# Control Barrier Function based Quadratic Programs Introduce Undesirable
Asymptotically Stable Equilibria

Matheus F. Reis, A. Pedro Aguiar and Paulo Tabuada
Matheus F. Reis and A. Pedro Aguiar are with the Department of Electrical and
Computer Engineering, Faculty of Engineering, University of Porto, Portugal,
<EMAIL_ADDRESS><EMAIL_ADDRESS>Paulo Tabuada is with the
Electrical and Computer Engineering Department at the University of
California, Los Angeles<EMAIL_ADDRESS>The work of the last author was
partially supported by the NSF award 1645824 and by the CONIX Research Center,
one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program
sponsored by DARPA.

###### Abstract

Control Lyapunov functions (CLFs) and control barrier functions (CBFs) have
been used to develop provably safe controllers by means of quadratic programs
(QPs), guaranteeing safety in the form of trajectory invariance with respect
to a given set. In this manuscript, we show that this framework can introduce
equilibrium points (particularly at the boundary of the unsafe set) other than
the minimum of the Lyapunov function into the closed-loop system. We derive
explicit conditions under which these undesired equilibria (which can even
appear in the simple case of linear systems with just one convex unsafe set)
are asymptotically stable. To address this issue, we propose an extension to
the QP-based controller unifying CLFs and CBFs that explicitly avoids
undesirable equilibria on the boundary of the safe set. The solution is
illustrated in the design of a collision-free controller.

##  I INTRODUCTION

While the design of asymptotically stabilizing controllers has been
extensively studied in control Lyapunov theory [8], the design of controllers
capable of enforcing invariance of a particular set of states has been the
subject of study in the context of control barrier functions (CBFs) [2]. The
concept of barrier functions was initially used in constrained optimization
[4] due to their ability to provably establish invariance properties of sets.
In [12] and [11], barrier certificates were introduced as a tool to formally
prove safety of nonlinear and hybrid systems. In the seminal work [3], the
concept of CBFs was introduced with a novel, less conservative form of barrier
constraint, allowing the barrier function value to grow when far away from the
boundary of the safe set. This extension allows the system trajectory to
approach the boundary of the safe set without ever leaving it. Additionally,
[3] also introduced the idea of unifying CBFs with Control Lyapunov Functions
(CLFs) [13] through the use of quadratic programs (QPs), effectively combining
safety and stabilization requirements in a single, elegant framework suitable
for control. The optimization-based framework introduced by [3] was followed
by a series of related works demonstrating its applicability, such as
collision-free control for multi-robot systems [15], persistent control for
teams of mobile robots [9], bipedal robot walking [5], safe learning of system
dynamics [16], and adaptive safety using CBFs in the presence of parametric
model uncertainty [14].

Figure 1: System trajectories for the QP-based control of the integrator with
a circular obstacle. The undesirable asymptotically stable equilibrium is
shown in red.

However, the QP-based framework shared across these works suffers from an
important limitation. While it guarantees invariance of the system
trajectories with respect to the safe set as a hard constraint, it softens the
stabilization objective in other to maintain the feasibility of the
constrained optimization problem everywhere. In this work, we demonstrate
using a simple example that this methodology can introduce equilibria other
the minimum of the CLF into the closed-loop system, and that these undesirable
equilibria can even be asymptotically stable111An illustration of this
phenomena can be seen in Fig. 1 for the integrator system with a convex
obstacle. Note that even for the initial condition $x_{0}=(4,4)$, which is far
from the obstacle, the trajectory $x(t)$ does not converge to the origin..
While [6] proposed a modification of the original QP problem seeking to
guarantee that the CLF is strictly negative when the barrier constraint is
inactive, it still does not address the problem of existence of other types of
asymptotically stable equilibria other then the CLF minimum, as we demonstrate
in this manuscript. In [10], a smooth control law for safety and stabilization
was proposed using a different method based on finding and combining different
weighted centroids of the feasible control set. However, it remains unclear if
this approach actually addresses the problem of undesired equilibria.

This paper adds to the literature in the following important ways. First, it
demonstrates, both theoretically and by means of numerical simulations, that
the QP-based controller with CLF-CBF constraints proposed by [3] introduces
undesired equilibria other than the CLF minimum into the resulting closed-loop
system. Secondly, a sufficient condition under which the resulting undesired
equilibria are asymptotically stable are explicitly derived for the integrator
system. Finally, we propose an extension to the QP-based controller unifying
CLFs and CBFs that explicitly avoids undesirable equilibria on the boundary of
the safe set. The solution is illustrated in the design of a collision-free
controller.

Notation. The operator
$\nabla\\!:C^{1}(\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ is defined as the
gradient $\frac{\partial}{\partial x}$ of a scalar-valued differentiable
function with respect to $x$. We use the notation $L_{f}V$ to denote the Lie
derivative of a differentiable function
$V:\mathbb{R}^{n}\rightarrow\mathbb{R}$ along the vector fields $f$ and $g$,
that is, $L_{f}V=\nabla V^{\mathsf{T}}f\in\mathbb{R}$ and $L_{g}V=\nabla
V^{\mathsf{T}}g\in\mathbb{R}^{1\times m}$, respectively. We denote linear
dependency between two vectors $v,w\in\mathbb{R}^{n}$ by $v\parallel w$.
Define the scaled orthogonal projection $\mathcal{P}_{v}=\lVert
v\rVert^{2}I_{n}-vv^{\mathsf{T}}\in\mathbb{R}^{n\times n}$ for a vector
$v\in\mathbb{R}^{n}$, which is a scaled version of the matrix represention for
the orthogonal projection operator defined over $\mathbb{R}^{n}$. It has the
following useful properties: (i)
$\mathcal{P}_{v}=\mathcal{P}^{\mathsf{T}}_{v}$ (symmetry), (ii)
$\mathcal{P}^{2}_{v}=\lVert v\rVert^{2}\mathcal{P}_{v}$; (iii) the spectrum of
$\mathcal{P}_{v}$ is composed of 0 and $\lVert v\rVert^{2}$ with algebraic
multiplicity 1 and $n-1$, respectively; (iv) $\mathcal{P}_{v}\,z=\lVert
v\rVert^{2}z$ for all $z\in\mathbb{R}^{n}$ on the projective subspace defined
by $v\in\mathbb{R}^{n}$ (that is, such that $z^{\mathsf{T}}v=0$); (v)
$\mathcal{P}_{v}\,w=0$ for all $w\in\mathbb{R}^{n}$ such that $v\parallel w$.
For a matrix $M\in\mathbb{R}^{n\times m}$, the set
$\mathcal{N}(M)=\\{x\in\mathbb{R}^{m}|Mx=0\\}$ denotes the kernel or nullspace
of $M$. The set $\mathcal{SO}(n)$ is the special orthogonal group of dimension
$n$, consisting of all orthogonal matrices $M$, i.e., $M^{\mathsf{T}}M=I_{n}$
of determinant $1$, and the set $\mathfrak{so}(n)$ consists of the
corresponding special orthogonal Lie algebra of $\mathcal{SO}(n)$. The
operator $\wedge:\mathbb{R}^{\frac{1}{2}n(n-1)}\rightarrow\mathfrak{so}(n)$ is
the skew-symmetric map from the real vector space of dimension
$\mathbb{R}^{\frac{1}{2}n(n-1)}$ to the corresponding Lie algebra
$\mathfrak{so}(n)$. For example, for $n=3$,
$\hat{\omega}\in\mathbb{R}^{3\times 3}$ represents the usual cross product
operator. The operator
$\mathcal{O}_{n}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times\frac{n}{2}(n-1)}$
is defined by $\hat{\omega}x=\mathcal{O}_{n}(x)\omega$. For $n=3$, using the
anticommutativity of the cross product, we have
$\hat{\omega}x=-\hat{x}\omega$, and therefore $\mathcal{O}_{3}(x)=-\hat{x}$.

##  II Quadratic Programs for Safety Critical Systems

Consider the nonlinear, control-affine system

$\displaystyle\dot{x}\\!=\\!f(x)\\!+\\!g(x)u$ (1)

where $x\in\mathbb{R}^{n}$ is the system state and $u\in\mathbb{R}^{m}$ is the
control input. The vector fields $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$,
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$ are locally Lipschitz. The
notation of $G(x)=g(x)g(x)^{\mathsf{T}}$ will also prove to be useful.

######  Definition II.1 (CLFs).

A positive definite function $V$ is a control Lyapunov function (CLF) for
system (1) if it satisfies:

$\displaystyle\inf_{u\in\mathbb{R}^{m}}\left[L_{f}V(x)+L_{g}V(x)u\right]\leq-\gamma(V(x))$

where $\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ is a class
$\mathcal{K}$ function [7].

