Dynamic Equivalence of Control Systems and Infinite Permutation Matrices

Dynamic Equivalence of Control Systems
and Infinite Permutation Matrices

Jeanne N. CLELLAND †, Yuhao HU † and Matthew W. STACKPOLE ‡

J.N. Clelland, Y. Hu and M.W. Stackpole

† Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO
80309-0395, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Maxar Technologies, 1300 W. 120th Ave, Westminster, CO 80234, USA
<EMAIL_ADDRESS>
Received February 05, 2019, in final form August 20, 2019; Published online
August 26, 2019

To each dynamic equivalence of two control systems is associated an infinite
permutation matrix. We investigate how such matrices are related to the
existence of dynamic equivalences.

dynamic equivalence; control systems

34H05; 58A15; 58A17

##  1 Introduction

A _control system_ is an underdetermined ODE system of the form

$\displaystyle\dot{\bf x}={\bf f}(t,{\bf x},{\bf u}),$

where ${\bf x}=(x_{i})$ are called the _state variables_ and ${\bf
u}=(u_{\alpha})$ the _control variables_. The meaning of “control” is clear:
Under suitable regularity conditions, specifying a control function ${\bf
u}(t)$ and an initial state ${\bf x}(t_{0})$ uniquely determines a local
“trajectory” ${\bf x}(t)$ that satisfies the ODE system and the initial state.
When ${\bf f}$ does not explicitly depend on $t$, the control system is said
to be _autonomous_ , which we will assume throughout this paper.

Let $\dot{\bf x}={\bf f}({\bf x},{\bf u})$ and $\dot{\bf y}={\bf g}({\bf
y},{\bf v})$ be two control systems. Suppose that there exist mappings
$\phi=\phi\big{(}{\bf x},{\bf u},\dot{\bf u},\ddot{\bf u},\dots,{\bf
u}^{(p)}\big{)}$ and $\psi=\psi\big{(}{\bf y},{\bf v},\dot{\bf v},\ddot{\bf
v},\dots,{\bf v}^{(q)}\big{)}$ such that,

  * •

for any solution $({\bf x}(t),{\bf u}(t))$ of the first system, the function
$({\bf y},{\bf v})=\phi\big{(}{\bf x},{\bf u},\dot{\bf u},\ddot{\bf
u},\dots,{\bf u}^{(p)}\big{)}$ is a solution of the second;

  * •

for any solution $({\bf y}(t),{\bf v}(t))$ of the second system, the function
$({\bf x},{\bf u})=\psi\big{(}{\bf y},{\bf v},\dot{\bf v},\ddot{\bf
v},\dots,\allowbreak{\bf v}^{(q)}\big{)}$ is a solution of the first;

  * •

moreover, applying $\phi$ and $\psi$ successively to a solution $({\bf
x}(t),{\bf u}(t))$ of the first system yields the same solution $({\bf
x}(t),{\bf u}(t))$, and, similarly, applying $\psi$ and $\phi$ successively to
a solution $({\bf y}(t),{\bf v}(t))$ of the second system yields the same
solution $({\bf y}(t),{\bf v}(t))$.

If all these conditions are satisfied, we say that the pair of maps
$(\phi,\psi)$ establishes a _dynamic equivalence_ between the two control
systems. Intuitively, a dynamic equivalence provides a one-to-one
correspondence between the solutions of one control system with those of the
other.

Fixing a dynamic equivalence $(\phi,\psi)$, one can always find the smallest
$p,q\geq 0$ so that $\phi=\phi\big{(}{\bf x},{\bf u},\dot{\bf u},\ddot{\bf
u},\dots,{\bf u}^{(p)}\big{)}$ and $\psi=\psi\big{(}{\bf y},{\bf v},\dot{\bf
v},\ddot{\bf v},\dots,{\bf v}^{(q)}\big{)}$. We call such a pair $(p,q)$ the
_height_ of the corresponding dynamic equivalence. A dynamic equivalence with
height $(0,0)$ is known as a _static $($feedback$)$ equivalence_, in which
case $\phi$, $\psi$ are inverses of each other as diffeomorphisms.

An immediate question is: _How much more general is the notion of dynamic
equivalence than that of static equivalence?_ Classical results (see [10, 12])
suggest that the answer depends on the number of control variables. In
particular, a dynamic equivalence between two control systems with a single
control variable is necessarily static. It is also well known that the number
of control variables is invariant under a dynamic equivalence. However, in the
cases of 2 or more controls, a precise answer to the question above remains
largely unknown.

In [14], the author considered all _control-affine_ systems with 3 states and
2 controls, proving that three statically non-equivalent systems are pairwise
dynamically equivalent at height $(1,1)$. In addition, he introduced a new
method of studying dynamic equivalences of two control systems. He found that
to each dynamic equivalence is associated an infinite permutation matrix.
Intuitively, such a matrix tells us how the ‘generating $1$-forms’ of certain
prolongations of the two control systems (viewed as Pfaffian systems), when
chosen appropriately, relate under a dynamic equivalence.

In the current work, we present further properties of dynamic equivalences
that can be derived using the associated infinite permutation matrices. First
we prove that there is a _rank matrix_ (Definition 3.6) associated to a
dynamic equivalence, which has a more ‘invariant’ nature than an associated
infinite permutation matrix (Proposition 3.7). Then we prove several
inequalities and equalities (Propositions 4.1 and 4.2) satisfied by the rank
matrix. Using these results, we prove an inequality satisfied by the height
$(p,q)$ of a dynamic equivalence (Theorem 4.3). In particular, this inequality
implies the

######  Theorem 4.5.

The height $(p,q)$ of a dynamic equivalence between two control systems with
$n_{1}$ and $n_{2}$ states, respectively, and $2$ controls must satisfy
$n_{1}+p=n_{2}+q$.

