parameter of the nonlinear
form. The gravitational part includes the Einstein-Hilbert term and the AdS
term, while the electromagnetic field part is characterized by a nonlinear
coupling $\mathcal{L}_{m}(s,a)$. By varying the action (2) with respect to
$g_{\mu\nu}$ and $A_{\mu}$, we derive the equations of field

$\displaystyle
R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R-\frac{3}{L^{2}}g_{\mu\nu}=T_{\mu\nu},\quad
T_{\mu\nu}\equiv\frac{1}{2}\left(-g_{\mu\nu}\mathcal{L}_{m}+4\frac{\partial\mathcal{L}_{m}}{\partial
s}F_{\mu\lambda}{F_{\nu}}^{\lambda}\right),$ (3)
$\displaystyle\nabla_{\mu}G^{\mu\nu}=0,\quad
G^{\mu\nu}\equiv\frac{\partial\mathcal{L}_{m}}{\partial s}F^{\mu\nu}.$ (4)

Assuming a static and spherically symmetric spacetime, and considering the
Dirac magnetic monopole, the line element can be expressed as

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},$ (5) $\displaystyle A=-q_{m}\cos\theta d\phi,$ (6)

Under this ansatz, Eq. (4) naturally holds. Furthermore, Eq. (3) yields

$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}+1-\frac{m}{r}+\frac{1}{2r}\int_{r}^{\infty}{dr\left(r^{2}\mathcal{L}_{m}\left(2q_{m}^{2}/r^{4},a\right)\right)}.$
(7)

It is evident that once the electromagnetic field part of the Lagrangian
$\mathcal{L}_{m}$ is given, $f(r)$ can be solved. Conversely, if $f(r)$ is
known, $\mathcal{L}_{m}$ can be reconstructed FanConReBH2016 . Note that in
Eq. (7), the integral with an upper limit of infinity gives us an opportunity
to calculate the physical quantities directly. However, it is essential to
note that the validity of Eq. (7) should relies on the condition that this
integral is finite.

On the time-like hypersurface $\Sigma$ at $t=t_{0}$, the electric and magnetic
charges are

$Q_{e}=\frac{1}{4\pi}\int_{\partial\Sigma}{\star G}=0,\quad
Q_{m}=\frac{1}{4\pi}\int_{\partial\Sigma}{F}=q_{m},$ (8)

where $\partial\Sigma$ denotes the boundary at $r=\infty$ for hypersurface
$\Sigma$. The results indicate that the electromagnetic field $F$ is sourced
by the magnetic charges. Moreover, the difference in magnetic potential
between the event horizon and infinity is represented as

$\displaystyle\Phi_{m}=\int_{r_{+}}^{\infty}{dr\frac{q_{m}}{r^{2}}\partial_{s}\mathcal{L}_{m}(2q_{m}^{2}/r^{4},a)}.$
(9)

In asymptotically AdS spacetime, the energy can be derived from the Komar mass
$-\frac{1}{8\pi}\int_{\partial\Sigma}{\star dK}$, where $K=\partial_{t}$ is a
time-like Killing vector field. Due to the AdS background, this energy is
divergent. However, by subtracting the background contribution, an effective
energy can be obtained as

$M=-\frac{1}{8\pi}\int_{\partial\Sigma}{\star dK}-M_{AdS}=\frac{m}{2}.$ (10)

###  II.2 Euclidean action and thermodynamics

It is widely acknowledged that the thermodynamics of black holes can be
derived from the partition function Gibbons1976ActionIntegrals

$\mathcal{Z}=\int{\mathcal{D}g\exp\left(-I_{E}\right)},$ (11)

For general gravitational theory, evaluating this path integral is
challenging. However, using the semi-classical approach, we only consider the
contributions from the on-shell solution, i.e.,

$\mathcal{Z}\simeq\exp\left(-I_{\text{on-shell}}\right).$ (12)

For the gravity coupled with nonlinear electromagnetic fields, the on-shell
action can be calculated by using Eqs. (5), (6), and (7). Before proceeding
the calculation, it is essential to note that the singularities may occur at
the event horizon in the Euclidean geometry of the black hole. Expanding the
metric near the event horizon $r=r_{+}$, we get

$\displaystyle
ds^{2}\simeq\rho^{2}d\left(f^{\prime}\left(r_{+}\right)t_{E}/2\right)^{2}+d\rho^{2}+r_{+}^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (13)
$\displaystyle\rho=\frac{2}{\sqrt{f^{\prime}\left(r_{+}\right)}}\sqrt{r-r_{+}}.$
(14)

The cone singularity arises at $r=r_{+}$ unless the Euclidean time has a
period $\beta={4\pi}/{f^{\prime}(r_{+})}$. Here, $\beta$ corresponds to the
black hole temperature as given by

$T_{H}=\frac{1}{\beta}=\frac{f^{\prime}\left(r_{+}\right)}{4\pi}=\frac{1}{4\pi
r_{+}}\left(1+3\frac{r_{+}^{2}}{L^{2}}-\frac{1}{2}r_{+}^{2}\mathcal{L}_{m}(2q_{m}^{2}/r_{+}^{4},a)\right).$
(15)

Furthermore, the on-shell Euclidean action is represented by

$I_{\text{on-
shell}}=-\frac{1}{16\pi}\int_{M}{d^{4}x\sqrt{g}\left(R+\frac{6}{L^{2}}-\mathcal{L}_{m}\right)}-\frac{1}{8\pi}\int_{\partial
M}{d^{3}x\sqrt{\gamma}K}+I_{ct},$ (16)

where the first, second, and third terms are for the bulk actions of
gravitational and electromagnetic fields, Gibbons-Hawking action, and the
counter-term acquired through the AdS renormalization Emparan:1999SurfaceTerms
; deHaro:2000HolographicReconstruction ;
Skenderis:2002HolographicRenormalization , respectively. Specifically, in four
dimensions, the counter-term can be represented as

$I_{ct}=\frac{1}{16\pi}\int_{\partial
M}{d^{3}x\sqrt{\gamma}\left(\frac{4}{L}+LR_{\partial M}\right)}.$ (17)

Combining them, we can derive the free energy as

$F=-\frac{1}{\beta}\ln\mathcal{Z}=I_{\mathrm{on}\\-\mathrm{shell}}/\beta=\frac{m}{2}-\frac{f^{\prime}(r_{+})}{4\pi}\pi
r_{+}^{2}.$ (18)

Subsequently, other thermodynamical quantities can be obtained by the free
energy,

$\displaystyle S=-\frac{\partial F}{\partial
T_{H}}=-\frac{\partial_{r_{+}}F}{\partial_{r_{+}}T_{H}}=\pi r_{+}^{2},$ (19)
$\displaystyle M=F+T_{H}S=\frac{m}{2},$ (20)
$\displaystyle\Phi_{m}=\frac{\partial F}{\partial
Q_{m}}=\int_{r_{+}}^{\infty}{dr\frac{Q_{m}}{r^{2}}\partial_{s}\mathcal{L}_{m}(2Q_{m}^{2}/r^{4},a)}.$
(21)