This definition means that there exists a set of controls that makes the CLF
strictly decreasing everywhere outside of the origin. CBFs can be used to
design controllers enforcing stability. The corresponding set of stabilizing
controls is

$\displaystyle
K_{clf}(x)\\!=\\!\\{u\\!\in\\!\mathbb{R}^{m}:L_{f}V(x)\\!+\\!L_{g}V(x)u\leq-\gamma(V(x))\\}\,.$

In constrast, safety can be framed in the context of enforcing invariance of a
particular set of states. Consider the safe set $\mathcal{C}$ defined as the
superlevel set of a continuously differentiable function
$h:\mathcal{D}\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$, as [2]

$\displaystyle\mathcal{C}$
$\displaystyle=\\{x\in\mathcal{D}\subset\mathbb{R}^{n}:h(x)\geq 0\\}$
$\displaystyle\partial\mathcal{C}$
$\displaystyle=\\{x\in\mathcal{D}\subset\mathbb{R}^{n}:h(x)=0\\}$
$\displaystyle\text{int}(\mathcal{C})$
$\displaystyle=\\{x\in\mathcal{D}\subset\mathbb{R}^{n}:h(x)>0\\}\,.$

Let $k(x)$ be a feedback controller such that the closed-loop system

$\displaystyle\dot{x}=f_{cl}(x):=f(x)+g(x)k(x)$ (2)

is locally Lipschitz. Then, for any initial condition $x(0)\in\mathcal{D}$
there exists a maximum interval of existence $I(x(0))=[0,\tau_{max})$ such
that $x(t)$ is the unique solution to (2) on $I(x(0))$ (if $f_{cl}$ is forward
complete, $\tau_{max}=\infty$).

######  Definition II.2 (Safety).

The set $\mathcal{C}$ is forward invariant if, for every $x(0)\in\mathcal{C}$,
$x(t)\in\mathcal{C}$ for all $t\in I(x(0))$. System (2) is safe with respect
to the set $\mathcal{C}$ if $\mathcal{C}$ is forward invariant.

######  Definition II.3 (CBFs).

Let $\mathcal{C}\subset\mathcal{D}\subset\mathbb{R}^{n}$ be the superlevel set
of a continuously differentiable function
$h:\mathcal{D}\rightarrow\mathbb{R}$, then $h$ is a Control Barrier Function
(CBF) for (1) if there exists a locally Lipschitz extended class
$\mathcal{K}_{\infty}$ function 222An extended class $\mathcal{K}_{\infty}$
function $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ is strictly increasing with
$\alpha(0)\\!=\\!0$. $\alpha$ such that

$\displaystyle\sup_{u\in\mathbb{R}^{m}}\left[L_{f}h(x)+L_{g}h(x)u\right]\geq-\alpha(h(x))\quad\forall
x\in\mathcal{D}\,.$

The definition simply means that a CBF $h(x)$ is only allowed to decrease on
the interior of the safe set $\text{int}(\mathcal{C})$, but not on its
boundary $\partial\mathcal{C}$. The set of control values that render
$\mathcal{C}$ forward invariant can be formally defined as

$\displaystyle
K_{cbf}(x)\\!=\\!\\{u\\!\in\\!\mathbb{R}^{m}:L_{f}h(x)\\!+\\!L_{g}h(x)u\\!+\\!\alpha(h(x))\geq
0\\}\,.$

###  II-A Quadratic Program Formulation

The minimum-norm controller proposed by [3] is

$\displaystyle\begin{bmatrix}\,k(x)^{\mathsf{T}}\\!\\!&\\!\\!\delta(x)\,\end{bmatrix}^{\mathsf{T}}$
$\displaystyle\\!\\!=\\!\\!\operatorname*{argmin}_{(u,w)\in\mathbb{R}^{m+1}}\,\lVert
u\rVert^{2}+pw^{2}$ (3) $\displaystyle s.t.\quad
L_{f}V(x)\\!+\\!L_{g}V(x)u\leq-\gamma(V(x))+w$
$\displaystyle\quad\quad\,\,\,L_{f}h(x)+L_{g}h(x)u\geq-\alpha(h(x))$

where $p$ is a positive constant. The objective is to minimize the norm of the
control signal and of an auxiliary relaxation variable $w$, while satisfying
the CLF and CBF constraints. The CBF constraint guarantees that $u\in
K_{cbf}(x)$, keeping the system trajectories invariant with respect to the
safe set. The relaxation variable in the CLF constraint softens the
stabilization objective, maintaining the feasibility of the QP.

In the next section, we show that the closed-loop system (2) with control
$k(x)$ from (3) has undesirable equilibrium points other than the origin, and
that these points can be asymptotically stable.

##  III Analysis of the closed-loop system with QP-based controller

In this section, we investigate some aspects regarding the existence of
equilibrium points on the closed-loop system (2) with $k(x)$ given by (3) and
their stability properties.

###  III-A Existence of equilibrium points

We now present an important result describing the closed-loop equilibria. The
proof is presented in Appendix VII-A.

######  Theorem 1.

The set $\mathcal{E}$ of equilibrium points of the closed-loop system
resulting from the application of the control law (3) into (1) is given by

$\displaystyle\mathcal{E}=\\{0\\}\cup\mathcal{E}_{\text{int}}\cup\mathcal{E}_{\partial\mathcal{C}}$

where $0\in\mathbb{R}^{n}$ is the origin of the state space and

$\displaystyle\mathcal{E}_{\text{int}}$
$\displaystyle\\!=\\!\Big{\\{}x\in{\bf\Omega_{\overline{cbf}}^{clf}}\\!\setminus\\!\\{0\\}|f(x)\\!=\\!p\gamma(V(x))G(x)\nabla
V(x)\Big{\\}}$ (4) $\displaystyle\mathcal{E}_{\partial\mathcal{C}}$
$\displaystyle\\!=\\!\Big{\\{}x\in{\bf\Omega_{cbf}^{clf}}\cap\partial\mathcal{C}\,|\,\mathcal{N}\\!\left(\\!\begin{bmatrix}f(x)^{\mathsf{T}}\\\
\nabla V(x)^{\mathsf{T}}G(x)\\\ \nabla
h(x)^{\mathsf{T}}G(x)\end{bmatrix}^{\\!\\!\mathsf{T}}\right)\\!\setminus\\!\\{0\\}\neq\emptyset\Big{\\}}$
(5)

where $G(x)=g(x)g(x)^{\mathsf{T}}$, $\mathcal{E}_{\text{int}}$ is the set of
interior equilibria and $\mathcal{E}_{\partial\mathcal{C}}$ is the set of
boundary equilibria. The set ${\bf\Omega_{\overline{cbf}}^{clf}}$ denotes the
states where the CLF constraint in (3) is active and the CBF constraint is
inactive, while ${\bf\Omega_{cbf}^{clf}}$ denotes the states where both CLF
and CBF constraints are active:

$\displaystyle{\bf\Omega_{\overline{cbf}}^{clf}}=\Big{\\{}x\in\mathbb{R}^{n}:L_{f}V\\!+\\!\gamma(V)\geq
0\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad$ $\displaystyle
L_{g}VL_{g}h^{\mathsf{T}}(\mathcal{L}_{f}V\\!+\\!\gamma(V))<(L_{f}h\\!+\\!\alpha(h))(p^{-1}\\!+\\!\lVert\mathcal{L}_{g}V\rVert^{2})\Big{\\}}$
$\displaystyle{\bf\Omega_{cbf}^{clf}}=\Big{\\{}x\in\mathbb{R}^{n}:L_{g}VL_{g}h^{\mathsf{T}}\left(\frac{L_{f}h+\alpha(h)}{L_{f}V+\gamma(V)}\right)\leq\lVert
L_{g}h\rVert^{2}\,,$ $\displaystyle
L_{g}VL_{g}h^{\mathsf{T}}\geq\left(\frac{L_{f}h\\!+\\!\alpha(h)}{L_{f}V\\!+\\!\gamma(V)}\right)(\lVert
L_{g}V\rVert^{2}\\!+\\!p^{-1})\Big{\\}}$

######  Remark III.1.

Regarding the boundary equilibria, in general, the existence of a nontrivial
null space for the matrix
$\begin{bmatrix}f(x)\\!&\\!g(x)\mathcal{L}_{g}V(x)^{\mathsf{T}}\\!&\\!g(x)\mathcal{L}_{g}h(x)^{\mathsf{T}}\end{bmatrix}$
implies collinearity among vectors $f(x)$,
$g(x)\mathcal{L}_{g}V(x)^{\mathsf{T}}$ and
$g(x)\mathcal{L}_{g}h(x)^{\mathsf{T}}$. For an integrator $\dot{x}=u$ with a
convex CLF, $\mathcal{E}_{\partial\mathcal{C}}$ is simply

$\mathcal{E}_{\partial\mathcal{C}}=\\{x\in{\bf\Omega_{cbf}^{clf}}\cap\partial\mathcal{C}\,|\,\nabla
V(x)\\!\parallel\\!\nabla h(x)\\}\,.$

###  III-B Stability of equilibrium points

It was already shown that, in general, the origin $x=0$ is not an unique
equilibrium point of the closed-loop system (2) with $k(x)$ given by (3). In
this section, the objective is to estabilish an example showing that some of
these equilibria can be asymptotically stable.

######  Theorem 2.

Consider the integrator $\dot{x}=u$ with $u=k(x)$ given by (3) with a convex
CLF $V(x)$. An equilibrium point
$x^{\star}\in\mathcal{E}_{\partial\mathcal{C}}$ is asymptotically stable if

$\displaystyle H_{V}(x^{\star})-cH_{h}(x^{\star})>0$

where $c\in\mathbb{R}_{\geq 0}$ is the constant satisfying $\nabla
V(x^{\star})=c\nabla h(x^{\star})$ and $H_{V}$, $H_{h}$ are the Hessian
matrices of the CLF and the CBF, respectively.