##  2 Control systems and dynamic equivalence

###  2.1 Control systems

######  Definition 2.1.

A _control system with $n$ states and $m$ controls_ is an underdetermined ODE
system

$\displaystyle\dot{\bf x}={\bf f}({\bf x},{\bf u}),$ (2.1)

where

$\displaystyle{\bf x}=(x_{i})\in\mathbb{R}^{n},\qquad{\bf
u}=(u_{\alpha})\in\mathbb{R}^{m},$

and ${\bf f}=(f_{i})\colon\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n}$ is a
smooth function satisfying $\operatorname{rank}\big{(}\frac{\partial
f_{i}}{\partial u_{\alpha}}\big{)}=m$ on some open domain111Since our study is
local, we henceforth assume that such a domain is the entire
$\mathbb{R}^{n+m}$. in $\mathbb{R}^{n+m}$. Here, $x_{i}$ are called the _state
variables_ , $u_{\alpha}$ the _control variables_.

For a control system, there is an equivalent geometric characterization. Let
$\mathcal{D}\subset\mathbb{R}^{n}$ be the domain of the state variables ${\bf
x}=(x_{i})$. The admissible $t$-derivatives of $x_{i}$, as imposed by the
control system, are given by specifying a submanifold $\Sigma\subset
T\mathcal{D}$ that submerses onto $\mathcal{D}$ with rank-$m$ fibers. Each
fiber is precisely parametrized by the control variables ${\bf
u}=(u_{\alpha})$. The submanifold $\Sigma$ induces an embedding

$\displaystyle\iota\colon\
\mathbb{R}\times\Sigma\hookrightarrow\mathbb{R}\times T\mathcal{D},$

which is the identity in the $\mathbb{R}$-factor (with coordinate $t$) and
$t$-independent in the $\Sigma$-factor. In coordinates, this embedding may be
written as

$\displaystyle\iota(t,{\bf x},{\bf u})=(t,{\bf x},{\bf f}({\bf x},{\bf u})),$

which satisfies

$\displaystyle\iota^{*}(\mathrm{d}x_{i}-\dot{x}_{i}\mathrm{d}t)=\mathrm{d}x_{i}-f_{i}({\bf
x},{\bf u})\mathrm{d}t.$

In other words, the system (2.1) corresponds to the Pfaffian system
$(M,\mathcal{C}_{M})$, where $M:=\mathbb{R}\times\Sigma$, and
$\mathcal{C}_{M}$ is the restriction to $M$ of the standard contact system
$\mathcal{C}=\langle\mathrm{d}x_{i}-\dot{x}_{i}\mathrm{d}t\rangle_{i=1}^{n}$
on the jet bundle $J^{1}(\mathbb{R},\mathcal{D})\cong\mathbb{R}\times
T\mathcal{D}$. Conversely, let $\Sigma\subset T\mathcal{D}$ be a submanifold
that submerses onto a domain $\mathcal{D}\subset\mathbb{R}^{n}$ with rank-$m$
fibers. The Pfaffian system $(M,\mathcal{C}_{M})$ corresponds to a control
system with $n$ states and $m$ controls.

###  2.2 Prolongations of a control system

######  Definition 2.2.

Let $(M,\mathcal{C}_{M})$ be a control system with $n$ states and $m$
controls. Let $(t,{\bf x},{\bf u})$ be coordinates on $M$. Suppose that
$\mathcal{C}_{M}$ is generated by $\mathrm{d}x_{i}-f_{i}({\bf x},{\bf
u})\mathrm{d}t$ $(i=1,\dots,n)$. By definition, the _first total prolongation_
222Geometrically, $M^{(1)}$ is known as the _space of integral line elements_
of $(M,\mathcal{I})$. See [3]. In particular, one can show that this
definition of $\big{(}M^{(1)},\mathcal{C}^{(1)}\big{)}$ is independent of the
choice of coordinates on $M$. of $(M,\mathcal{C}_{M})$ is the Pfaffian system
$\big{(}M^{(1)},\mathcal{C}^{(1)}\big{)}$, where
$M^{(1)}=M\times\mathbb{R}^{m}$ with the coordinates $\big{(}t,{\bf x},{\bf
u},{\bf u}^{(1)}\big{)}$; $\mathcal{C}^{(1)}$ is the Pfaffian system generated
by

$\displaystyle\mathrm{d}x_{i}-f_{i}({\bf x},{\bf
u})\mathrm{d}t,\qquad\mathrm{d}u_{\alpha}-u_{\alpha}^{(1)}\mathrm{d}t,\qquad
i=1,\dots,n,\qquad\alpha=1,\dots,m.$

Let $k$ be a positive integer. One can start from a control system
$(M,\mathcal{C}_{M})$ with $n$ states and $m$ controls and generate total
prolongations successively for $k$ times. The result is called the _$k$ -th
total prolongation of $(M,\mathcal{C})$_, denoted as
$\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$, where $M^{(k)}$ has the coordinates
$\big{(}t,{\bf x},{\bf u},{\bf u}^{(1)},\dots,{\bf u}^{(k)}\big{)}$ and
$\mathcal{C}^{(k)}$ is generated by the $1$-forms

$\displaystyle\mathrm{d}x_{i}-f_{i}({\bf x},{\bf
u})\mathrm{d}t,\qquad\mathrm{d}u_{\alpha}-u_{\alpha}^{(1)}\mathrm{d}t,\qquad\mathrm{d}u_{\alpha}^{(\ell)}-u_{\alpha}^{(\ell+1)}\mathrm{d}t,$
$\displaystyle i=1,\dots,n,\qquad\alpha=1,\dots,m,\qquad\ell=1,\dots,k-1.$

When $k=0$, we simply let $\big{(}M^{(0)},\mathcal{C}^{(0)}\big{)}$ denote
$(M,\mathcal{C}_{M})$.

It is clear that $\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$ is a control system
with $n+km$ states and $m$ controls.

###  2.3 Dynamic equivalence

Given two control systems, it is natural to regard them as equivalent if one
can establish a one-to-one correspondence between their solutions.