Clearly, the magnetic potential and energy exactly satisfy Eqs. (9) and (10).
If the cosmological constants $L$ and parameter $a$ are treated as variables,
the extended first law of thermodynamics can be formulated as

$dM=T_{H}dS+\Phi_{m}dQ_{m}+\mathcal{A}da+\frac{\Theta}{8\pi}d\Lambda,$ (22)

with $\Theta=-4/3\pi r_{h}^{3}$, $\Lambda=-3/L^{2}$, and

$\mathcal{A}=\frac{1}{16\pi}\int_{r_{+}}^{\infty}{d^{4}x\sqrt{-g}\partial_{a}\mathcal{L}_{m}(2q_{m}^{2}/r^{4},a)}.$
(23)

The parameter $a$ serves two main purposes: it characterizes the coupling
parameter of the theory and regulates the dimension of the nonlinear
electromagnetic fields. This ensures that there is a certain $\sigma$ such
that the functional relation $M(\lambda^{2}S,\lambda
Q_{m},\lambda^{-2}\Lambda,\lambda^{\sigma}a)=\lambda M(S,Q_{m},\Lambda,a)$
holds, thereby indicating the Smarr relation,

$M=2T_{H}S-2\frac{\Theta}{8\pi}\Lambda+\Phi_{m}Q_{m}+\sigma\mathcal{A}a,$ (24)

where $\sigma$ can be interpreted as the dimension of the parameter $a$. As
demonstrated above, one can see that the construction of thermodynamics for
black holes with general nonlinear electromagnetic fields is self-consistent.
Nevertheless, this differs from the case of the Bardeen (-AdS) black hole,
where the energy is related to the coupling constant in the Lagrangian, and we
will discuss it in the next section.

##  III Bardeen-AdS-class black hole and thermodynamics

The Bardeen-AdS black hole can be viewed as a special solution in the model of
Einstein gravity coupled with a nonlinear magnetic monopole. However, as
previously mentioned, the Bardeen-AdS black hole is affected by the “coupling
constant issues”. Therefore, to avoid this problem, the Lagrangian of the
nonlinear electromagnetic field shall be revised as

$\mathcal{L}_{m}=\frac{6m_{0}}{q_{0}^{3}}\left(\frac{\sqrt{q_{0}^{2}s/2}}{1+\sqrt{q_{0}^{2}s/2}}\right)^{5/2},\
s=F_{\mu\nu}F^{\mu\nu}.$ (25)

In this revision, the original $m$ and $|g|$ in Eq. (1) has been replaced by
$m_{0}/2$ and $q_{0}$, respectively. This action form aligns with the “Bardeen
class” for “$\mu=3$” in Ref. FanConReBH2016 , while with different parameters.
It should be noted that with these parameters, the physical interpretation
becomes more straightforward. Using Eqs. (6) and (7), the solutions for the
gravity and electromagnetic field are given by

$\displaystyle
f\left(r\right)=\frac{r^{2}}{L^{2}}+1-\frac{m}{r}+m_{0}\left(\frac{q_{m}}{q_{0}}\right)^{3/2}\left(\frac{1}{r}-\frac{r^{2}}{\left(r^{2}+q_{m}q_{0}\right)^{3/2}}\right),$
(26)

and

$F=q_{m}\sin\theta d\theta\wedge d\phi.$ (27)

The metric (26) differs from the Bardeen-AdS black hole unless $m=m_{0}$ and
$q=q_{0}$. Thus, this black hole solution should be considered as an extension
of the Bardeen-AdS black hole within the framework of the nonlinear
electromagnetic field model. Following Ref. FanConReBH2016 , we refer to this
spacetime with event horizons as the Bardeen-AdS-class black hole. Notably,
the energy and magnetic charge are given by

$M=\frac{1}{2}m,\quad Q_{m}=q_{m}.$ (28)

By expanding $f(r)$ near $r=0$, we obtain

$f\left(r\right)=m_{0}\left(\left(\frac{q_{m}}{q_{0}}\right)^{3/2}-\frac{m}{m_{0}}\right)\frac{1}{r}+1+r^{2}\left(\frac{1}{L^{2}}-\frac{m_{0}}{q_{0}^{3}}\right)+\mathcal{O}\left(r^{3}\right).$
(29)

When $(q_{m}/q_{0})^{3/2}=m/m_{0}$, the curvature singularity at $r=0$
disappears. Moreover, Eq. (29) also indicates that, at small scale, the
effects of the nonlinear magnetic charge and mass counterbalance each other,
potentially avoiding a spacetime singularity when in equilibrium.

Another noteworthy point is that the spacetime defined by metric (26) can have
multiple horizons. Through a simple analysis, we confirm that the Bardeen-AdS-
class black hole can have one, two, or three horizons. To illustrate it, let
$r_{h}$ and $r_{+}$ denote the radii of the horizons and outer horizon (event
horizon), respectively. For the black hole solution, we have

$f\left(r_{h},m\right)=0\quad\Longrightarrow\quad m=m\left(r_{h}\right),$ (30)

which relates the parameter $m$ to $r_{h}$. There are several important points
to clarify. First, $r_{h}$ may not necessarily equal $r_{+}$ and can represent
the inner horizon radius, even though both $r_{h}$ and $r_{+}$ satisfy the
equation $f(r,m)=0$. Second, for the Bardeen-AdS-class black holes, the range
of $r_{h}$ is $(0,+\infty)$, but $r_{+}$ is often restricted to a smaller
range. Third, for black holes characterized by a single-valued $m$ (with other
parameters fixed), different values of $r_{h}$ may correspond to the same $m$,
implying redundant representations of the black hole if one uses $r_{h}$
instead of $m$. However, this issue does not arise when using the outer
horizon radius $r_{+}$. Despite this fact, discussing $r_{h}$ is still helpful
because the range of $r_{+}$ can be inferred from the analysis of $r_{h}$. For
further discussion, we introduce

$T\left(r_{h}\right)=\frac{1}{4\pi}\left.\partial_{r}f\left(r,m\left(r_{h}\right)\right)\right|_{r=r_{h}},$
(31)

which is proportional to the gradient of $f(r,m)$ at $r=r_{h}$. Even though
the expressions of the function $T(r_{h})$ and the black hole temperature
$T_{H}$ are identical, there is a subtle difference because $r_{h}$ is not
always the radius of the outer horizon. Some characteristic curves of
$T(r_{h})$ are shown in Fig. 1, where $r_{*}$ is given by

$r_{*}=\frac{\sqrt{q_{0}q_{m}}}{\sqrt{30}}\sqrt{-3L^{2}/\left(q_{0}q_{m}\right)+\sqrt{\left(3L^{2}/\left(q_{0}q_{m}\right)+40\right)3L^{2}/\left(q_{0}q_{m}\right)}}.$
(32)

The equation $T(r_{h})=0$ has two real roots if

$\frac{m_{0}q_{m}}{q_{0}^{2}}\geq\frac{\left(30-3L^{2}/\left(q_{0}q_{m}\right)+\sqrt{9L^{4}/\left(q_{0}q_{m}\right)^{2}+120L^{2}/\left(q_{0}q_{m}\right)}\right)^{7/2}}{225\sqrt{30}\left(-3L^{2}/\left(q_{0}q_{m}\right)+\sqrt{9L^{4}/\left(q_{0}q_{m}\right)^{2}+120L^{2}/\left(q_{0}q_{m}\right)}\right)^{2}}.$
(33)

Considering the continuity of $f(r)$ for $r\in(0,\infty)$, we can conclude
that the equation $f(r)=0$ has at most three solutions. By substituting
several specific parameter values, it is easy to find that $f(r)=0$ can have
zero, one, two, or three roots, corresponding to the number of the horizons.