The proof of this result is presented in Appendix VII-B. In Fig. 1, we
illustrate Theorem 2 by means of an example. We use the CLF
$V(x)=0.5\lambda_{1}x_{1}^{2}+0.5\lambda_{2}x_{2}^{2}$,
$\lambda_{1}>\lambda_{2}$ and CBF $h(x)=0.5\lVert x-x_{c}\rVert^{2}-0.5r^{2}$,
whose superlevel set $h(x)=0$ models the boundary of a circular obstacle with
radius $r$ centered on $x_{c}\in\mathbb{R}^{2}$. The set of boundary
equilibria is given by
$\mathcal{E}_{\partial\mathcal{C}}=\\{x\in{\bf\Omega_{cbf}^{clf}}\cap\partial\mathcal{C}\,|\,\Lambda
x=c(x-x_{c}),\,c\in\mathbb{R}\\}$ in this case. System trajectories starting
close to the top of the obstacle converge to the asymptotically stable
equilibrium point shown in red. Observing the CLF level set at the equilibrium
and the boundary $\partial\mathcal{C}$, note that the local curvature of the
CBF is smaller than the local curvature of $H_{V}(x)$ at the equilibrium, as
expected from Theorem 2.

##  IV Lyapunov Shaping for QP-based Controllers

We now seek to design a stabilizing controller $k(x)$ such that the resulting
closed-loop system (2) does not contain certain types of undesired equilibria.
Consider that a positive-definite, non-radial 333 A radial function
$r:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is defined by the property
$r(Qx)=r(x)$ for all $Q\in\mathcal{SO}(n)$. That is, $r$ is invariant under
rotations around the origin.  reference CLF
$V_{r}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is given.

######  Remark IV.1.

As an example, the reference CLF could be the quadratic form
$V_{r}(x)=\frac{1}{2}x^{\mathsf{T}}\Lambda x$, where
$\Lambda>0\in\mathbb{R}^{n\times n}$ is a diagonal matrix with at least a pair
of distinct eigenvalues.

Next, define another CLF
$V:\mathbb{R}^{n}\times\mathcal{SO}(n)\rightarrow\mathbb{R}$ as

$\displaystyle V(x,Q)=V_{r}(Qx)$ (6)

with $Q\in\mathcal{SO}(n)$. The time derivative of $Q$ is

$\displaystyle\dot{Q}=Q\,\hat{\omega}$ (7)

where $\omega\in\mathbb{R}^{\frac{1}{2}n(n-1)}$ is a virtual control signal
with the dimension of $\mathfrak{so}(n)$. Using (2) and the properties of the
skew-symmetric map $\wedge$, it is possible to show that the time derivative
of $V(x,Q)$ is affine with respect to $u$ and $\omega$.

As pointed out by Remark III.1, the existence of boundary equilibria is
connected to the existence of sets where vectors $f(x)$, $G(x)\nabla V(x,Q)$
and $G(x)\nabla h(x)$ are pairwise collinear. Motivated by this fact, we
design a function that measures the proximity of the trajectories to these
sets:

$\displaystyle\mathcal{D}(x,Q)$ $\displaystyle\\!=\\!\frac{1}{2}\nabla
V(x,Q)^{\mathsf{T}}G\\!\left(\mathcal{P}_{f}\\!+\\!\mathcal{P}_{G\nabla
h}\right)\\!G\nabla V(x,Q)$ (8)

######  Remark IV.2.

Using property (v) of the scaled orthogonal projection, note that (8) is zero
for all the combinations of collinearity conditions under which
$\begin{bmatrix}f(x)\\!\\!&\\!\\!G(x)\nabla V(x,Q)\\!\\!&\\!\\!G(x)\nabla
h(x)\end{bmatrix}$ has a nontrivial null space, which includes both points on
$\mathcal{E}_{\partial\mathcal{C}}$ and on $\mathcal{E}_{\text{int}}$.
Therefore, $\mathcal{D}(x,Q)$ is zero for all
$x\in\mathcal{E}_{\partial\mathcal{C}}\,\cup\,\mathcal{E}_{\text{int}}$,
$Q\in\mathcal{SO}(n)$, and strictly positive elsewhere.

Aiming at avoiding the collinearity conditions imposed by the set
$\mathcal{D}(x,Q)=0$ when the trajectories approach the boundary
$\partial\mathcal{C}$, we define the barrier function candidate

$\displaystyle
h_{\mathcal{D}}(x,Q)=\sigma(h(x))\left(\mathcal{D}(x,Q)-\epsilon\right)$ (9)

where $\epsilon$ is a small positive constant and
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a smooth, positive semi-definite
function such that: (i) $\sigma(0)>0$ and (ii)
$\lim_{t\rightarrow\infty}\sigma(t)=0$. The reason for this selection is to
ensure that $\mathcal{D}(x,Q)\geq\epsilon>0$ when $x$ is close to
$\partial\mathcal{C}$.

In a similar way than for the CLF (6), the time derivative of
$h_{\mathcal{D}}(x,Q)$ is also affine with respect to $u$ and $\omega$. This
fact allows the corresponding inequality constraint on the dynamics of
$h_{\mathcal{D}}(x,Q)$ to be written as an affine inequality constraint in
both $u$ and $\omega$, allowing the use of QPs.

Finally, we propose a modification on the QP-based approach by [3] to achieve
stabilization and safety for (2) without the existence of boundary equilibria.

######  Theorem 3.

Consider the nonlinear system with dynamics given by (1) and full rank $g(x)$.
Assume a reference, positive definite, non-radial CLF $V_{r}(x)$, and the CLF
$V(x,Q)$ given by (6), where $Q\in\mathcal{SO}(n)$ is virtual state with
dynamics given by (7) and $Q(0)=I_{n}$. Additionally, assume a convex CBF
$h(x)$ and $h_{\mathcal{D}}(x,Q)$ given by (9). The QP

$\displaystyle\begin{bmatrix}\,k(x)^{\mathsf{T}}\\!\\!\\!&\\!\\!\\!\omega(x)^{\mathsf{T}}\\!\\!\\!&\\!\\!\\!\delta(x)\,\end{bmatrix}^{\mathsf{T}}\\!\\!=\operatorname*{argmin}_{\mathclap{\begin{subarray}{c}(u,v,w)\in\\\
\mathbb{R}^{m+\frac{1}{2}(n^{2}-n+2)}\end{subarray}}}\,\lVert
u\rVert^{2}\\!+\\!q\lVert v\rVert^{2}\\!+\\!pw^{2}$ (10) $\displaystyle
s.t.\,\,\dot{V}(x,Q,u,v)+\gamma(V(x,Q))$ $\displaystyle\leq w$
$\displaystyle\dot{h}(x,u)+\alpha(h(x))$ $\displaystyle\geq 0$
$\displaystyle\dot{h}_{\mathcal{D}}(x,Q,u,v)+\beta(h_{\mathcal{D}}(x,Q))$
$\displaystyle\geq 0$

with $p,q>0$ and function $\beta\in\mathcal{K}_{\infty}$ renders the set
$\mathcal{C}$ forward invariant and guarantees that no boundary equilibria
exist.

The proof of this result is presented in Appendix VII-C. The first and second
constraints on (10) are the usual CLF and CBF constraints from (3),
guaranteeing stabilization and safety with respect to set $\mathcal{C}$ as
soft and hard constraints, respectively. The third constraint guarantees that
$h_{\mathcal{D}}(x,Q)\geq 0$ everywhere. Since $\omega(x)$ determines the
dynamics of $Q$ (as given by (7)), its effect is of rotating the reference,
non-radial CLF around the origin, ensuring that $\mathcal{D}(x,Q)\geq\epsilon$
when the trajectories are close to the boundary $\partial\mathcal{C}$ (since
$\sigma(h(x))>0$ in this case).

##  V Simulation Results

In this section, we present numerical examples of our approach for the
integrator and different nonlinear systems. We use the same CLF used before as
the reference CLF $V_{r}(x)=0.5\lambda_{1}x_{1}^{2}+0.5\lambda_{2}x_{2}^{2}$,
$\lambda_{1}=6$ and $\lambda_{2}=1$ and CBF $h(x)=0.5\lVert
x-x_{c}\rVert^{2}-0.5r^{2}$, with
$x_{c}=\begin{bmatrix}\,0\\!\\!&\\!\\!3\,\end{bmatrix}^{\mathsf{T}}$ and
$r=1.5$. For the proposed controller on Theorem 3, we have used $p=q=5$,
$\gamma=\alpha=\beta=1$ and $\epsilon=0.1$.

In the integrator case, $f(x)=0$ and $g(x)=I_{n}$, and (8) simplifies to
$\mathcal{D}(x,Q)=\frac{1}{2}\nabla V_{r}(Qx)^{\mathsf{T}}Q\mathcal{P}_{\nabla
h}Q^{\mathsf{T}}\nabla V_{r}(Qx)$. Note that in this case,
$\mathcal{D}(x,Q)=0$ consists of the set of points where the gradients $\nabla
V(x,Q)=Q^{\mathsf{T}}\nabla V_{r}(Qx)$ and $\nabla h(x)$ are collinear. Figure
2 shows different system trajectories for the same initial conditions shown in
Fig. 1. The red equilibrium point on top of the obstacle does not exist for
the closed-loop system with the proposed controller, and all trajectories are
attracted the stable origin instead. We also show the level set of the CLF for
a particular state on a particular trajectory, illustrating that $V(x,Q)$ is
actually a rotated version of the reference CLF $V_{r}(x)$. Our proposed
control rotates $V_{r}(x)$ in order to avoid the trajectory to approach the
set defined by $\mathcal{D}(x,Q)=0$.