Of course, two control systems $\dot{\bf x}={\bf f}({\bf x},{\bf u})$ and
$\dot{\bf y}={\bf g}({\bf y},{\bf v})$ are equivalent in the sense above when
they can be transformed into each other by a change of variables of the form
${\bf y}=\phi({\bf x})$, ${\bf v}=\xi({\bf x},{\bf u})$ and ${\bf
x}=\phi^{-1}({\bf y})$, ${\bf u}=\rho({\bf y},{\bf v})$. This notion of
equivalence is called _static equivalence_ , which, in particular, requires
the two equivalent control systems to have same number of states and the same
number of controls. However, it is possible for two systems with different
numbers of states to have a one-to-one correspondence between their solutions,
as is indicated by the following standard property of jet bundles.

######  Proposition 2.3.

Let $(M,\mathcal{C}_{M})$ be a control system. Let $\pi\colon
M^{(k)}\rightarrow M$ be the canonical projection from its $k$-th total
prolongation. Any integral curve $\tau\colon\mathbb{R}\rightarrow M$ of
$(M,\mathcal{C}_{M})$ has a unique lifting
$\tau^{(k)}\colon\mathbb{R}\rightarrow M^{(k)}$ $($i.e., satisfying
$\pi\circ\tau^{(k)}=\tau)$ to an integral curve of
$\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$. In addition, for each integral
curve $\sigma\colon\mathbb{R}\rightarrow M^{(k)}$ of
$\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$, its projection $\pi\circ\sigma$ is
an integral curve of $(M,\mathcal{C}_{M})$.

In other words, given two control systems, a one-to-one correspondence between
their solutions may involve differentiation. This motivates the following
notion of equivalence.

######  Definition 2.4.

Two control systems $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ are said
to be _dynamically equivalent_ if there exist integers $p,q\geq 0$ and
submersions $\Phi\colon M^{(p)}\rightarrow N$ and $\Psi\colon
N^{(q)}\rightarrow M$ that satisfy333A more careful definition would set the
domains of $\Phi$ and $\Psi$ to be open subsets of $M^{(p)}$ and $N^{(q)}$,
respectively. See, for example, [12]. Since our results are local, for the
economy of notations, we will be content with the definition presented here.

  1. (i)

$\Phi$, $\Psi$ preserve the $t$-variable and are $t$-independent in the state
and control components;

  2. (ii)

$\Phi$ cannot factor through any $M^{(k)}$ for $k<p$; $\Psi$ cannot factor
through any $N^{(\ell)}$ for $\ell<q$;

  3. (iii)

for each integral curve $\tau\colon\mathbb{R}\rightarrow M$ of
$(M,\mathcal{C}_{M})$, $\Phi\circ\tau^{(p)}$ is an integral curve of
$(N,\mathcal{C}_{N})$; for each integral curve
$\sigma\colon\mathbb{R}\rightarrow N$ of $(N,\mathcal{C}_{N})$,
$\Psi\circ\sigma^{(q)}$ is an integral curve of $(M,\mathcal{C}_{M})$;

  4. (iv)

letting $\tau$ and $\sigma$ be as in (iii), we have

$\displaystyle\tau=\Psi\circ\big{(}\Phi\circ\tau^{(p)}\big{)}^{(q)},\qquad\sigma=\Phi\circ\big{(}\Psi\circ\sigma^{(q)}\big{)}^{(p)}.$

For the convenience of the reader, we present the commutative diagram:

${M^{(p)}}$${N^{(q)}}$${\mathbb{R}}$${M}$${N}$${\mathbb{R}.}$$\scriptstyle{\phantom{aa}\Phi}$$\scriptstyle{\pi}$$\scriptstyle{\Psi\phantom{aa}}$$\scriptstyle{\pi}$$\scriptstyle{\tau^{(p)}}$$\scriptstyle{\tau}$$\scriptstyle{\sigma^{(q)}}$$\scriptstyle{\sigma}$

######  Remark 2.5.

We observe the following:

  1. (a)

It is easy to verify that Definition 2.4 defines an equivalence relation.

  2. (b)

By this definition, a control system $(M,\mathcal{C}_{M})$ is dynamically
equivalent to each of its total prolongations
$\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$.

  3. (c)

A dynamic equivalence with $p=q=0$ is a static equivalence. To see this, let
$(t,{\bf x},{\bf u})$ and $(t,{\bf y},{\bf v})$ be coordinates on $M$ and $N$,
respectively. Represent $\Phi$ in local coordinates as $(t,{\bf y},{\bf
v})=\big{(}t,\phi^{s}({\bf x},{\bf u}),\phi^{c}({\bf x},{\bf u})\big{)}$.
(Here the superscripts ‘$s$’ and ‘$c$’ of $\phi$ stand for ‘state’ and
‘control’, respectively.) Since $\Phi$ maps integral curves of
$(M,\mathcal{C}_{M})$ to integral curves of $(N,\mathcal{C}_{N})$, it is
necessary that, for each $\mathrm{d}y_{i}-g_{i}({\bf y},{\bf
v})\mathrm{d}t\in\mathcal{C}_{N}$, its pull-back

$\displaystyle\Phi^{*}(\mathrm{d}y_{i}-g_{i}({\bf y},{\bf
v})\mathrm{d}t)=\mathrm{d}\big{(}\phi^{s}_{i}({\bf x},{\bf
u})\big{)}-g_{i}\big{(}\phi^{s}({\bf x},{\bf u}),\phi^{c}({\bf x},{\bf
u})\big{)}\mathrm{d}t$

is contained in $\mathcal{C}_{M}$. It follows that $\phi^{s}({\bf x},{\bf u})$
is independent of ${\bf u}$. A similar argument applies to $\Psi$. Finally,
Condition (iv) in Definition 2.4 implies that $\Phi$ and $\Psi$ are inverses
of each other.

  4. (d)

This definition of dynamic equivalence corresponds to the notion of
_endogenous transformation_ in the control literature (see [10]). In broader
contexts, it is related to the notion of _Lie–Bäcklund equivalences_ (see [1,
6, 9]), that of _$\mathcal{C}$ -transformations_ (see [5, 8]) and equivalences
between differential algebras (see [7]).

######  Definition 2.6.

We call the pair of integers $(p,q)$ in Definition 2.4 the _height_ of the
corresponding dynamic equivalence.