Figure 1: The sketch of the curves for $T(r_{h})$. Here, we set $L=1$ and
$q_{m}=q_{0}=1$. The relationship between $r_{h}$ and $T$ is shown for
$m_{0}=4.06$, $m_{0}=5.00$, and $m_{0}=3.00$, depicted as an orange thick
line, a blue thin line, and a green dashed line, respectively.

Although the number of horizons is known, we still need to clarify the range
of $r_{+}$. From Fig. 1, we ascertain the following results for the black hole
solutions:

  * •

$T(r_{*})\geqslant 0$. Only one horizon exists, implying $r_{+}=r_{h}$ and the
range of $r_{+}$ is $(0,\infty)$.

  * •

$T(r_{*})<0$. For $r_{h}\in[r_{2},\infty)$ and $r_{h}\in(r_{1},r_{2})$, we
have $r_{+}=r_{h}$ and $r_{+}\neq r_{h}$, respectively. For
$r_{h}\in(0,r_{1}]$, if $f_{2}>0$, $r_{h}=r_{+}$, while $f_{2}\leqslant 0$,
$r_{h}\neq r_{+}$, where
$f_{2}=\left.f\left(r,m\left(r_{h}\right)\right)\right|_{r=r_{2}}$. As a
result, the range of $r_{+}$ is
$\left[r_{2},\infty\right)\cup\left\\{r_{+}\middle|0<r_{+}\leq
r_{1}\land\,\,f\left(r_{2},m(r_{+})\right)>0\right\\}$.

It must be noted that not all $r_{h}$ satisfying $T(r_{h})>0$ represent the
radius of the event horizon. The signs of $T(r_{*})$ and $f_{2}$ also need
further examination. By considering these aspects, we can determine whether
$r_{+}=r_{h}$ holds and establish the range of $r_{+}$. Based on the general
analysis above, the range of $r_{+}$ for Bardeen-AdS-class black holes may
appear complex. However, this is not the case. In Sec. IV, we will provide the
range of $r_{+}$ and employ a more intuitive approach.

According to the discussion in Sec. II.2, the Hawking temperature is
determined from the geometry near the event horizon, resulting in

$T_{H}=\frac{1}{4\pi}\left(\frac{1}{r_{+}}+3\frac{r_{+}}{L^{2}}-3m_{0}r_{+}\sqrt{\frac{q_{m}^{5}}{q_{0}\left(q_{0}q_{m}+r_{+}^{2}\right)^{5}}}\right).$
(34)

The corresponding energy is given by

$M=\frac{1}{2}m=\frac{1}{2}\left(\frac{r_{+}^{3}}{L^{2}}+r_{+}+m_{0}\left(\frac{q_{m}}{q_{0}}\right)^{3/2}\left(1-\frac{r_{+}^{3}}{\left(r_{+}^{2}+q_{0}q_{m}\right)^{3/2}}\right)\right).$
(35)

Furthermore, the extended first law of black hole thermodynamics is expressed
as

$dM=T_{H}dS+\frac{\Theta}{8\pi}d\Lambda+\Phi_{m}dQ_{m}+\phi_{m}dm_{0}+\phi_{q}dq_{0},$
(36)

where $\phi_{m}$ and $\phi_{q}$ arise from variations of $M$ with respect to
$q_{m}$ and $q_{0}$, respectively. The corresponding thermodynamical
quantities for the black holes are described by

$\displaystyle S=\pi r_{+}^{2},$ (37) $\displaystyle\Theta=-\frac{4}{3}\pi
r_{+}^{3},$ (38)
$\displaystyle\Phi_{m}=\frac{3m_{0}}{4q_{0}}\sqrt{\frac{q_{m}}{q_{0}}}\left(1-\frac{r_{+}^{5}}{\left(r_{+}^{2}+q_{0}q_{m}\right)^{5/2}}\right),$
(39)
$\displaystyle\phi_{m}=\frac{1}{2}\left(\frac{q_{m}}{q_{0}}\right)^{3/2}\left(1-\frac{r_{+}^{3}}{\left(r_{+}^{2}+q_{0}q_{m}\right)^{3/2}}\right),$
(40)
$\displaystyle\phi_{q}=\frac{3m_{0}}{4q_{0}}\left(\frac{q_{m}}{q_{0}}\right)^{3/2}\left(-1+\frac{r_{+}^{5}+2q_{0}q_{m}r_{+}^{3}}{\left(r_{+}^{2}+q_{0}q_{m}\right)^{5/2}}\right).$
(41)

It is noteworthy that both $m_{0}$ and $q_{0}$ serve as coupling constants in
the Lagrangian, distinct from the integration constants $m$ and $q_{m}$.
However, for the Bardeen-AdS black hole, $m$ and $q_{m}$ are equal to $m_{0}$
and $q_{0}$, respectively, leading to specific thermodynamical issues. From
Eq. (36), it becomes evident that identifying $m$ with $m_{0}$ leads to a
contradiction between the entropy-area law and the Hawking temperature, as
well as inconsistencies with other thermodynamical quantities and the first
law. Similarly, identifying $q_{m}$ and $q_{0}$ may introduce inconsistencies
between $\Phi_{m}$ and Eq. (9). Furthermore, setting $m=m_{0}$ and $q=q_{0}$
aligns the first law with that found in Ref. Zhang2016FirstLawandSmarrformula
; Guo:2023RecoverThermodynamicsyRegularBlackHoles . All these observations
suggest that the thermodynamics of Bardeen-AdS black holes differs from that
of general nonlinear electromagnetic field models.

On the other hand, directly incorporating certain integral constants ($m$ and
$q_{m}$) as coupling constants in the action is not natural. To reproduce the
Bardeen-AdS black hole, one must set $m=m_{0}$ and $q_{m}=q_{0}$ in the
Bardeen-AdS-class metric (26). Nonetheless, it is essential to keep in mind
that $m$ relates to the energy, while $m_{0}$ serves as a coupling constant.
Therefore, we regard Bardeen-AdS black holes as specific states where the
energy $M$ of the Bardeen-AdS-class black hole precisely equals $m_{0}/2$
(when $q_{m}=q_{0}$). Hence, we argue that the Bardeen-AdS black holes
represent only particular states within Bardeen-AdS-class black holes, as
discussed in Ref. Huang:2023IntermediateStates .