Figure 2: System trajectories for the closed-loop system with the proposed QP-
based control for the integrator.

In Figs. 3 and 4, we show the results for two different nonlinear systems with
$f(x)$ given by $f_{1}(x)=0.1\lVert
x\rVert\begin{bmatrix}\,1\\!&\\!1\,\end{bmatrix}^{\mathsf{T}}$ and
$f_{2}(x)=0.1(\lVert
x\rVert-x^{\mathsf{T}}x)\begin{bmatrix}\,1\\!&\\!1\,\end{bmatrix}^{\mathsf{T}}$,
respectivelly, both with $g(x)=I_{n}$. For the same initial conditions, the
trajectories show the evolution of the closed-loop system state for both
controllers. As before, the nominal QP-based controller proposed by [3]
introduces the same undesired equilibrium point in the closed-loop system,
while the trajectories for our proposed controller are attracted towards the
origin. This behavior is obtained by the rotation of the CLF around the origin
induced by the QP-based controller described in Theorem 3.

Figure 3: System trajectories for the closed-loop system with nominal and
proposed QP-based controller for $f(x)=f_{1}(x)$ and $g(x)=I_{n}$ with a
circular obstacle. Figure 4: System trajectories for the closed-loop system
with nominal and proposed QP-based controller $f(x)=f_{2}(x)$ and $g(x)=I_{n}$
with a circular obstacle.

##  VI CONCLUSIONS

In the present work, we have formally demonstrated general conditions for the
formation of undesired equilibria on the closed-loop system with the QP-based
controller unifying CLFs and CBFs as proposed by [3], and showed that these
equilibria can be asymptotically stable even for simple systems. Additionally,
we have proposed a modification of the controller proposed in [3] using a
strategy that effectivelly avoids the conditions for the formation of boundary
equilibria. Future works include the extension of the proposed controller for
non-convex obstacles.

##  VII APPENDIX

###  VII-A Proof of Theorem 1

The Lagrangian associated to control law (3) is given by

$\displaystyle\mathcal{L}=\frac{1}{2}\lVert u\rVert^{2}+\frac{1}{2}pw^{2}+$
$\displaystyle\lambda_{1}(L_{f}V+L_{g}Vu+\gamma(V)-w)$ $\displaystyle-$
$\displaystyle\lambda_{2}(L_{f}V+L_{g}h\,u+\alpha(h))$ (11)

where the dependency of $V(x)$, $L_{f}V(x)$, $L_{g}V(x)$, $h(x)$, $L_{f}h(x)$,
$L_{g}h(x)$ and $u(x)$ on the state was dropped for compactness of notation.
Here, $\lambda_{1},\lambda_{2}$ are the Karush-Kuhn-Tucker (KKT) multipliers,
and the KKT conditions are

$\displaystyle\frac{\partial\mathcal{L}}{\partial
u}=u+\lambda_{1}L_{g}V-\lambda_{2}L_{g}h=0$ (12)
$\displaystyle\frac{\partial\mathcal{L}}{\partial w}=pw-\lambda_{1}=0$ (13)
$\displaystyle\lambda_{1}(L_{f}V+L_{g}Vu+\gamma(V)-w)=0$ (14)
$\displaystyle\lambda_{2}(L_{f}V+L_{g}h\,u+\alpha(h))=0$ (15)

with $\lambda_{1},\lambda_{2}\geq 0$. At this point, we need to distinguish
four different cases depending on the activation of each constraint.

Case 1. In this case, both constraints are inactive, that is
$L_{f}V+L_{g}Vu+\gamma(V)-w<0$ and $L_{f}h+L_{g}h\,u+\alpha(h)>0$ and
$\lambda_{1}=\lambda_{2}=0$. From the KKT conditions, $k(x)=0$, $\omega(x)=0$.
However, since $\gamma(V)<0$ in this case, this solution never holds.

Case 2. In this case, only the CLF constraint is active, that is
$L_{f}V+L_{g}V^{\mathsf{T}}u+\gamma(V)=\delta$ and
$L_{f}h+L_{g}h^{\mathsf{T}}u+\alpha(h)>0$, and $\lambda_{1}\geq
0,\lambda_{2}=0$. Using (12)-(13), the solution $k(x)$ is

$\displaystyle k(x)=-\frac{L_{f}V+\gamma(V)}{p^{-1}+\lVert
L_{g}V\rVert^{2}}L_{g}V^{\mathsf{T}}$ (16)

Since only the CBF is inactive in this case, (16) holds for
${\bf\Omega_{\overline{cbf}}^{clf}}$ as defined in Theorem 1. From the closed-
loop system (2), the equilibrium points are given by

$\displaystyle f_{cl}(x)=f(x)-\frac{L_{f}V+\gamma(V)}{p^{-1}+\lVert
L_{g}V\rVert^{2}}G\nabla V=0$ (17)

There are two possible solutions for (17):

  1. (i)

$f(x)=0$ and $G(x)\nabla V(x)=0$ for some
$x\in{\bf\Omega_{\overline{cbf}}^{clf}}$.

  2. (ii)

$f(x)=\kappa G(x)\nabla V(x)$: substituting this expression into (17) yields
$\kappa=p\gamma(V)$. Then, any $x\in{\bf\Omega_{\overline{cbf}}^{clf}}$ such
that $f(x)=p\gamma(V)G\nabla V$ is an equilibrium point of (2).

Both solutions hold strictly on $\text{int}(C)$. This motivates the definition
of the set of interior equilibria $\mathcal{E}_{\text{int}}$ on (4).

Case 3. In this case, only the CBF constraint is active, that is
$L_{f}h+L_{g}h\,u+\alpha(h)=0$ and $L_{f}V+L_{g}V\,u+\gamma(V)-\delta<0$, with
$\lambda_{1}=0,\lambda_{2}\geq 0$. Using the KKT conditions, the corresponding
solution for $k(x)$ is given by

$\displaystyle k(x)=-\lVert
L_{g}h\rVert^{-2}\left(L_{f}h+\alpha(h)\right)L_{g}h^{\mathsf{T}}$

Since the CLF is inactive in this case, this solution holds for

$\displaystyle{\bf\Omega_{cbf}^{\overline{clf}}}=\Big{\\{}x\in\mathbb{R}^{n}:L_{f}h+\alpha(h)\leq
0\,,$ $\displaystyle
L_{g}VL_{g}h^{\mathsf{T}}\left(L_{f}h+\alpha(h)\right)>\left(L_{f}V+\gamma(V)\right)\lVert
L_{g}h\rVert^{2}\Big{\\}}$

The equilibrium points are given by $f_{cl}(x)=0$, yielding

$\displaystyle f_{cl}(x)=f(x)-\lVert
L_{g}h\rVert^{-2}\\!\left(L_{f}h+\alpha(h)\right)G\nabla h=0$ (18)

Similarly to the previous case, the solutions of (18) are:

  1. (i)

$f(x)=0$ and $G(x)\nabla h(x)=0$ for some
$x\in{\bf\Omega_{cbf}^{\overline{clf}}}$.

  2. (ii)

$f(x)=\kappa G(x)\nabla h(x)$: substituting this expression into (18) yields
$\alpha(h)=0$, which means that this condition only happens for
$x\in\partial\mathcal{C}$.

However, points satisfying these conditions are not compatible with the first
condition of ${\bf\Omega_{cbf}^{\overline{clf}}}$. Therefore, no equilibria
exists in this case.

Case 4. This is the case where both constraints are active, that is
$L_{f}V+L_{g}V^{\mathsf{T}}u+\gamma(V)-\delta=0$ and
$L_{f}h+L_{g}h^{\mathsf{T}}u+\alpha(h)=0$. Therefore,
$\lambda_{1},\lambda_{2}\geq 0$ and we have to solve the KKT conditions
(12)-(13) simultaneously for both $\lambda_{1}$ and $\lambda_{2}$, yielding
the following matrix equation:

$\displaystyle\begin{bmatrix}p^{-1}+\lVert
L_{g}V\rVert^{2}\\!\\!&\\!\\!-L_{g}VL_{g}h^{\mathsf{T}}\\\
L_{g}VL_{g}h^{\mathsf{T}}\\!\\!&\\!\\!-\lVert
L_{g}h\rVert^{2}\end{bmatrix}\\!\\!\begin{bmatrix}\lambda_{1}\\\
\lambda_{2}\end{bmatrix}\\!\\!=\\!\\!\begin{bmatrix}L_{f}V+\gamma(V)\\\
L_{f}h+\alpha(h)\end{bmatrix}$ (19)

The determinant of the matrix on the left-side of (19) is

$\displaystyle\Delta=(L_{g}VL_{g}h^{\mathsf{T}})^{2}-(p^{-1}+\lVert
L_{g}V\rVert^{2})\lVert L_{g}h\rVert^{2}$

Note that $\Delta\leq 0$ as long as $p>0$. Consider the two cases:

  1. (i)

$\Delta=0$: in this case, the matrix on the left-side of (19) loses rank when
$L_{g}h=0$, and a solution can only exist for $L_{f}h+\alpha(h)=0$. In this
case, $k(x)$ is given by (16).