######  Remark 2.7.

Keeping the notations from the above, $p$ is the highest derivative of ${\bf
u}$ that $\Phi$ depends on, and $q$ is the highest derivative of ${\bf v}$
that $\Psi$ depends on. On the other hand, one may find the highest derivative
of each $v_{\beta}$ that $\Psi$ depends on and sum over $\beta$. Of course,
this depends on the choice of coordinates on $N$. Such a sum is related to the
notion of _differential weight_ (of $\Psi$) defined in [11], where its
relation with differential flatness is investigated.

Given two dynamically equivalent control systems $(M,\mathcal{C}_{M})$ and
$(N,\mathcal{C}_{N})$, if needed, one could always apply a _partial
prolongation_ (for details, see [14]) to one of them such that the resulting
systems have the same number of states and are still dynamically equivalent.
When this is achieved, the proposition below would become applicable.

######  Proposition 2.8.

Let $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ be control systems with
the same number of states. The height $(p,q)$ of a dynamic equivalence between
them must satisfy either $p=q=0$ or $p,q>0$.

###### Proof.

Suppose that the following commutative diagram represents a dynamic
equivalence of height $(p,0)$ $(p>0)$ between $(M,\mathcal{C}_{M})$ and
$(N,\mathcal{C}_{N})$:

${\big{(}M^{(p)},{\mathcal{C}^{(p)}}\big{)}}$${(M,\mathcal{C}_{M})}$${(N,\mathcal{C}_{N}).}$$\scriptstyle{\displaystyle\pi}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$

By the assumption,
$\operatorname{rank}(\mathcal{C}_{M})=\operatorname{rank}(\mathcal{C}_{N})$.
Since

$\displaystyle\pi^{*}\mathcal{C}_{M}=(\Psi\circ\Phi)^{*}\mathcal{C}_{M}=\Phi^{*}(\Psi^{*}\mathcal{C}_{M})\subseteq\Phi^{*}\mathcal{C}_{N},$

and since $\pi$, $\Phi$, $\Psi$ are all submersions, it is necessary that
$\langle\pi^{*}\mathcal{C}_{M}\rangle=\langle\Phi^{*}\mathcal{C}_{N}\rangle$.
Now, $\Phi$ is constant along the Cauchy characteristics of
$\Phi^{*}\mathcal{C}_{N}$, since the Cartan system (see [3]) of
$\mathcal{C}_{N}$ generates the entire cotangent bundle of $N$. On the other
hand, the Cauchy characteristics of $\pi^{*}\mathcal{C}_{M}$ are precisely the
fibres of $\pi$. This proves that $\Phi$ factors through $M$, a contradiction
to the choice of $p$. The case when $p=0$, $q>0$ is similar. ∎

##  3 Infinite permutation matrices associated
to a dynamic equivalence

In this section, we assume $p,q>0$ unless otherwise noted.

Given a control system $(M,\mathcal{C}_{M})$, one automatically obtains a
system of projections

$\displaystyle\pi_{k,j}\colon\ M^{(k)}\rightarrow M^{(j)},\qquad k\geq j.$

The inverse limit of this projective system is denoted as

$\displaystyle M^{(\infty)}:=\varprojlim_{k}M^{(k)}.$

Let $\pi_{k}\colon M^{(\infty)}\rightarrow M^{(k)}$ be the canonical
projections. Since $\pi_{k,j}^{*}\mathcal{C}^{(j)}\subseteq\mathcal{C}^{(k)}$
for all $k\geq j\geq 0$, one can define $\mathcal{C}^{(\infty)}$ to be the
differential system generated by $\bigcup_{k\geq
0}\pi_{k}^{*}\mathcal{C}^{(k)}$.

In coordinates, if $(M,\mathcal{C}_{M})$ corresponds to the system $\dot{\bf
x}={\bf f}({\bf x},{\bf u})$, then $M^{(\infty)}$ has the standard coordinates
$\big{(}t,{\bf x},{\bf u},{\bf u}^{(1)},\dots\big{)}$;
$\mathcal{C}^{(\infty)}$ is generated by the $1$-forms

$\displaystyle\mathrm{d}x_{i}-f_{i}({\bf x},{\bf
u})\mathrm{d}t,\qquad\mathrm{d}u_{\alpha}-u_{\alpha}^{(1)}\mathrm{d}t,\qquad\mathrm{d}u_{\alpha}^{(\ell)}-u_{\alpha}^{(\ell+1)}\mathrm{d}t,$
$\displaystyle i=1,\dots,n,\qquad\alpha=1,\dots,m,\qquad\ell\geq 1.$

The pair $\big{(}M^{(\infty)},\mathcal{C}^{(\infty)}\big{)}$ is called the
_infinite prolongation_ of $(M,\mathcal{C}_{M})$.

Now suppose that $(M,\mathcal{C}_{M})$ $(\dot{\bf x}={\bf f}({\bf x},{\bf
u}))$ and $(N,\mathcal{C}_{N})$ $(\dot{\bf y}={\bf g}({\bf y},{\bf v}))$ are
two control systems with $n_{1}$, $n_{2}$ states and $m_{1}$, $m_{2}$
controls, respectively. (Note that, at this point, we do not assume that
either the number of states or the number of controls is the same for both
systems). Furthermore, suppose that a dynamic equivalence of height $(p,q)$
between $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ is given by

$\displaystyle\Phi\colon\ M^{(p)}\rightarrow N,\qquad\Psi\colon\
N^{(q)}\rightarrow M.$

It can be shown that $\Phi$ and $\Psi$ induce, respectively, maps

$\displaystyle\Phi^{(\infty)}\colon\ M^{(\infty)}\rightarrow
N^{(\infty)},\qquad\Psi^{(\infty)}\colon\ N^{(\infty)}\rightarrow
M^{(\infty)},$

which satisfy

$\displaystyle\Phi^{(\infty)}\circ\Psi^{(\infty)}={\rm
Id},\qquad\Psi^{(\infty)}\circ\Phi^{(\infty)}={\rm Id}.$