Based on the self-consistency of the first law, the fundamental
thermodynamical properties of black holes without the extension of phase space
are of significant concern. To this end, we fix $L$, $m_{0}$, $q_{0}$, and
focus on the ensemble with fixed $Q_{m}=q_{0}$ for simplification. Thus, the
metric takes the form of

$\displaystyle
ds^{2}=-f\left(r\right)dt^{2}+\frac{1}{f\left(r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ $\displaystyle
f\left(r\right)=\frac{r^{2}}{L^{2}}+1-\frac{m-m_{0}}{r}-\frac{m_{0}r^{2}}{\left(r^{2}+q_{0}^{2}\right)^{3/2}}.$
(42)

The first law of thermodynamics for the Bardeen-AdS-class black hole then
reduces to

$dM=T_{H}dS\quad\Longleftrightarrow\quad dF=-SdT_{H},$ (43)

where

$T_{H}=\frac{1}{4\pi}\left(\frac{1}{r_{+}}+3\frac{r_{+}}{L^{2}}-3m_{0}r_{+}\sqrt{\frac{q_{0}^{4}}{\left(q_{0}^{2}+r_{+}^{2}\right)^{5}}}\right).$
(44)

It is important to note that if $Q_{m}$ and $q_{0}$ are regarded as identical,
the magnetic potential becomes $\Phi_{m}+\phi_{q}$, which deviates from the
conventional definition (9). However, under this identification, the reduced
first law of thermodynamics (43) is self-consistent because that $Q_{m}$ and
$q_{m}$ remain unchanged. Furthermore, black hole states and phase transitions
can be properly discussed, and we present intriguing results in Sec. IV.

##  IV Phase States and Phase Transitions of Bardeen-AdS-class Black Holes

When $Q_{m}=q_{0}$, the Bardeen-AdS black hole emerges from the Bardeen-AdS-
class black holes provided that the condition $M=m_{0}/2$ is satisfied. This
condition highlights that Bardeen-AdS black holes can be identified by
specific points (satisfying $m=m_{0}$) on certain thermodynamical
characteristic curves of the Bardeen-AdS-class black holes. However, it is
important to note that $m=m_{0}$ could correspond to a non-singular spacetime
without event horizon, known as a pure Bardeen-AdS spacetime (state). To
address this distinction, we examine the relevant equation

$f\left(r,m_{0}\right)=\frac{r^{2}}{L^{2}}+1-\frac{m_{0}r^{2}}{\left(r^{2}+q_{0}^{2}\right)^{3/2}}=0.$
(45)

Let us first list a critical parameter

$\frac{\hat{m}_{01}}{q_{0}}=\frac{\sqrt{6}\left(6-L^{2}/q_{0}^{2}+\sqrt{\left(L^{2}/q_{0}^{2}\right)^{2}+24L^{2}/q_{0}^{2}}\right)^{5/2}}{9\left(-L^{2}/q_{0}^{2}+\sqrt{\left(L^{2}/q_{0}^{2}\right)^{2}+24L^{2}/q_{0}^{2}}\right)^{2}},$
(46)

which delineates the “boundary” between the Bardeen-AdS black hole state and
pure Bardeen-AdS state. Additionally, the relation $m_{0}=\hat{m}_{01}$ is
shown in Fig. 2 and Fig. 4. Specifically, when $m_{0}\geq\hat{m}_{01}$, the
case $m=m_{0}$ signifies the presence of the Bardeen-AdS black hole.
Otherwise, for $m_{0}<\hat{m}_{01}$, it denotes the pure Bardeen-AdS state. In
general, different values of $m_{0}$ give rise to two distinct types of
Bardeen-AdS-class black holes. Type I contains the Bardeen-AdS black hole and
type II contains the pure Bardeen-AdS spacetime. Subsequent studies will
separately consider and explore the thermodynamics of these two types.

Figure 2: The parameter regions for the characteristic of the Bardeen-AdS-
class black hole. The parameter space is divided into three regions by two
curves. The blue bold curve and the green thin curve represent
$m_{0}=\hat{m}_{01}$ given by Eq. (46) and $m_{0}=\hat{m}_{02}$ given by Eq.
(56), respectively.

###  IV.1 Type I

To explore the thermodynamical properties of the Bardeen-AdS-class black
holes, let us examine $f(r)=0$, i.e.,

$\frac{r_{h}^{2}}{L^{2}}+1-\frac{m_{0}r_{h}^{2}}{\left(r_{h}^{2}+q_{0}^{2}\right)^{3/2}}=\frac{m-m_{0}}{r_{h}}.$
(47)

Consequently, $r_{h}$ is identified by the intersection of curves $A_{1}(r)$
and $A_{2}(r)$ given by

$A_{1}(r)=\frac{r^{2}}{L^{2}}+1-\frac{m_{0}r^{2}}{\left(r^{2}+q_{0}^{2}\right)^{3/2}},\
A_{2}\left(r\right)=\frac{m-m_{0}}{r}.$ (48)

A schematic representation of $A_{1}(r)$ and $A_{2}(r)$ is depicted in Fig. 3.
Since the Bardeen-AdS-class black holes contain the Bardeen-AdS black holes,
equation $A_{1}(r)=0$ must have roots. Hence, the black curve in Fig. 3
intersect with the $r/q_{0}$-axis. For $m-m_{0}\neq 0$, $A_{2}(r)$ behaves as
an inversely proportional function, suggesting that the blue curve ($m>m_{0}$)
may have one, two, or three intersection points, while the red curve
($m<m_{0}$) may have zero, one, or two intersection points with the black
curve. Moreover, as $m$ varies, there can be two instances where curve
$A_{1}(r)$ tangentially intersects curve $A_{2}(r)$, indicating that
$T(r_{h})$ will exhibit two zero points. Through analysis, it is deduced that
the maximum zero point, denoted as $r_{+m}$, also represents the minimum value
of the outer horizon radius $r_{+}$. Consequently, the allowed range for
$r_{+}$ can be expressed as $[r_{+m},\infty)$. Combining with all these
results, the Hawking temperature reads

$\displaystyle
T_{H}\left(r_{+}\right)=\left.T\left(r_{h}\right)\right|_{r_{h}=r_{+}}=\frac{1}{4\pi}\left(\frac{1}{r_{+}}+3\frac{r_{+}}{L^{2}}-3m_{0}r_{+}\sqrt{\frac{q_{0}^{4}}{\left(q_{0}^{2}+r_{+}^{2}\right)^{5}}}\right),r_{+}\geq
r_{+m}.$ (49)

Figure 3: The sketch to characterize the number of the horizon. The black
curve represents the function $A_{1}(r)$. The red, green, and blue curves
represent the function $A_{2}(r)$ for $m-m_{0}<0$, $m-m_{0}=0$, and
$m-m_{0}>0$, respectively. Specifically, we set $m_{0}=6q_{0}$ and $L=2q_{0}$.
The red, green, and blue curves are plotted by $m=5.5q_{0}$, $6q_{0}$, and
$6.5q_{0}$, respectively.