  2. (ii)

$\Delta<0$: in this case, the solution is given by

$\displaystyle k(x)$
$\displaystyle=-\lambda_{1}L_{g}V^{\mathsf{T}}+\lambda_{2}L_{g}h^{\mathsf{T}}$
(20)

with $\lambda_{1}$ and $\lambda_{2}$ drawn from

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{\Delta}\left((L_{f}h\\!+\\!\alpha(h))L_{g}VL_{g}h^{\mathsf{T}}-(L_{f}V\\!+\\!\gamma(V))\lVert
L_{g}h\rVert^{2}\right)$ $\displaystyle\lambda_{2}$
$\displaystyle=\frac{1}{\Delta}\big{(}(L_{f}h\\!+\\!\alpha(h))\left(\lVert
L_{g}V\rVert^{2}\\!+\\!p^{-1}\right)-$ (21)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(L_{f}V\\!+\\!\gamma(V))L_{g}VL_{g}h^{\mathsf{T}}\big{)}$

For both $\Delta=0$ or $\Delta>0$, this solution holds for

$\displaystyle{\bf\Omega_{cbf}^{clf}}=\Big{\\{}$ $\displaystyle
x\in\mathbb{R}^{n}:L_{g}VL_{g}h^{\mathsf{T}}\left(\frac{L_{f}h+\alpha(h)}{L_{f}V+\gamma(V)}\right)\\!\leq\\!\lVert
L_{g}h\rVert^{2}\,,$ $\displaystyle
L_{g}VL_{g}h^{\mathsf{T}}\geq\left(\frac{L_{f}h\\!+\\!\alpha(h)}{L_{f}V\\!+\\!\gamma(V)}\right)\left(\lVert
L_{g}V\rVert^{2}\\!+\\!p^{-1}\right)\Big{\\}}$

Using solution (20) on the closed-loop system (2), the equilibrium condition
$f_{cl}(x)=0$ is given by

$\displaystyle f_{cl}(x)=f(x)-\lambda_{1}G\nabla V+\lambda_{2}G\nabla h=0$
(22)

with $\lambda_{1},\lambda_{2}$ drawn from (21) ($\Delta<0$). In case
$\Delta=0$, $\lambda_{2}$ is not defined, but the last term of (22) is zero
anyway since $L_{g}h=0$. By carefully looking at (22), the following general
conditions for the occurrence of a valid solution arise.

  1. (i)

$f(x)=0$ for all $x\in\mathbb{R}^{n}$, and $\nabla V(x)||\nabla h(x)$

  2. (ii)

$\nabla h=0$ or $\nabla h\in\mathcal{N}(G)$, and $f(x)||G(x)\nabla V(x)$

  3. (iii)

$\nabla V=0$ or $\nabla V\in\mathcal{N}(G)$, and $f(x)||G(x)\nabla h(x)$

  4. (iv)

$\nabla V(x)||\nabla h(x)$, and $f(x)||G(x)\nabla h(x)$

Note that solution (ii) is valid for $\Delta=0$, with solution given by (16).
In this case, if $f(x)=\kappa G(x)\nabla V(x)$ for some constant
$\kappa\in\mathbb{R}$, replacing it on (22) yields
$\kappa=p\gamma(V)\in\mathbb{R}_{>0}$. Therefore, every
$x\in{\bf\Omega_{cbf}^{clf}}$ such that $G\nabla h(x)=0$ with
$f(x)=p\gamma(V(x))G(x)\nabla V(x)$ is an equilibrium point. Since this
solution is valid for $L_{f}\nabla h+\alpha(h)=0$, and $L_{f}\nabla h=0$ on
the equilibrium points, we conclude that these equilibria occur on the
boundary $h(x)=0$ of the safe set. For solutions (iii) and (iv), using a
similar reasoning, it is possible to show that these equilibria also can only
occur on $\partial\mathcal{C}$. Note that any $x\in\mathbb{R}^{n}$ such that
the null space of matrix $\begin{bmatrix}f(x)\\!\\!&\\!\\!G(x)\nabla
V(x)\\!\\!&\\!\\!G(x)\nabla h(x)\end{bmatrix}$ is nontrivial satisfies at
least one of the conditions (i), (ii), (iii) or (iv). This motivates the
definition of the set of boundary equilibria
$\mathcal{E}_{\partial\mathcal{C}}$ on (5). ∎

###  VII-B Proof of Theorem 2

For the integrator system, the Jacobian matrix of $f_{cl}(x^{\star})$ in (2)
using solution (20) (valid for a
$x^{\star}\in\mathcal{E}_{\partial\mathcal{C}}$) is given by

$\displaystyle J_{cl}(x^{\star})$ $\displaystyle\\!=\\!-\lVert\nabla
h\rVert^{-2}\\!\left(p\gamma(V)\mathcal{P}_{\nabla
h}(H_{V}\\!-\\!cH_{h})\\!+\\!\alpha^{\prime}(0)\nabla h\nabla
h^{\mathsf{T}}\right)$

Left-multiplying the eigenvalue equation
$J_{cl}(x^{\star})v_{i}=\lambda_{i}v_{i}$ by $\nabla h(x^{\star})$ and using
property (v) of the projection matrix yields $\lambda_{i}\nabla
h^{\mathsf{T}}(x^{\star})v_{i}=-\alpha^{\prime}(0)\nabla
h^{\mathsf{T}}(x^{\star})v_{i}$, which shows that all eigenvalues associated
to eigenvectors such that $v^{\mathsf{T}}_{i}\nabla h(x^{\star})\neq 0$ are
given by $\lambda_{i}=-\alpha^{\prime}(0)$, and therefore are strictly
negative (since $\alpha^{\prime}(0)>0$). In this case, the eigenvalue equation
can be rewritten as

$\displaystyle p\gamma(V)\mathcal{P}_{\nabla
h}\left(H_{V}\\!-\\!cH_{h}\\!-\\!\frac{\alpha^{\prime}(0)}{p\gamma(V)}I_{n}\right)v_{i}\\!=\\!0$
(23)

Equation (23) shows that the vector on which $\mathcal{P}_{\nabla h}$ operates
lies in $\mathcal{N}(\mathcal{P}_{\nabla h})$. However, from property (iii) of
the scaled projection matrices, the null space of $\mathcal{P}_{\nabla h}$ is
a one dimensional subspace generated by $\nabla h(x^{\star})$. Therefore, the
corresponding engenvector $v_{i}$ must be unique, meaning that
$\lambda_{i}=-\alpha^{\prime}(0)$ is an unique eigenvalue of
$J_{cl}(x^{\star})$ such that $v_{i}^{\mathsf{T}}\nabla h(x^{\star})\neq 0$.
All remaining $n-1$ eigenvectors $v_{1},v_{2},\cdots,v_{n-1}\in\mathbb{R}^{n}$
must lie in the ($n-1$)-dimensional projective hyperplane with normal given by
$\nabla h(x^{\star})$. Therefore, the stability of
$x^{\star}\in\mathcal{E}_{\partial\mathcal{C}}$ is completely determined by
the $n-1$ eigenvalues of $J_{cl}(x^{\star})$ associated to these eigenvectors.
Then, $x^{\star}\in\mathcal{E}_{\partial\mathcal{C}}$ is asymptotically stable
if

$\displaystyle v_{i}^{\mathsf{T}}J_{cl}(x^{\star})v_{i}=\lambda_{i}\lVert
v_{i}\rVert^{2}<0\,,\quad v_{i}^{\mathsf{T}}\nabla h(x)=0$

Using property (iv) of the projection matrix, yields

$\displaystyle\lambda_{i}\lVert v_{i}\rVert^{2}$
$\displaystyle=-p\gamma(V)\lVert\nabla
h\rVert^{-2}v_{i}^{\mathsf{T}}\mathcal{P}_{\nabla
h}(H_{V}(x^{\star})-cH_{h}(x^{\star}))v_{i}$
$\displaystyle=-p\gamma(V)\lVert\nabla
h\rVert^{-4}v_{i}^{\mathsf{T}}(H_{V}(x^{\star})-cH_{h}(x^{\star}))v_{i}<0$

Since $p\gamma(V)\lVert\nabla h\rVert^{-4}>0$, the equilibrium point
$x^{\star}\in\mathcal{E}_{\partial\mathcal{C}}$ is asymptotically stable if
$H_{V}(x^{\star})-cH_{h}(x^{\star})>0$. ∎

###  VII-C Proof of Theorem 3

First, note that the CBF $h(x)$ defining the set $\partial\mathcal{C}$ is a
zeroing barrier function [1]. Therefore, for any $x\in\partial\mathcal{C}$,
$\dot{h}(x)\geq-\alpha(h(x))=0$. Then, as established by [1], the set
$\mathcal{C}$ is forward invariant.

Similarly, since $h_{\mathcal{D}}(x)$ is also a zeroing barrier function, for
any $x\in\mathbb{R}^{n}$ such that $h_{\mathcal{D}}(x)=0$,
$\dot{h}_{\mathcal{D}}(x)\geq-\beta(h_{\mathcal{D}}(x))=0$, which establishes
that the set defined by
$\mathcal{C}_{\mathcal{D}}=\\{x\in\partial\mathcal{C}:\mathcal{D}(x,Q)\geq\epsilon\\}$
is also forward invariant. To prove that no boundary equilibrium exists, first
note that the equilibrium conditions are given by $f_{cl}(x)=0$ and
$\omega(x)=0$, since the system state consists of
$\\{x,Q\\}\in\mathbb{R}^{n}\times\mathcal{SO}(n)$. Next, we have to consider
all possible solutions for $k(x)$ and $\omega(x)$ imposed by (10) and show
that no solution for these equilibrium conditions is possible on the boundary
of $\mathcal{C}$.