Moreover, if we let

$\displaystyle{\boldsymbol{\omega}}^{0}=\mathrm{d}{\bf x}-{\bf f}({\bf x},{\bf
u})\mathrm{d}t,\qquad{\boldsymbol{\omega}}^{k}=\mathrm{d}{\bf u}^{(k-1)}-{\bf
u}^{(k)}\mathrm{d}t,\qquad k\geq 1,$
$\displaystyle{\boldsymbol{\eta}}^{0}=\mathrm{d}{\bf y}-{\bf g}({\bf y},{\bf
v})\mathrm{d}t,\qquad{\boldsymbol{\eta}}^{k}=\mathrm{d}{\bf v}^{(k-1)}-{\bf
v}^{(k)}\mathrm{d}t,\qquad k\geq 1,$

then there exist matrices $A^{i}_{j}$, $B^{i}_{j}$ $(i,j\geq 0)$
satisfying444Here we have assumed $p,q>0$, otherwise, these properties may not
hold. For instance, consider the standard dynamic equivalence between an
arbitrary $(M,\mathcal{C}_{M})$ and $\big{(}M^{(k)},\mathcal{C}^{(k)}\big{)}$.

  1. P1.

_for $k\geq 0$, we have_

$\displaystyle{\Phi^{(\infty)}}^{*}{\boldsymbol{\eta}}^{k}=A^{k}_{0}{\boldsymbol{\omega}}^{0}+A^{k}_{1}{\boldsymbol{\omega}}^{1}+\cdots+A^{k}_{p+k}{\boldsymbol{\omega}}^{p+k},$
(3.1)
$\displaystyle{\Psi^{(\infty)}}^{*}{\boldsymbol{\omega}}^{k}=B^{k}_{0}{\boldsymbol{\eta}}^{0}+B^{k}_{1}{\boldsymbol{\eta}}^{1}+\cdots+B^{k}_{q+k}{\boldsymbol{\eta}}^{q+k},$
(3.2)

where $A^{k}_{p+k},B^{k}_{q+k}\neq 0$ for all $k\geq 0$.

  2. P2.

_All $A_{p+k}^{k}$ are equal for $k\geq 1$; similarly for $B_{q+k}^{k}$.
Hence, we denote_

$\displaystyle A_{\infty}:=A_{p+k}^{k},\qquad B_{\infty}:=B_{q+k}^{k},\qquad
k\geq 1.$

  3. P3.

$\operatorname{rank}\big{(}A^{0}_{p}\big{)}=\operatorname{rank}(A_{\infty})>0$,
$\operatorname{rank}\big{(}B^{0}_{q}\big{)}=\operatorname{rank}(B_{\infty})>0$.

In particular, P1 follows from the definition of a dynamic equivalence of
height $(p,q)$; P2 can be seen by taking the exterior derivative of (3.1) on
both sides for $k\geq 1$ then reducing modulo
${\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{p+k}$ and similarly
for (3.2). For more details of these proofs, see [14].555The proofs in [14]
were made under the assumptions: $n_{1}=n_{2}$ and $m_{1}=m_{2}$, but they can
be easily modified to work for the situation being considered here.

To prove P3, note that there exists an $n_{2}\times m_{2}$ matrix $F$ such
that

$\displaystyle\mathrm{d}{\boldsymbol{\eta}}^{0}$ $\displaystyle\equiv
F{\boldsymbol{\eta}}^{1}\wedge\mathrm{d}t\mod{\boldsymbol{\eta}}^{0}$
$\displaystyle\equiv
FA^{1}_{p+1}{\boldsymbol{\omega}}^{p+1}\wedge\mathrm{d}t\mod\boldsymbol{{\boldsymbol{\omega}}}^{0},{\boldsymbol{\omega}}^{1},\dots,{\boldsymbol{\omega}}^{p}.$

On the other hand,

$\displaystyle\mathrm{d}{\boldsymbol{\eta}}^{0}$
$\displaystyle=\mathrm{d}\big{(}A^{0}_{0}{\boldsymbol{\omega}}^{0}+\cdots+A^{0}_{p}{\boldsymbol{\omega}}^{p}\big{)}$
$\displaystyle\equiv-A^{0}_{p}{\boldsymbol{\omega}}^{p+1}\wedge\mathrm{d}t\mod{\boldsymbol{\omega}}^{0},{\boldsymbol{\omega}}^{1},\dots,{\boldsymbol{\omega}}^{p}.$

Consequently,

$\displaystyle FA^{1}_{p+1}=-A^{0}_{p}.$

By Definition 2.1, $F$ has full rank. Therefore,
$\operatorname{rank}\big{(}A^{1}_{p+1}\big{)}=\operatorname{rank}\big{(}A^{0}_{p}\big{)}$.
The case for $B^{i}_{j}$ is similar. Property P3 follows.

###  3.1 Two classical theorems

Concerning dynamic equivalences between two control systems, the following two
classical theorems are of fundamental importance.

######  Theorem 3.1 ([10, 12]).

Two dynamically equivalent control systems must have the same number of
control variables.

######  Theorem 3.2 ([4, 13]).

A dynamic equivalence between two control systems with $n$ states and single
control can only have height $(p,q)=(0,0)$ $($i.e., the equivalence is
static$)$.

######  Remark 3.3.

The original result of Cartan takes a different form. In particular, the
notion of equivalence considered in [4] is _absolute equivalence_. On the
other hand, it is a consequence of Corollary 26 in [13] that two control
systems are absolutely equivalent (in an appropriate sense as $t$-independence
of maps is assumed) if and only if they are dynamically equivalent. Thus,
Theorem 3.2 may be regarded as a version of Cartan’s result (see also
Corollary 12 in [13]).