For $r_{+}\rightarrow\infty$ and $r_{+}\rightarrow r_{+m}$, the asymptotic
behaviors of $T_{H}(r_{+})$ are

$\displaystyle\ T_{H}\rightarrow\frac{3r_{+}}{4\pi L^{2}},$ (50)
$\displaystyle\
T_{H}\rightarrow\left.\partial_{r_{+}}T_{H}\right|_{r_{+}=r_{+m}}\left(r_{+}-r_{+m}\right),$
(51)

respectively. Meanwhile $\partial_{r_{+}}T_{H}>0$ at $r_{+}=r_{+m}$ is always
satisfied. This implies that the number of stable black hole phase states
exceeds the number of unstable states by one. To further explore the black
hole states, let us focus on the critical points determined by the following
equations

$\partial_{r_{h}}T\left(r_{h}\right)=0,\quad\partial_{r_{h}}^{2}T\left(r_{h}\right)=0,$
(52)

which gives

$L=\frac{r_{h}^{2}\sqrt{60r_{h}^{2}-45q_{0}^{2}}}{\sqrt{2q_{0}^{4}-21q_{0}^{2}r_{h}^{2}+12r_{h}^{4}}},\quad
m_{0}=\frac{2\left(q_{0}^{2}+r_{h}^{2}\right)^{9/2}}{60q_{0}^{2}r_{h}^{6}-45q_{0}^{4}r_{h}^{4}}.$
(53)

Thus, the critical relationship among parameters $L$, $m_{0}$, and $q_{0}$ is
clarified, and the curve representing this relationship is plotted as the
black curve in Fig. 4. This curve, delineated by critical points, divides the
parameter space into two regions: the orange and white regions. This partition
indicates that the number of stationary points of $T(r_{h})$ varies between
the parameters of the orange and white regions.

A potential issue arises: since the curve $r_{+}-T_{H}$ represents only a
segment of $r_{h}-T(r_{h})$, the occurrence of critical point behavior on the
$r_{+}-T_{H}$ curve cannot be guaranteed. Nonetheless, $r_{h}$ satisfying Eq.
(52) must be greater than $r_{+m}$, as discussed in Appendix A. Therefore, the
critical point appears exclusively on the $r_{+}-T_{H}$ curve.

Figure 4: Parameter regions for the characteristics of $T(r_{h})$. There are
three stationary points for $T(r_{h})$ in the orange area, and one stationary
point in the white area. The dashed blue curve is plotted by
$m_{0}=\hat{m}_{01}$ (given by Eq. (46)). The region above the dashed blue
curve indicates the Bardeen-AdS black hole within the Bardeen-AdS-class black
hole, while the region below represents the pure Bardeen-AdS black hole. The
dot-dashed green curve is plotted for $m_{0}=\hat{m}_{02}$ (given by Eq.
(56)).

To further examine the thermodynamical behavior within the white and orange
regions in Fig. 4, we set $m_{0}=3q_{0}$ and vary $L$ near $L_{0}\equiv
9.32234q_{0}$ (parameters chosen at the boundary of these two regions in Fig.
4). The corresponding curves of $r_{h}-T$ and $T_{H}-F$ are plotted in Figs. 5
and 6, respectively. Only the solid curves truly represent the Hawking
temperature ($r_{+}-T_{H}$ curve), while the dashed curves represent the
“temperature” for the inner horizons. For the $r_{h}-T(r_{h})$ curves, it
demonstrates that when $L=1.2L_{0}$ (in the orange region), there are three
stationary points, whereas at $L=0.8L_{0}$ (in the white region), there is
only one stationary point. These observations suggest that in the orange
region, the $r_{h}-T(r_{h})$ curve exhibits three stationary points, while in
the white region, there is only one stationary point. Concerning the
$r_{+}-T_{H}$ curve, the orange region corresponds to two stationary points,
while the white region to none. Similar to the Reissner-Nordstrom (RN) AdS
black hole Chamblin:1999CatastrophicHolography ; Chamblin:1999ChargedAdSBH ;
Kubiznak:2012PVCritical ; WeiLiuSci , the Bardeen-AdS-class black hole can
exist in three states: a stable small black hole, an unstable intermediate
black hole, and a stable large black hole. Furthermore, as the temperature
increases from zero, there is a phase transition from the small black hole
state to the large black hole state.

Figure 5: The $r_{h}-T(r_{h})$ curves of Bardeen-AdS-class black hole. Here,
we set $m_{0}=3q_{0}$ and $L_{0}\equiv 9.32234q_{0}$. The solid curves
represent the Hawking temperature, while the dashed curves represent the
“temperature” for the inner horizon. The red point on the curve denotes the
Bardeen-AdS black hole state.  Figure 6: The $T_{H}-F$ curves of Bardeen-AdS-
class black hole. Here, we set $m_{0}=3q_{0}$ and $L_{0}\equiv 9.32234q_{0}$.
The red points on the curves denote the Bardeen-AdS black hole states.

In Figs. 5 and 6, it is evident that the Bardeen-AdS black hole state is
depicted as the small black hole state. However, the Bardeen-AdS black hole
can also exist in intermediate or large black hole states, as illustrated in
Fig. 7. Thus, the Bardeen-AdS black hole represents an intermediate state
along the thermodynamical characteristic curves, as discussed in Ref.
Huang:2023IntermediateStates . Moreover, in Fig. 3, a positive correlation
between $m$ and $r_{+}$ is observed. Consequently, it naturally follows that
Bardeen-AdS black holes exist at the junction of Bardeen-AdS-class black holes
with $m>m_{0}$ and $m<m_{0}$.

Figure 7: The $r_{h}-T(r_{h})$ curves of Bardeen-AdS-class black hole. Here,
we set $L=15q_{0}$, and plot three cases with $m_{0}=10q_{0},\ 6.5q_{0}\text{
and }3q_{0}$. The solid curves denote the Hawking temperature, while the
dashed curves represent the “temperature” for the inner horizon. The black
points on the curves signify the presence of the Bardeen-AdS black hole state.

Next, let us briefly explore the thermodynamical properties of the Bardeen-AdS
black hole. As an intermediate state, its thermodynamical properties should be
derived from the constraints of the Bardeen-AdS-class black hole. Reduced from
Eq. (44), the Hawking temperature of Bardeen-AdS black hole is given by

$T_{\text{Bardeen}}=\frac{3r_{+}^{4}/L^{2}+r_{+}^{2}-2q_{0}^{2}}{4\pi
r_{+}\left(q_{0}^{2}+r_{+}^{2}\right)},$ (54)

where $r_{+}$ is the largest solution of Eq. (45). In addition, $r_{+}$ has a
lower bound
$\frac{1}{6}\sqrt{-L^{2}+\sqrt{L^{2}\left(L^{2}+24q_{0}^{2}\right)}}$, which
corresponds to the extreme Bardeen-AdS black hole. The heat capacity of the
Bardeen-AdS black hole can be derived as