The solutions for the QP (10) can be divided into two major groups, depending
on the activation of the third constraint. If the third constraint is
inactive, we have $\dot{h}_{\mathcal{D}}+\beta(h_{\mathcal{D}})>0$ and
$\lambda_{3}=0$, and the first equilibrium condition $f_{cl}(x)=0$ occurs on
the same conditions as those discussed in Section VII-C. Assume that a
boundary equilibrium point $x^{\star}\in\partial\mathcal{C}$ exists in this
case. Then, $\mathcal{D}(x^{\star},Q)=0$, which is a contradiction with the
fact that $\mathcal{C}_{\mathcal{D}}$ is forward invariant.

A similar reasoning can be done by studying the equilibrium conditions
resulting from the solutions of the QP (10) when the third constraint is
active. In this case, we have $\dot{h}_{\mathcal{D}}+\beta(h_{\mathcal{D}})=0$
and $\lambda_{3}\geq 0$. Once again, we have to consider all cases, depending
on the activation of the first (CLF) and second (CBF) constraints. Using the
KKT conditions to compute the general solutions along with the time derivative
of (9), and using (2), the general equilibrium conditions holding for all
cases are given by

$\displaystyle f\\!-\\!\lambda_{1}G\nabla V\\!+\\!\bar{\lambda}_{2}G\nabla
h\\!+\\!\lambda_{3}\sigma(h)G\nabla\mathcal{D}$ $\displaystyle=0$ (24)
$\displaystyle\lambda_{1}\nabla
V^{\mathsf{T}}\mathcal{O}_{n}(x)-\lambda_{3}\sigma(h)\nabla_{\\!Q}\mathcal{D}$
$\displaystyle=0$ (25)

where
$\bar{\lambda}_{2}=\lambda_{2}+\lambda_{3}\sigma^{\prime}(h)(\mathcal{D}-\epsilon)$.
The gradients $\nabla\mathcal{D}\in\mathbb{R}^{n}$ and
$\nabla_{\\!Q}\mathcal{D}\in\mathbb{R}^{\frac{1}{2}n(n-1)}$ are computed from
the time derivative of (8), and their expressions are:

$\displaystyle\nabla\mathcal{D}$
$\displaystyle=(H_{V}G+\Gamma^{\mathsf{T}}_{g,\nabla
V})(\mathcal{P}_{f}\\!+\\!\mathcal{P}_{G\nabla h})G\nabla V+$
$\displaystyle\quad\,\,(H_{h}G+\Gamma^{\mathsf{T}}_{g,\nabla
h})\mathcal{P}_{G\nabla V}G\nabla h+\nabla f^{\mathsf{T}}\mathcal{P}_{G\nabla
V}f$ $\displaystyle\nabla_{\\!Q}\mathcal{D}$
$\displaystyle=(H_{V}\mathcal{O}_{n}(x)-\mathcal{O}_{n}(\nabla
V))^{\mathsf{T}}G(\mathcal{P}_{f}\\!+\\!\mathcal{P}_{G\nabla h})G\nabla V$

where the matrix $\Gamma_{g,v}\in\mathbb{R}^{n\times n}$ for
$v\in\mathbb{R}^{n}$ is defined as

$\displaystyle\Gamma_{g,v}=\sum^{m}_{i=1}(g_{i}^{\mathsf{T}}vI_{n}+g_{i}v^{\mathsf{T}})\nabla
g_{i}$

where the $g_{i}(x)\in\mathbb{R}^{n}$ are the columns of
$g(x)\in\mathbb{R}^{n\times m}$.

Note that, in general, (24) holds for $f(x)=\kappa_{1}G\nabla V$ and $G\nabla
h=\kappa_{2}G\nabla V$. Assume that a boundary equilibrium point
$x^{\star}\in\partial\mathcal{C}$ exists in this case. Then,
$\mathcal{D}(x^{\star},Q)=0$, which is a contradiction with the fact that
$\mathcal{C}_{\mathcal{D}}$ is forward invariant. In other cases where
$\lambda_{1}$ or $\lambda_{2}$ are zero, we conclude that (24) never holds or
the solution is incompatible with the set where the corresponding QP solution
is valid, for all $x\in\partial\mathcal{C}$. Therefore, we conclude that no
boundary equilibrium can exist on the closed-loop system.

∎

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# A proof of Néron–Ogg–Shafarevich criterion via its archimedean analogue

Gyujin Oh

###### Abstract.

In this short note, we deduce the classical Néron–Ogg–Shafarevich criterion on
good reduction of abelian varieties from its _archimedean analogue_ : a
holomorphic family of abelian varieties over a punctured disc extends to the
whole unit disc if and only if the topological monodromy representation is
trivial.

###### Contents

  1. 1 Introduction
  2. 2 Archimedean Néron–Ogg–Shafarevich criterion
  3. 3 Warm-up: the case of elliptic curves
  4. 4 A proof of Néron–Ogg–Shafarevich criterion

##  1\. Introduction

There is a folklore analogy between the ramification theory of $\ell$-adic
Galois representations and the singularity of vector bundles with integrable
connections. A general dictionary is that the unramified, tamely ramified, and
wildly ramified Galois representations correspond to the removable, regular,
and irregular singularities, respectively. Furthermore, an integrable
connection with at worst regular singularities is encoded by its topological
monodromy representation by the Riemann–Hilbert corresopndence. Thus, one
expects that the unramifiedness of $\ell$-adic monodromy representation is
related to the triviality of topological monodromy representation.

On the other hand, the classical Néron–Ogg–Shafarevich criterion relates the
good reduction of an abelian variety with the unramifiedness of its
$\ell$-adic monodromy representation. Guided by the viewpoint of the previous
paragraph, we first formulate and prove the corresponding criterion regarding
the triviality of topological monodromy representation of a family of abelian
varieties, which we call the _archimedean analogue_ of the
Néron–Ogg–Shafarevich criterion. To avoid confusion, from now on, we will
refer to the classical Néron–Ogg–Shafarevich criterion as the $\ell$-adic
Néron–Ogg–Shafarevich criterion.

The main purpose of the note is to deduce the $\ell$-adic
Néron–Ogg–Shafarevich criterion from the archimedean Néron–Ogg–Shafarevich
criterion. The proof crucially uses the arithmetic toroidal compactification
of the moduli of principally polarized abelian varieties as in the work of
Faltings–Chai [FC]. This new proof of the $\ell$-adic criterion explicitly
realizes the folklore analogy between the ramification theory of $\ell$-adic
Galois representations and the singularities of topological monodromy
representations.

The idea of the proof is to use the “degeneration of degenerations.” Namely,
we use the fact that Mumford’s degeneration of abelian varieties can realize
all semi-abelian degeneration of abelian varieties, and also that such
degeneration can be put into a family. Perhaps the most well-known instance of
this fact is that, for elliptic curves, the Tate uniformization can be put
into a universal family, called the _universal Tate elliptic curve_ of
Raynaud. The idea of relating $\ell$-adic and topological monodromy via a
family of degenerations was prominently used in [Oda], which studies the
Galois representation of a higher-genus curve using the information about its
topological monodromy.

The article is organized as follows. In §2, we will first precisely formulate
and prove the archimedean Néron–Ogg–Shafarevich criterion. Then, in §3, we
will first deduce the $\ell$-adic Néron–Ogg–Shafarevich criterion from the
archimedean criterion in the case of elliptic curves, using the more familiar
universal Tate elliptic curve. In §4, we will end with the proof of the
$\ell$-adic criterion in the general case by using Mumford’s construction of
degeneration of abelian varieties.

### Acknowledgements

We thank John Halliday for pointing out a mistake in a previous version.

##  2\. Archimedean Néron–Ogg–Shafarevich criterion

We first formulate the “archimedean analogue” of the Néron–Ogg–Shafarevich
criterion, which is considerably easier to prove than the $\ell$-adic
Néron–Ogg–Shafarevich criterion.

######  Proposition 2.1 ((Archimedean Néron–Ogg–Shafarevich criterion)).

Let $f:A\rightarrow D^{\times}$ be a holomorphic family of abelian varieties
of dimension $g$ over the punctured disc $D^{\times}$. Then, $f$ extends to a
family of abelian varieties over $D$ if and only if the monodromy
representation
$\rho:\pi_{1}(D^{\times},t_{0})\cong\mathbb{Z}\rightarrow\operatorname{Aut}H^{1}(A_{t_{0}},\mathbb{Z})$
is trivial, where $t_{0}\in D^{\times}$ is a fixed base point.

###### Proof.

If the family extends to the unit disc $D$, the monodromy representation
factors through $\pi_{1}(D,t_{0})$, which is trivial. Conversely, suppose the
monodromy representation is trivial. We first prove this direction assuming
that the family $f:A\rightarrow D^{\times}$ is a family of _principally
polarized_ abelian varieties. Then, the family $f$ defines a period morphism
from $D^{\times}$ modulo the monodromy to the Siegel upper half plane
$\mathbb{H}_{g}$. As the monodromy is trivial, the period morphism is a
holomorphic map $p:D^{\times}\rightarrow\mathbb{H}_{g}$. As $\mathbb{H}_{g}$
is conformally equivalent to a bounded domain, $0=D-D^{\times}$ is a removable
singularity of $p$.