It turns out that these two theorems are consequences of the fact that the
following matrices are inverses of each other, after taking into account the
maps $\Phi^{(\infty)}$ and $\Psi^{(\infty)}$:

$\displaystyle\boldsymbol{A}=\left(\begin{matrix}A^{0}_{0}&\cdots&A^{0}_{p}&\boldsymbol{0}&\cdots\\\
A^{1}_{0}&\cdots&A^{1}_{p}&A_{\infty}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right),\qquad\boldsymbol{B}=\left(\begin{matrix}B^{0}_{0}&\cdots&B^{0}_{q}&\boldsymbol{0}&\cdots\\\
B^{1}_{0}&\cdots&B^{1}_{p}&B_{\infty}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right).$ (3.3)

In fact, suppose that $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ have
$n_{1}$, $n_{2}$ states and $m_{1}$, $m_{2}$ controls, respectively. We can
arrange that $n_{1}=n_{2}=:n$ by partially prolonging one of the systems, if
necessary. Moreover, since partial prolongations preserve the number of
controls, this will have no effect on the values of $m_{1}$ or $m_{2}$. Then
by Proposition 2.8, there are two possible cases: $p=q=0$ or $p,q>0$. In the
former case, we have static equivalence. In the latter case, the form of
matrix $\boldsymbol{A}$ implies that the $n+rm_{2}$ linearly independent
components of ${\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{r}$ are all
linear combinations of the $n+(r+p)m_{1}$ components of
${\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{r+p}$. When
$m_{2}>m_{1}$, this is impossible because $n+(r+p)m_{1}<n+rm_{2}$ as long as

$\displaystyle r>\frac{m_{1}p}{m_{2}-m_{1}}.$

Theorem 3.1 follows. Furthermore, when $p,q>0$, $\boldsymbol{A}$ and
$\boldsymbol{B}$ being inverses of each other requires that either
$A_{\infty}B_{\infty}=\boldsymbol{0}$ or
$B_{\infty}A_{\infty}=\boldsymbol{0}$. This is impossible when
$m_{1}=m_{2}=1$, in which case $A_{\infty}$ and $B_{\infty}$ are just
nonvanishing functions. Theorem 3.2 is an immediate consequence.

More generally, we have the

######  Lemma 3.4.

Suppose that $\mathcal{E}$ is a dynamic equivalence of height $(p,q)$
$(p,q>0)$ between two control systems with $m$ controls and not necessarily
the same number of states. The associated matrices $A_{\infty}$ and
$B_{\infty}$ must satisfy

$\displaystyle
2\leq\operatorname{rank}(A_{\infty})+\operatorname{rank}(B_{\infty})\leq m.$
(3.4)

###### Proof.

This is because $p,q>0$ implies $A_{\infty}B_{\infty}=\boldsymbol{0}$.∎

###  3.2 The infinite permutation matrix $\boldsymbol{\mathcal{S}}$

Let $(M,\mathcal{C}_{M})$, $(N,\mathcal{C}_{N})$, ${\boldsymbol{\omega}}^{i}$,
${\boldsymbol{\eta}}^{i}$, $\boldsymbol{A}$, $\boldsymbol{B}$ be as above. In
[14], it is proved that there exist transformations

$\displaystyle\left(\begin{matrix}\bar{\boldsymbol{\omega}}^{0}\\\
\bar{\boldsymbol{\omega}}^{1}\\\
\vdots\end{matrix}\right)=\left(\begin{matrix}\boldsymbol{g}^{0}_{0}&\boldsymbol{0}&\dots\\\
\boldsymbol{g}^{1}_{0}&\boldsymbol{g}^{1}_{1}&\dots\\\
\vdots&\vdots&\ddots\end{matrix}\right)\left(\begin{matrix}{\boldsymbol{\omega}}^{0}\\\
{\boldsymbol{\omega}}^{1}\\\
\vdots\end{matrix}\right),\qquad\left(\begin{matrix}\bar{\boldsymbol{\eta}}^{0}\\\
\bar{\boldsymbol{\eta}}^{1}\\\
\vdots\end{matrix}\right)=\left(\begin{matrix}\boldsymbol{h}^{0}_{0}&\boldsymbol{0}&\dots\\\
\boldsymbol{h}^{1}_{0}&\boldsymbol{h}^{1}_{1}&\dots\\\
\vdots&\vdots&\ddots\end{matrix}\right)\left(\begin{matrix}{\boldsymbol{\eta}}^{0}\\\
{\boldsymbol{\eta}}^{1}\\\ \vdots\end{matrix}\right),$ (3.5)

where $\boldsymbol{g}^{i}_{i}=\boldsymbol{g}^{i+1}_{i+1}$,
$\boldsymbol{h}^{i}_{i}=\boldsymbol{h}^{i+1}_{i+1}$ for all $i\geq 1$, such
that, pointwise,

  1. C1.

$\begin{cases}\operatorname{span}\big{\\{}{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{k}\big{\\}}=\operatorname{span}\big{\\{}\bar{\boldsymbol{\omega}}^{0},\dots,\bar{\boldsymbol{\omega}}^{k}\big{\\}},\\\
\operatorname{span}\big{\\{}{\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{k}\big{\\}}=\operatorname{span}\big{\\{}\bar{\boldsymbol{\eta}}^{0},\dots,\bar{\boldsymbol{\eta}}^{k}\big{\\}},\end{cases}k\geq
0$;

  2. C2.

$\begin{cases}\mathrm{d}\bar{\boldsymbol{\omega}}^{\ell}=-\bar{\boldsymbol{\omega}}^{\ell+1}\wedge\mathrm{d}t\mod\bar{\boldsymbol{\omega}}^{0},\dots,\bar{\boldsymbol{\omega}}^{\ell},\\\
\mathrm{d}\bar{\boldsymbol{\eta}}^{\ell}=-\bar{\boldsymbol{\eta}}^{\ell+1}\wedge\mathrm{d}t\mod\bar{\boldsymbol{\eta}}^{0},\dots,\bar{\boldsymbol{\eta}}^{\ell},\end{cases}\ell\geq
1$;

  3. C3.

${\Phi^{(\infty)}}^{*}\bar{\boldsymbol{\eta}}=\bar{\boldsymbol{A}}{\bar{\boldsymbol{\omega}}}$,
${\Psi^{(\infty)}}^{*}\bar{\boldsymbol{\omega}}=\bar{\boldsymbol{B}}{\bar{\boldsymbol{\eta}}}$,
where $\bar{\boldsymbol{A}}$, $\bar{\boldsymbol{B}}$ take the same form as
(3.3) (in particular, all $\bar{A}^{k}_{p+k}$ are equal for $k\geq 1$, etc.),
and both $\bar{\boldsymbol{A}}$ and $\bar{\boldsymbol{B}}$ are _infinite
permutation matrices_ , that is, each row/column of $\bar{\boldsymbol{A}}$ and
$\bar{\boldsymbol{B}}$ contains a single $1$ with the rest of the entries
being all $0$.