$C_{\text{Bardeen}}=\left.T_{H}\frac{\partial S}{\partial
T_{H}}\right|_{m_{0}=m\left(r_{+}\right)}=\frac{2\pi
r_{+}^{2}\left(r_{+}^{2}+q_{0}^{2}\right)\left(3r_{+}^{4}+L^{2}r_{+}^{2}-2L^{2}q_{0}^{2}\right)}{3r_{+}^{6}+\left(18q_{0}^{2}-L^{2}\right)r_{+}^{4}+10L^{2}q_{0}^{2}r_{+}^{2}-4L^{2}q_{0}^{4}},$
(55)

with a typical curve shown in Fig. 8. Due to the inequality
$T_{\text{Bardeen}}>0$, the numerator of Eq. (55) must be non-negative, so
only the denominator needs to be considered. Upon analyzing the denominator,
we find that $\hat{L}=12.32q_{0}$ is a critical value. When $L\leq\hat{L}$,
the heat capacity is always positive, indicating stable states for the
Bardeen-AdS black hole, which can only exist as small or large black hole
states. Conversely, when $L>\hat{L}$, the heat capacity can be either positive
or negative, depending on the radius of the Bardeen-AdS black hole. It is also
important to note that the singularity of the heat capacity does not imply a
phase transition of black holes. This is because in the preceding discussion
of phase transitions, $m_{0}$ remains unchanged, while the curves in Fig. 8
are plotted by simultaneously varying both $m_{0}$ and $r_{+}$.

Figure 8: The heat capacity for the Bardeen-AdS black hole. Here,
$\hat{L}\approx 12.32q_{0}$ is obtained by analyzing the denominator on the
right-hand side of Eq. (55). Specifically, the curves corresponding to
$L=1.05\hat{L}$, $L=\hat{L}$, and $L=0.95\hat{L}$ are depicted as the orange
thick curve, blue thin curve, and green dashed curve, respectively.

###  IV.2 Type II

As shown above, the Bardeen-AdS black holes can be treated as intermediate
states among the Bardeen-AdS-class black holes. However, for the Bardeen-AdS-
class black holes with $m_{0}<\hat{m}_{01}$, the pure Bardeen-AdS state will
replace the Bardeen-AdS black hole and become the “vacuum state”. In this
section, we will discuss these special black hole from thermodynamical
perspective.

First, to determine the range of $r_{+}$, we need to examine the horizon of
the black hole. This requires studying the solution of Eq. (47) or
$A_{1}(r)=A_{2}(r)$. The Bardeen-AdS black hole state does not belong to any
state within the Bardeen-AdS-class black hole, implying that the equation
$A_{1}(r)=0$ has no roots. This ultimately defines the parameter range of
$m_{0}$ as $m_{0}<\hat{m}_{01}$. In this case, by solving the equations
$A_{1}(r)=A_{2}(r)$, $\partial_{r}A_{1}(r)=\partial_{r}A_{2}(r)$, and
$\partial_{r}^{2}A_{1}(r)=\partial_{r}^{2}A_{2}(r)$, we identify another
critical value $\hat{m}_{02}$, which is expressed as

$\frac{\hat{m}_{02}}{q_{0}}=\frac{\left(30-3L^{2}/q_{0}^{2}+\sqrt{3L^{2}/q_{0}^{2}\left(3L^{2}/q_{0}^{2}+40\right)}\right)^{7/2}}{225\sqrt{30}\left(-3L^{2}/q_{0}^{2}+\sqrt{3L^{2}/q_{0}^{2}\left(3L^{2}/q_{0}^{2}+40\right)}\right)^{2}}.$
(56)

The relations $m_{0}=\hat{m}_{01}$ and $m_{0}=\hat{m}_{02}$ are shown in Fig.
2 and Fig. 4. Utilizing these two critical values, $\hat{m}_{01}$ and
$\hat{m}_{02}$, we provide schematic diagrams of the event horizon shown in
Fig. 9. The absence of intersections between the black curve for $A_{1}(r)$
and the $r/q_{0}$-axis suggests that the behavior of the horizon is quite
complex. However, some conclusions are clear: only when $m>m_{0}$, the
spacetime solution is a black hole, and the horizon requires further
discussion; when $m=m_{0}$, the spacetime manifests as a non-singular state,
i.e., pure Bardeen-AdS state; when $m<m_{0}$, naked singularities exist in the
spacetime, which we will not consider.

Let us turn to the situation of the horizon for the Bardeen-AdS-class black
hole shown in Fig. 9. In the case of $\hat{m}_{02}<m_{0}<\hat{m}_{01}$, this
black hole can exhibit one, two, or three horizons. Importantly, the radius
range for the outer event horizon is $(0,x_{1}q_{0})\cup(x_{3}q_{0},\infty)$
rather than $(0,\infty)$, where $x_{1}$ and $x_{3}$ are the intersection and
tangent points of the blue curve and the black curve, respectively.
Conversely, in the case of $0<m_{0}\leq\hat{m}_{02}$, this black hole can only
possess a single horizon, with $r_{+}$ ranging from 0 to $\infty$. Given these
distinctions, it is possible to address these two scenarios for black hole
thermodynamics separately.

Figure 9: The sketch to characterize the number of horizon. The black curves
represent the function $A_{1}(r)$, while the red and blue curves represent the
function $A_{2}(r)$. Specifically, we set $L_{0}=2q_{0}$, and we can derive
$\hat{m}_{02}\approx 2.51q_{0}$ and $\hat{m}_{01}\approx 3.53q_{0}$ by Eqs.
(46) and (56). (a) $m_{0}=1.2\hat{m}_{02}$. We plot the black curve for
$A_{1}(r)$, and the red and blue curves for $A_{2}(r)$ corresponding to
$m\approx 3.27q_{0}$ and $m\approx 3.19q_{0}$, respectively. (b)
$m_{0}=\hat{m}_{02}$. The black curve is for $A_{1}(r)$, and the blue curve is
for $A_{2}(r)$ with $m\approx 2.82q_{0}$. (c) $m_{0}=0.9\hat{m}_{02}$. The
black curve is for $A_{1}(r)$, and the blue curve is for $A_{2}(r)$ with
$m\approx 2.82q_{0}$. Additionally, the green lines along the axis represents
the allowable range for $r_{+}$.