Now let us prove the converse direction in the general case. By Zahrin’s
trick, there is a family of principally polarized abelian varieties
$A^{\prime}\rightarrow D^{\times}$ whose topological monodromy representation
is trivial and which contains $A\subset A^{\prime}$ as a family of abelian
subvarieties. By the previous paragraph, $A^{\prime}\rightarrow D^{\times}$
extends to a family of principally polarized abelian varieties
$\widetilde{A^{\prime}}\rightarrow D$. Let $\widetilde{A}$ be the closure of
$A$ in $\widetilde{A^{\prime}}$, with its reduced scheme structure. We claim
that $\widetilde{A}\rightarrow D$ is a family of abelian varieties, or that
$\widetilde{A}_{0}$ is an abelian variety. Note that the fiberwise group
structure
$\mu:\widetilde{A^{\prime}}\times_{D}\widetilde{A^{\prime}}\rightarrow\widetilde{A^{\prime}}$
restricts to
$\mu^{\prime}:\widetilde{A}\times_{D}\widetilde{A}\rightarrow\widetilde{A^{\prime}}$.
Since $\mu^{\prime}\rvert_{A\times_{D^{\times}}A}$ factors through
$A\subset\widetilde{A^{\prime}}$, $\mu^{\prime}$ factors through
$\widetilde{A}$. This shows that $\widetilde{A}_{0}$ is a reduced closed
algebraic subgroup of the abelian variety $\widetilde{A^{\prime}}_{0}$. Since
a reduced connected closed algebraic subgroup of an abelian variety over a
characteristic zero field is an abelian variety, it is now sufficient to show
that $\widetilde{A}_{0}$ is connected, which is true as it is the continuous
image of the closure of a connected set $A\times_{D^{\times}}A$, which is
itself connected. ∎

Using this, we would like to prove the $\ell$-adic Néron–Ogg–Shafarevich
criterion. From now on, we fix a finite extension $K/\mathbb{Q}_{p}$.

######  Theorem 2.2 (($\ell$-adic Néron–Ogg–Shafarevich criterion)).

Let $A/K$ be an abelian variety. Let
$\rho:G_{K}\rightarrow\operatorname{GL}(T_{\ell}A)$ be the $\ell$-adic
monodromy representation for $\ell\neq p$. Then, $A$ has good reduction if and
only if $\rho\rvert_{I_{K}}$ is trivial.

The rest of the article will be dedicated to proving Theorem 2.2. As one
direction is immediate, we are left to prove the other nontrivial direction,
namely proving the good reduction of $A$ assuming that $\rho\rvert_{I_{K}}$ is
trivial.

As the first reduction step, we note that both sides of the statement of
Theorem 2.2 are invariant under an unramified base change. Furthermore, by
Zahrin’s trick, one can assume that $A$ is principally polarized.

Next, we note that only semistable abelian varieties need to be considered,
using the geometry of the arithmetic toroidal compactification of the moduli
of abelian varieties with level structures.

######  Proposition 2.3.

Let $A/K$ be a principally polarized abelian variety, and $\rho$ be the
$\ell$-adic monodromy representation of $A$ for $\ell\neq p$. If
$\rho\rvert_{I_{K}}$ is trivial, then $A$ has at worst semistable reduction.

###### Proof.

Let $g$ be the dimension of $A$. By assumption, $A[\ell^{n}]$ is unramified
for any $n\geq 1$. Take a finite unramified extension $L/K$ over which
$A[\ell^{3}]$ splits. Then,
$A[\ell^{3}](L)\cong(\mathbb{Z}/\ell^{3}\mathbb{Z})^{2g}$. Fix one such
isomorphism $\iota$. Then, $(A,\iota)$ defines a point $P\in
A_{g,\ell^{3}}(L)$, where $A_{g,\ell^{3}}$ is the moduli space of principally
polarized abelian varieties of dimension $g$ with full level $\ell^{3}$
structure. By [Mum, Theorem 7.10], it is known that $A_{g,\ell^{3}}$ is
represented by a smooth quasi-projective scheme over $\mathbb{Z}[1/\ell]$.
Furthermore, by [FC, Theorem IV.6.7], for a good enough choice of auxiliary
data, there is a smooth projective $\mathbb{Z}[1/\ell]$-scheme
$\overline{A}_{g,n}$, which contains $A_{g,n}$ as a dense open subscheme. By
the valuative criterion for properness, the point $P\in A_{g,\ell^{3}}(L)$
extends to an $\mathcal{O}_{L}$-point
$\mathcal{P}\in\overline{A}_{g,\ell^{3}}(\mathcal{O}_{L})$.

Recall that there is a semi-abelian scheme
$\overline{G}\rightarrow\overline{A}_{g,\ell^{3}}$, extending the universal
abelian scheme $G\rightarrow A_{g,n}$. After pulling back this family via the
$\mathcal{O}_{L}$-point
$\mathcal{P}:\operatorname{Spec}\mathcal{O}_{L}\rightarrow\overline{A}_{g,\ell^{3}}$,
we obtain a semi-abelian scheme over $\mathcal{O}_{L}$ whose generic fiber is
$A_{L}$. This implies that $A_{L}$ has at worst semistable reduction. As the
reduction type is invariant under an unramified base change, we conclude that
$A$ has at worst semistable reduction, as desired. ∎

##  3\. Warm-up: the case of elliptic curves

Mumford’s construction of degenerating abelian varieties in the case of
elliptic curves is exactly the _Tate uniformization_. By the means of
Raynaud’s universal Tate elliptic curve, one even knows that the degeneration
of elliptic curves can be put into a one-dimensional family. In this section,
we show how to prove Theorem 2.2 using the universal Tate elliptic curve, to
illustrate the usefulness of a family of degenerations.

###### Proof of Theorem 2.2, in the case of elliptic curves.

Suppose that an elliptic curve $A$ over $K$ has bad reduction but also has
trivial monodromy over the inertia group $I_{K}$. As we know from Proposition
2.3, $A$ has semistable reduction. We will leverage the fact that $A$ has the
Tate uniformization, and furthermore that the Tate uniformization is obtained
from the _universal Tate elliptic curve_ of Raynaud. Our reference for the
construction is [Bos, §9.2] and [Con, §2.5].

Let $K^{\operatorname{nr}}$ be the maximal unramified extension of $K$. Then,
the _universal Tate elliptic curve_ is a family
$f:\mathfrak{T}\rightarrow\mathfrak{S}=\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}[[q]]$
such that, if $E$ is an elliptic curve over $K^{\operatorname{nr}}$ with
semistable bad reduction, then the (semistable) Néron model $\mathfrak{E}$ of
$E$ over $\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}$ is obtained
by pulling back $\mathfrak{T}$ along the map
$\mathcal{O}_{K^{\operatorname{nr}}}[[q]]\rightarrow\mathcal{O}_{K^{\operatorname{nr}}}$,
$q\mapsto q(E)$, the $q$-parameter of $E$. The family $\mathfrak{T}$ used here
is obtained by taking the base-change of $\underline{\mathrm{Tate}}_{1}$ in
[Con, §2.5], which is in turn the algebraization of the (formal) universal
Tate elliptic curve in [Bos, §9.2], from $\mathbb{Z}[[q]]$ to
$\mathcal{O}_{K^{\operatorname{nr}}}[[q]]$.

Let $\mathfrak{V}=\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}$ be
the closed subscheme of $\mathfrak{S}$ defined by the map
$\mathcal{O}_{K^{\operatorname{nr}}}[[q]]\xrightarrow{q\mapsto
q(A)}\mathcal{O}_{K^{\operatorname{nr}}}$, and let $\eta$ be the generic point
of $\mathfrak{V}$. Let $i:\mathfrak{V}\hookrightarrow\mathfrak{S}$ be the
closed embedding. Then,

$f\rvert_{\mathfrak{V}}:i^{*}\mathfrak{T}\rightarrow\mathfrak{V},$

is a semistable model of $A_{K^{\operatorname{nr}}}$. This in particular means
that
$f\rvert_{\mathfrak{V}_{\eta}}:(i^{*}\mathfrak{T})_{\eta}\rightarrow\mathfrak{V}_{\eta}$
is identified with
$A_{K^{\operatorname{nr}}}\rightarrow\operatorname{Spec}(K^{\operatorname{nr}})$.
Upon choosing a geometric generic point
$\overline{\eta}:\operatorname{Spec}(\overline{K})\rightarrow\operatorname{Spec}(K^{\operatorname{nr}})\rightarrow\mathfrak{V}$,
the pro-$\ell$-part of the monodromy representation

$\rho_{\mathfrak{V}_{\eta},\overline{\eta}}:\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{V}_{\eta},\overline{\eta})_{\ell}\rightarrow\operatorname{Aut}(R^{1}(f\rvert_{{\overline{\eta}}})_{*}\underline{{\mathbb{Q}}_{\ell}}),$

is identified with the $\ell$-adic monodromy representation of $A$ restricted
to the pro-$\ell$-part of the inertia group $(I_{K})_{\ell}$,

$\rho\rvert_{(I_{K})_{\ell}}:(I_{K})_{\ell}\rightarrow\operatorname{Aut}(H^{1}(A_{\overline{K}},{\mathbb{Q}}_{\ell})).$

On the other hand, as
$A_{K^{\operatorname{nr}}}\rightarrow\operatorname{Spec}(K^{\operatorname{nr}})$
is put in a larger family, the above monodromy representation
$\rho_{\mathfrak{V}_{\eta},\overline{\eta}}$ factors through the monodromy
over a larger base $\mathfrak{U}=\\{q\neq
0\\}=\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}((q))\subset\mathfrak{V}$:

---
$\textstyle{\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{V}_{\eta},\overline{\eta})_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\mathfrak{V}_{\eta},\overline{\eta}}}$$\textstyle{\operatorname{Aut}(R^{1}(f\rvert_{\mathfrak{V}_{\overline{\eta}}})_{*}\underline{\mathbb{Q}_{\ell}})}$$\textstyle{\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{U},\overline{\eta})_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\mathfrak{U},\overline{\eta}}}$

namely, the left arrow is the natural map
$\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{V}_{\eta},\overline{\eta})_{\ell}\rightarrow\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{U},\overline{\eta})_{\ell}$
via $\mathcal{O}_{K^{\operatorname{nr}}}((q))\xrightarrow{q\mapsto
q(A)}K^{\operatorname{nr}}$. Note that, by Abhyankar’s lemma, all étale
$\ell$-covers of both $\mathfrak{U}$ and $\mathfrak{V}_{\eta}$ are Kummer
covers, so both
$\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{V}_{\eta},\overline{\eta})_{\ell}$
and $\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{U},\overline{\eta})_{\ell}$
are isomorphic to $\mathbb{Z}_{\ell}$. After identifying both with
$\mathbb{Z}_{\ell}$, the natural map is multiplication by $\ell^{n}$ for some
$n\geq 0$.111This was pointed out to us by John Halliday. By assumption,
$\rho_{\mathfrak{V}_{\eta},\overline{\eta}}=\rho\rvert_{(I_{K})_{\ell}}$ is
trivial, so we conclude that $\rho_{\mathfrak{U},\overline{\eta}}$ has finite
image.

On the other hand, $\mathfrak{T}$ is smooth and proper over $\mathfrak{U}$, so
the specialization map of étale fundamental groups identifies the monodromy
over $\mathfrak{U}$ at $\overline{\eta}$ is identified with the monodromy over
$\mathfrak{U}$ at $\overline{\iota}$, where
$\overline{\iota}=\operatorname{Spec}\overline{K^{\operatorname{nr}}((q))}$ is
a geometric generic point of
$\mathfrak{U}=\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}((q))$.
Thus, the monodromy representation

$\rho_{\mathfrak{U},\overline{\iota}}:\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{U},\overline{\iota})_{\ell}\rightarrow\operatorname{Aut}(R^{1}(f\rvert_{\overline{\iota}})_{*}\underline{\mathbb{Q}_{\ell}}),$

has finite image. Now
$\pi_{1,\operatorname{\acute{e}t}}(\mathcal{O}_{K^{\operatorname{nr}}})=1$, so
$\rho_{\mathfrak{U},\overline{\iota}}$ will stay the same even if we base
change $\mathcal{O}_{K^{\operatorname{nr}}}$ to a characteristic zero
algebraically closed field. We fix an abstract field embedding
$j:K_{\operatorname{nr}}\hookrightarrow\mathbb{C}$ and consider the base-
change to $\mathbb{C}$ using this embedding. Therefore, the monodromy of the
$\mathbb{C}$-base-change

$\rho_{\mathfrak{U}_{\mathbb{C}},\overline{\iota}_{\mathbb{C}}}:\pi_{1,\operatorname{\acute{e}t}}(\mathfrak{U}_{\mathbb{C}},\overline{\iota}_{\mathbb{C}})_{\ell}=\operatorname{Gal}(\overline{\mathbb{C}((q))}/\mathbb{C}((q)))_{\ell}\rightarrow\operatorname{Aut}(R^{1}(f\rvert_{\overline{\iota}_{\mathbb{C}}})_{*}\underline{\mathbb{Q}_{\ell}}),$

has finite image.

The crucial point is that the family
$\mathfrak{T}_{\mathbb{C}[[q]]}\rightarrow\operatorname{Spec}\mathbb{C}[[q]]$,
obtained by base-changing $f:\mathfrak{T}\rightarrow\mathfrak{S}$ via
$\mathcal{O}_{K^{\operatorname{nr}}}\hookrightarrow
K^{\operatorname{nr}}\xrightarrow{j}\mathbb{C}$, is the formal germ of a
complex-analytic family $f^{\prime}:\mathcal{T}\rightarrow D$ of complex
analytic varieties over the unit disk. One could see this by noticing that the
power series defining the Tate uniformization has coefficients in $\mathbb{Z}$
and can be verbatim used to define a complex-analytic family over the
punctured unit disk. Another way to realize this is that the aforementioned
family induces a map $p:\operatorname{Spec}\mathbb{C}[[q]]\rightarrow X(1)$,
where $X(1)$ is the moduli of elliptic curves. As $X(1)$ is an algebraic
space, one could take the open unit disk around the point $p\in
X(1)^{\operatorname{an}}$, where $X(1)^{\operatorname{an}}$ is a Moishezon
manifold.

Using the above observation, we see that
$\rho_{\mathfrak{U}_{\mathbb{C}},\overline{\iota}_{\mathbb{C}}}$ is the
$\ell$-completion of the topological monodromy
$\rho_{f^{\prime}}:\pi_{1}(D^{\times},t_{0})\rightarrow\operatorname{Aut}H^{1}(A_{t_{0}},\mathbb{Z})$,
where $t_{0}\in D^{\times}$. As the family $\mathcal{T}\rightarrow D$ is
semistable, this local topological monodromy is unipotent. However, any
unipotent matrix in $\operatorname{GL}_{2}(\mathbb{Z})$ is either trivial or
has infinite order, so this concludes that the topological monodromy over
$D^{\times}$ is trivial. By the archimedean Néron–Ogg–Shafarevich criterion,
Proposition 2.1, $\mathcal{T}\rvert_{D^{\times}}\rightarrow D^{\times}$ can be
extended to a family of abelian varieties $\mathcal{T}^{\prime}\rightarrow D$.
As $\mathcal{T}\rvert_{D^{\times}}\rightarrow D^{\times}$ can be extended to
the semistable family $\mathcal{T}\rightarrow D$, this contradicts the fact
that there is at most one way of extending a family of elliptic curves over
$D^{\times}$ to $D$.∎

##  4\. A proof of Néron–Ogg–Shafarevich criterion

The crucial part of the proof of the previous section is that you can put a
semistable elliptic curve into a family of semi-abelian varieties over
$\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}[[q]]$ whose base-
change to $\mathbb{C}[[q]]$ is the formal germ of a complex-analytic family of
semi-abelian varieties. With this in mind, we can now prove Theorem 2.2 in the
general case.

###### Proof of Theorem 2.2.

As in the proof of Proposition 2.3, there is a point $P\in
A_{g,\ell^{3}}(K^{\operatorname{nr}})$ such that $G_{P}\cong
A_{K^{\operatorname{nr}}}$, and there is an extension
$i:\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}\rightarrow\overline{A}_{g,\ell^{3}}$.
Let $\overline{P}\in A_{g,\ell^{3}}(\overline{\mathbb{F}}_{p})$ be the point
in the special fiber of $i$. As $\overline{A}_{g,\ell^{3}}$ is smooth over
$\mathbb{Z}[1/\ell]$, the local ring of $\overline{A}_{g,\ell^{3}}$ at
$\overline{P}$ is

$\mathcal{O}_{\overline{A}_{g,\ell^{3}},\overline{P}}\cong
W(\overline{\mathbb{F}}_{p})[[X_{1},\cdots,X_{d}]],$

where $d=\frac{g(g+1)}{2}$. The morphism
$i:\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}\rightarrow\overline{A}_{g,\ell^{3}}$
then induces
$\operatorname{Spf}\mathcal{O}_{K^{\operatorname{nr}}}\rightarrow\operatorname{Spf}\mathcal{O}_{\overline{A}_{g,\ell^{3}},\overline{P}}$,
or a map
$f:W(\overline{\mathbb{F}}_{p})[[X_{1},\cdots,X_{d}]]\rightarrow\mathcal{O}_{K^{\operatorname{nr}}}$.
As its mod $p$ reduction is $\overline{P}$, if we choose a uniformizer
$\pi\in\mathcal{O}_{K^{\operatorname{nr}}}$, then $f(X_{i})$ is divisible by
$\pi$. Thus, $f$ factors through

$W(\overline{\mathbb{F}}_{p})[[X_{1},\cdots,X_{d}]]\xrightarrow{X_{i}\mapsto\frac{f(X_{i})}{\pi}X}\mathcal{O}_{K^{\operatorname{nr}}}[[X]]\xrightarrow{X\mapsto\pi}\mathcal{O}_{K^{\operatorname{nr}}}.$

Let
$p:\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}[[X]]\rightarrow\overline{A}_{g,\ell^{3}}$
be the corresponding morphism, and consider
$p^{*}\overline{G}\rightarrow\operatorname{Spec}\mathcal{O}_{K^{\operatorname{nr}}}[[X]]$.
This is, as before, a one-dimensional family of semi-abelian schemes, which
has the following properties.

  * •

It specializes to $A_{K^{\operatorname{nr}}}$ at a certain point.

  * •

It is smooth away from the $X=0$ locus.

  * •

Its base-change to $\mathbb{C}[[X]]$ is the formal germ of a complex-analytic
semistable family of abelian varieties over a unit disc.

The proof then proceeds exactly as in the proof in §3, where the only
difference is that we have to use the fact that a non-identity unipotent
matrix in $\operatorname{GL}_{n}(\mathbb{Z})$ has infinite order for any $n$.
∎

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