In addition, we have

######  Proposition 3.5.

Assume that $p,q>0$. The matrices $\bar{\boldsymbol{A}}$ and
$\bar{{\boldsymbol{B}}}$ satisfy
$\bar{\boldsymbol{A}}=\bar{{\boldsymbol{B}}}^{T}.$ In particular, the infinite
permutation matrix $\bar{\boldsymbol{A}}$ must be of the form

$\displaystyle\mathcal{S}:=\bar{\boldsymbol{A}}=\left(\begin{matrix}\bar{A}^{0}_{0}&\bar{A}^{0}_{1}&\cdots&\bar{A}^{0}_{p}&\boldsymbol{0}&\cdots\\\
\bar{A}^{1}_{0}&\bar{A}^{1}_{1}&\cdots&\bar{A}^{1}_{p}&\bar{A}_{\infty}&\ddots\\\
\vdots&\vdots&\ddots&\vdots&&\ddots\\\
\bar{A}^{q}_{0}&\bar{A}^{q}_{1}&\cdots&\bar{A}^{q}_{p}\\\
\boldsymbol{0}&\bar{B}_{\infty}^{T}&&\\\
\vdots&\ddots&\ddots\end{matrix}\right);$ (3.6)

in other words, $\bar{A}^{q+k}_{k}=\bar{B}_{\infty}^{T}$ $(k\geq 1)$, and
$\bar{A}^{q+\ell}_{k}=\boldsymbol{0}$ for all $\ell>k$.

###### Proof.

The fact $\bar{\boldsymbol{A}}=\bar{{\boldsymbol{B}}}^{T}$ follows from
$\bar{\boldsymbol{A}}\bar{{\boldsymbol{B}}}=\bar{\boldsymbol{B}}\bar{\boldsymbol{A}}=\operatorname{diag}(1,1,\dots)$
and Property C3 above. Consequently,
$\bar{A}^{q+k}_{k}=\big{(}\bar{B}^{k}_{q+k}\big{)}^{T}=\bar{B}_{\infty}^{T}$
for $k\geq 1$. For a similar reason, $\bar{A}^{q+\ell}_{k}=\boldsymbol{0}$ for
all $\ell>k$.∎

######  Definition 3.6.

Let $\mathcal{S}$, taking the form of (3.6), be an infinite permutation matrix
obtained from a dynamic equivalence with height $(p,q)$ $(p,q>0)$ between two
control systems. Let
$r^{i}_{j}=\operatorname{rank}\big{(}\bar{A}^{i}_{j}\big{)}$. We define the
_rank matrix_ associated to $\mathcal{S}$ to be

$\displaystyle\mathcal{R}(\mathcal{S}):=\big{(}r^{i}_{j}\big{)}.$

Given a dynamic equivalence, an associated matrix $\mathcal{S}$ may depend on
the choice of the transformations
$\bar{\boldsymbol{\omega}}=\boldsymbol{G}{\boldsymbol{\omega}}$ and
$\bar{\boldsymbol{\eta}}=\boldsymbol{H}{\boldsymbol{\eta}}$. However, we have

######  Proposition 3.7.

If $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are two infinite permutation
matrices obtained from the same dynamic equivalence, then their rank matrices
satisfy

$\displaystyle\mathcal{R}(\mathcal{S}_{1})=\mathcal{R}(\mathcal{S}_{2}).$

###### Proof.

Suppose that the underlying dynamic equivalence has height $(p,q)$ $(p,q>0)$.
One can write $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ in block forms

$\displaystyle\mathcal{S}_{1}=\left(\begin{matrix}U^{0}_{0}&\cdots&U^{0}_{p}&\boldsymbol{0}&\cdots\\\
\vdots&\ddots&\vdots&&\ddots\\\ U^{q}_{0}&\cdots&U^{q}_{p}\\\
\boldsymbol{0}&&&\ddots\\\
\vdots&\ddots\end{matrix}\right),\qquad\mathcal{S}_{2}=\left(\begin{matrix}V^{0}_{0}&\cdots&V^{0}_{p}&\boldsymbol{0}&\cdots\\\
\vdots&\ddots&\vdots&&\ddots\\\ V^{q}_{0}&\cdots&V^{q}_{p}\\\
\boldsymbol{0}&&&\ddots\\\ \vdots&\ddots\end{matrix}\right).$

Let $u^{i}_{j}:=\operatorname{rank}\big{(}U^{i}_{j}\big{)}$ and
$v^{i}_{j}:=\operatorname{rank}\big{(}V^{i}_{j}\big{)}$. Since
$\mathcal{S}_{1}$, $\mathcal{S}_{2}$ arise from the same dynamic equivalence,
there exist invertible block lower triangular matrices666Respectively, the
block sizes of $\boldsymbol{K}$ and $\boldsymbol{L}$ are the same as those of
$\boldsymbol{G}$ and $\boldsymbol{H}$.