  * •

Case 1: $\hat{m}_{02}<m_{0}<\hat{m}_{01}$. Not all values of
$r_{h}\in(0,\infty)$ correspond to the radius of the outer horizon. The range
of $r_{+}$ is determined by the intersection points of the blue and black
curves, specifically $(0,x_{1}q_{0})\cup(x_{3}q_{0},+\infty)$. For
$r_{+}\in(0,x_{1}q_{0})$, the black hole temperature monotonically decreases
with $r_{+}$, as discussed in Appendix A; for $r_{+}\in(x_{3}q_{0},+\infty)$,
the temperature exhibits the asymptotic behaviors described by

$\displaystyle\ T_{H}\rightarrow\frac{3r_{+}}{4\pi L^{2}}\ \text{and}\
\partial_{r_{+}}T_{H}>0,$ (57) $\displaystyle\
T_{H}\rightarrow\left.\partial_{r_{+}}T_{H}\right|_{r_{+}=x_{3}q_{0}}\left(r_{+}-x_{3}q_{0}\right)\
\text{and}\ \partial_{r_{+}}T_{H}>0,$ (58)

as $\ r_{+}\rightarrow\infty$ or $x_{3}q_{0}$, respectively. Thus, for
$r_{+}\geq x_{3}q_{0}$, the number of stable black hole states exceeds that of
unstable states by one, contingent upon the critical points detailed in Eq.
(52). The $r_{h}-T(r_{h})$ curves are illustrated in Fig. 10, revealing four
distinct states: the unstable tiny black hole state (red curve), the stable
small black hole state (orange curve), the unstable intermediate black hole
state (green curve), and the stable large black hole state (blue curve).
Additionally, there exists a pure Bardeen-AdS state without an event horizon,
characterized by spacetime structure given in

$ds^{2}=-\left(\frac{r^{2}}{L^{2}}+1-\frac{m_{0}r^{2}}{\left(r^{2}+q_{0}^{2}\right)^{3/2}}\right)dt^{2}+\left(\frac{r^{2}}{L^{2}}+1-\frac{m_{0}r^{2}}{\left(r^{2}+q_{0}^{2}\right)^{3/2}}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (59)

By performing a Euclidean on-shell integration, the free energy of the pure
Bardeen-AdS state is determined as

$F_{\text{Bardeen}}=\frac{1}{2}m_{0}.$ (60)

Notably, as $m_{0}$ approaches zero, the system reduces to the pure AdS
spacetime with the free energy tending to zero. The difference in the free
energy between the Bardeen-AdS-class black hole state and the pure Bardeen-AdS
state is expressed

$\Delta
F=\frac{1}{4}\left(r_{+}-\frac{r_{+}^{3}}{L^{2}}-\frac{m_{0}r_{+}^{3}\left(2r_{+}^{2}-q_{0}^{2}\right)}{\left(r_{+}^{2}+q_{0}^{2}\right)^{5/2}}\right).$
(61)

Here, $\Delta F=0$ signifies a possible phase transition between the pure
Bardeen-AdS state and the black hole state. Additionally, several phase
transition diagrams are presented in Fig. 10. As the temperature increases
from zero, two types of phase transitions are observed: one similar to the
Hawking-Page transition from the pure Bardeen-AdS state to the black hole
state, and another resembling the transition from small to large black hole
states, similar to RN-AdS black holes. Notably, at zero temperature, there are
no the unstable tiny black hole states. Additionally, the points for zero
temperature on the dashed segment indicate the coincidence of two inner
horizons.

(a) $L=10q_{0}$, $m_{0}\approx 2.05q_{0}$ (b) $L=8q_{0}$, $m_{0}\approx
2.05q_{0}$ (c) $L=7q_{0}$, $m_{0}\approx 2.05q_{0}$ (d) $L=10q_{0}$,
$m_{0}\approx 2.05q_{0}$

(e) $L=8q_{0}$, $m_{0}\approx 2.05q_{0}$ (f) $L=7q_{0}$, $m_{0}\approx
2.05q_{0}$

Figure 10: The characteristic curve ($r_{h}-T_{H}$ and $T_{H}-\Delta F$) of
the Bardeen-AdS-class black hole phase states. The solid curves represent the
relationship between $r_{+}$ and the Hawking temperature, while the dashed
curves are just the $T(r_{h})$ functions. The red, orange, green, and blue
curves represent the tiny, small, medium, and large black hole states,
respectively. The black curves represent the Bardeen-AdS states. Here, the
parameters are $L=8q_{0}$ and $m_{0}\approx 2.05q_{0}$ satisfying Eq. (52) for
critical points.

  * •

Case 2: $0<m_{0}\leq\hat{m}_{02}$. Bardeen-AdS-class black holes possess only
one horizon, thus the range of $r_{+}=r_{h}$ is $(0,\infty)$. The boundary
behaviors of $T_{H}$ are described by

$\displaystyle\ T_{H}\rightarrow\frac{3r_{+}}{4\pi L^{2}}\ \text{and}\
\partial_{r_{+}}T_{H}>0,$ (62) $\displaystyle\ T_{H}\rightarrow\frac{1}{4\pi
r_{+}}\ \text{and}\ \partial_{r_{+}}T_{H}=-\frac{1}{4\pi r_{+}^{2}},$ (63)

for $\ r_{+}\rightarrow\infty$ and $0^{+}$, respectively. This indicates that
the number of stable and unstable black hole states is equal. The specific
number of states depends on Eq. (52) for the critical points. Similar to the
case $\hat{m}_{02}<m_{0}<\hat{m}_{01}$, there can be up to four black hole
states and one pure Bardeen-AdS state. Utilizing Eq. (52), we find that the
number of stationary points of the function $T_{H}(r_{+})$ can be one, two, or
three. The temperature and free energy are displayed in Fig. 11.

Unlike the situation where $\hat{m}_{02}<m_{0}<\hat{m}_{01}$, the
$r_{+}-T_{H}$ and $T_{H}-\Delta F$ curves are continuous and complete. This is
because, in this scenario, $r_{h}$ and $r_{+}$ are entirely consistent.
Furthermore, at the threshold point $m_{0}=\hat{m}_{02}$, both the tiny black
hole phase and the small black hole phase emerge at zero temperature.
Moreover, in this scenario, we have $T(r_{h})=\partial_{r_{h}}T(r_{h})=0$,
implying the overlap of the black hole’s three horizons. Additionally, when
$m_{0}$ is strictly less than $\hat{m}_{02}$, the black hole state will only
be excited from finite temperatures.

(a) $L\approx 7.67q_{0}$, $m_{0}\approx 1.39q_{0}$ (b) $L\approx 7.67q_{0}$,
$m_{0}\approx 1.85q_{0}$ (c) $L\approx 7.67q_{0}$, $m_{0}\approx 1.00q_{0}$
(d) $L\approx 7.67q_{0}$, $m_{0}\approx 1.39q_{0}$

(e) $L\approx 7.67q_{0}$, $m_{0}\approx 1.85q_{0}$ (f) $L\approx 7.67q_{0}$,
$m_{0}\approx 1.00q_{0}$

Figure 11: The characteristic curve ($r_{h}-T_{H}$ and $T_{H}-\Delta F$) for
the Bardeen-AdS-class black hole phase states. The solid curves represent the
relationship between $r_{+}$ and the Hawking temperature, while the dashed
curves represent the $T(r_{h})$ functions. The red, orange, green, and blue
curves represent the tiny, small, medium, and large black hole states,
respectively. The black curves represent the Bardeen-AdS state. Here, the
parameters are $L\approx 7.67q_{0}$ and $m_{0}\approx 1.85q_{0}$ satisfying
Eq. (52) and $m_{0}=\hat{m}_{02}$.

From the discussions of above two cases, it emerges that in the absence of the
Bardeen-AdS black hole, the Bardeen-AdS-class black hole can manifest an
additional tiny black hole phase and the pure Bardeen-AdS state. This tiny
black hole phase exhibits similar thermodynamical characteristics to the
Schwarzschild black hole, specifically a negative heat capacity. This behavior
originates from the properties of the metric (26) near $r=0$ for $m>m_{0}$.

##  V Conclusion and Discussion

The Bardeen black hole, a significant model of regular black holes, can be
explained via nonlinear electromagnetic fields. However, its thermodynamic
properties have been the subject of various controversies. To address this
issue, more general black holes need to be studied, with regular black holes
considered as intermediate black hole states or as pure regular spacetimes
without event horizon.

In our work, by using new parameters, we re-modified the action, leading to
the derivation of the metric for the Bardeen-AdS-class black hole. This
generalized black hole is a standard solution of nonlinear electromagnetic
fields and does not involve the “coupling constant issue”, allowing for
consistent construction of the thermodynamics. We focused on the ensemble with
fixed $Q_{m}=q_{0}$. In this context, the Bardeen-AdS-class black hole with
the parameter $m=m_{0}$ corresponds to a regular spacetime. Interestingly,
this regular spacetime acts as an intermediate phase state of the Bardeen-AdS-
class black hole in two distinct ways: one is the Bardeen-AdS black hole
corresponding to the parameter range $m_{0}\geq\hat{m}_{01}$, and the other is
the pure Bardeen-AdS spacetime (without an event horizon) corresponding to the
parameter range $0<m_{0}<\hat{m}_{01}$. As an RN gravitational system, the
latter represents a novel spacetime state, bearing some resemblance to the
pure AdS spacetime.

Moreover, whether in the presence of the Bardeen-AdS black hole or the pure
Bardeen-AdS spacetime, the thermodynamics of Bardeen-AdS-class black holes
exhibits significant differences. In the former case, there are three black
hole states: the large, intermediate, and small states. A phase transition
occurs between the small and large black hole states, similar to that of the
RN-AdS black holes. In the latter scenario, there are four black hole phase
states and one regular spacetime state: the tiny, small, intermediate, and
large black hole states, along with the pure Bardeen-AdS state. Besides the
phase transition between the small and large black holes, there is an
additional phase transition between the pure Bardeen-AdS state and black hole
states. This phase transition resembles the Hawking-Page transition, which
occurs between a non-black hole phase and a black hole phase. From the AdS/CFT
perspective, such a phase transition may correspond to a confinement-
deconfinement transition Witten:1998Confinement .

There is another critical value, $\hat{m}_{02}$, for the Bardeen-AdS-class
black holes with the pure Bardeen-AdS state. This critical value characterizes
the connectedness of the domain for the radius of the outer horizon. When
$\hat{m}_{02}<m_{0}<\hat{m}_{01}$, the domain of $r_{+}$ splits into two
separate regions. In this case, the $r_{+}-T_{H}$ and $T_{H}-F$ curves are
discontinuous, indicating that the thermodynamical topology
Wei:2022ThermodynamicDefects  of black holes may undergo transitions. When
$0<m_{0}<\hat{m}_{02}$, the domain of $r_{+}$ remains connected. This
characteristic arises from the nature of the multiple horizons in black holes
and is not unique to Bardeen-AdS-class black holes.

The Bardeen-AdS-class black hole is an extension of the Bardeen-AdS black
hole, designed to ensure the naturalness and self-consistency of its
thermodynamics. The presence of the Bardeen-AdS black hole as a state within
the Bardeen-AdS-class black hole framework significantly affects the
properties of the horizon and the thermodynamical characteristics. This
underscores the importance of the Bardeen-AdS black hole, which may require
further explanation and investigation.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Grants No. 12075103 and No. 12247101)

##  Appendix A The critical point for $r_{h}-T(r_{h})$ curve

In this appendix, we present a proposal using Fig. 1. In the context of the
Bardeen-AdS-class black hole ($q_{m}=q_{0}$ case), given that $T(r_{h})=0$ has
two roots, $r_{1}$ and $r_{2}$($r_{1}<r_{2}$), the radius of horizon $r_{h}$,
which satisfies the equations $\partial_{r_{h}}T(r_{h})=0$ and
$\partial_{r_{h}}^{2}T(r_{h})=0$ (see Eq. (52)), is greater than $r_{2}$.

Here is the proof of this proposal. In Sec. IV.1, Eq. (52) can be simplified
to Eq. (53), where we observe that the range for $r_{hc}$, the value of
$r_{h}$ satisfying Eq. (52), is given by

$r_{hc}>\sqrt{\frac{1}{24}\left(21+\sqrt{345}\right)}q_{0},$ (64)

since $m_{0}$ and $L$ are positive real numbers. At the critical point, the
temperature is given by

$T_{c}=T(r_{hc})=\frac{3r_{hc}^{2}-4q_{0}^{2}}{2\pi
r_{hc}\left(4r_{hc}^{2}-3q_{0}^{2}\right)},$ (65)

Due to Eq. (64), we have $T_{c}>0$. Moreover, through analysis, it can be
determined that Eq. (32) provides the range of $r_{1}$, specifically
$r_{1}\leq r_{*}$. By analyzing of Eq. (32), we find that $r_{*}$ has an upper
bound $\sqrt{6}q_{0}/{3}$. Consequently, we have

$r_{1}\leq
r_{*}<\frac{\sqrt{6}}{3}q_{0}<\sqrt{\frac{1}{24}\left(21+\sqrt{345}\right)}q_{0}<r_{hc},$
(66)

which implies the root of Eq. (52) is not in $(0,r_{1})$. Moreover,
considering the fact $T_{c}>0$, the root of Eq. (52) is also not within
$[r_{1},r_{2}]$. Therefore, the root of Eq. (52) can only be within
$(r_{2},\infty)$. This completes our proof.

Using this proposal, we can determine the monotonicity of the $r_{h}-T(r_{h})$
curve in the interval $(0,r_{1})$ shown in Fig. 1. Considering the behavior of
$T(r_{h})$ at $r_{h}=0$ and $r=r_{1}$, as given by,

$\displaystyle\left.T\left(r_{h}\right)\right|_{r_{h}\rightarrow
0^{+}}=+\infty,\left.\partial_{r_{h}}T\left(r_{h}\right)\right|_{r_{h}\rightarrow
0^{+}}=-\infty,$ (67)
$\displaystyle\left.T\left(r_{h}\right)\right|_{r_{h}\rightarrow
r_{1}}=0,\left.\partial_{r_{h}}T\left(r_{h}\right)\right|_{r_{h}\rightarrow
r_{1}}<0.$ (68)

and in conjunction with this proposal, we see that the number of stationary
points of $T(r_{h})$ in the interval $(0,r_{1})$ remains unchanged under
continuous variation of the parameters $m_{0}$, $q_{0}$, and $L$. In Fig. 4,
the parameter region above the green line indicates that $T(r_{h})$ has two
zeros. Since this parameter region is path-connected (continuous transitions
between different parameter values), we can confirm the monotonicity of
$T(r_{h})$ in the interval $(0,r_{1})$ by examining a specific point within
it. Moreover, considering the monotonic behavior of $T(r_{h})$ with $m=3$ and
$L=5$, we can conclude that $T(r_{h})$ monotonically decreases in the interval
$(0,r_{1})$.

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