$\displaystyle\boldsymbol{K}=\left(\begin{matrix}\boldsymbol{k}^{0}_{0}&\boldsymbol{0}&\dots\\\
\boldsymbol{k}^{1}_{0}&\boldsymbol{k}^{1}_{1}&\dots\\\
\vdots&\vdots&\ddots\end{matrix}\right),\qquad\boldsymbol{L}=\left(\begin{matrix}\boldsymbol{\ell}^{0}_{0}&\boldsymbol{0}&\dots\\\
\boldsymbol{\ell}^{1}_{0}&\boldsymbol{\ell}^{1}_{1}&\dots\\\
\vdots&\vdots&\ddots\end{matrix}\right),$

where $\boldsymbol{k}^{i}_{i}=\boldsymbol{k}^{i+1}_{i+1}$,
$\boldsymbol{\ell}^{i}_{i}=\boldsymbol{\ell}^{i+1}_{i+1}$ for all $i\geq 1$,
such that

$\displaystyle\mathcal{S}_{1}\boldsymbol{K}=\boldsymbol{L}\mathcal{S}_{2}=\left(\begin{matrix}W^{0}_{0}&\cdots&W^{0}_{p}&\boldsymbol{0}&\cdots&\cdots&\cdots&\cdots\\\
W^{1}_{0}&\cdots&\cdots&W^{1}_{p+1}&\ddots\\\ \vdots&&&&\ddots&\ddots\\\
W^{q}_{0}&\cdots&\cdots&\cdots&\cdots&W^{q}_{p+q}&\boldsymbol{0}&\cdots\\\
\vdots&&&&&&\ddots&\ddots\end{matrix}\right).$

As results of the forms of $\boldsymbol{K}$ and $\boldsymbol{L}$, we have

  1. (i)

$u^{i}_{p+i}=v^{i}_{p+i}$ for all $i\geq 0$. This is because
$W^{i}_{p+i}=U^{i}_{p+i}\boldsymbol{k}^{p+i}_{p+i}=\boldsymbol{\ell}^{i}_{i}V^{i}_{p+i}$,
where both $\boldsymbol{k}^{p+i}_{p+i}$ and $\boldsymbol{\ell}^{i}_{i}$ are
invertible.

  2. (ii)

$u^{0}_{i}=v^{0}_{i}$ for all $0\leq i<p$. To see why this is true, consider
the submatrix $\big{(}W^{0}_{i}W^{0}_{i+1}\cdots W^{0}_{p}\big{)}$. For each
$i<p$, its row rank equals to $v^{0}_{i}+v^{0}_{i+1}+\cdots+v^{0}_{p}$, which
must be equal to its column rank $u^{0}_{i}+u^{0}_{i+1}+\cdots+u^{0}_{p}$.

  3. (iii)

$u^{1}_{i}=v^{1}_{i}$ for all $0\leq i<p+1$. To see this, consider the
submatrix
$\left(\begin{smallmatrix}W^{0}_{i}&\cdots&W^{0}_{p}&\boldsymbol{0}\\\
W^{1}_{i}&\cdots&W^{1}_{p}&W^{1}_{p+1}\end{smallmatrix}\right)$. Its row rank
$\big{(}v^{0}_{i}+\cdots+v^{0}_{p}\big{)}+\big{(}v^{1}_{i}+\cdots+v^{1}_{p+1}\big{)}$
must be equal to its column rank
$\big{(}u^{0}_{i}+\cdots+u^{0}_{p}\big{)}+\big{(}u^{1}_{i}+\cdots+u^{1}_{p+1}\big{)}$.
Let $i$ decrease from $p$ and use $(ii)$. The desired result follows.

  4. (iv)

$u^{j}_{i}=v^{j}_{i}$ for $j\geq 2$, $0\leq i<p+j$. This can be verified by a
similar comparison between the column and row ranks of the submatrices

$\displaystyle\left(\begin{matrix}W^{0}_{i}&\cdots&W^{0}_{p}&\boldsymbol{0}&\cdots&\boldsymbol{0}\\\
W^{1}_{i}&\cdots&W^{1}_{p}&W^{1}_{p+1}&\ddots&\vdots\\\
\vdots&&&&\ddots&\boldsymbol{0}\\\
W^{j}_{i}&\cdots&W^{j}_{p}&W^{j}_{p+1}&\cdots&W^{j}_{p+j}\end{matrix}\right),\qquad
i<p+j.$

This completes the proof.∎

As a consequence of Proposition 3.7, we have

######  Corollary 3.8.

If $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are two infinite permutation
matrices obtained from a dynamic equivalence with height $(p,q)$ $(p,q>0)$,
then there exist block diagonal matrices

$\displaystyle\boldsymbol{K}=\operatorname{diag}\big{(}\boldsymbol{k}^{i}_{i}\big{)}_{i\geq
0},\qquad\boldsymbol{L}=\operatorname{diag}\big{(}\boldsymbol{\ell}^{i}_{i}\big{)}_{i\geq
0},$

where $\boldsymbol{k}^{i}_{i}$ and $\boldsymbol{\ell}^{i}_{i}$ are usual
permutation matrices of appropriate sizes,777That is, the sizes of
$\boldsymbol{k}^{i}_{i}$ and $\boldsymbol{\ell}^{i}_{i}$ are consistent with
the product of block matrices $\boldsymbol{L}\mathcal{S}_{2}\boldsymbol{K}$.
such that

$\displaystyle\mathcal{S}_{1}=\boldsymbol{L}\mathcal{S}_{2}\boldsymbol{K}.$

###### Proof.

This is because $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ (i) are permutation
matrices; and (ii) have the same rank in each pair of corresponding blocks.∎

######  Corollary 3.9.

Let $\mathcal{S}$ be an infinite permutation matrix obtained from a dynamic
equivalence of height $(p,q)$ $(p,q>0)$. Using the notations in (3.3), the
associated rank matrix $\mathcal{R}(\mathcal{S})=\big{(}r^{i}_{j}\big{)}$
satisfies

$\displaystyle r^{k}_{p+k}=\operatorname{rank}(A_{\infty}),\qquad
r^{q+k}_{k}=\operatorname{rank}(B_{\infty}),\qquad k\geq 0.$

###### Proof.

To see why this is true, first notice that a transformation (3.5) preserves
the ranks of $A_{\infty}$ and $B_{\infty}$; then use Property P3.∎

######  Remark 3.10.

A dynamic equivalence may be viewed as an invertible differential operator [2,
5, 9]. In particular, one may, according to [5], define the associated _$d$
-scheme of squares_. It is interesting to observe how the relations (3)–(5) in
[5] resemble the conditions that $r^{i}_{j}$ satisfy for our rank matrix, but
we will not pursue this relation further in the current article.

##  4 The height of a dynamic equivalence

Given two control systems $(M,\mathcal{C}_{M})$ ($n_{1}$ states, $m$ controls)
and $(N,\mathcal{C}_{N})$ ($n_{2}$ states, $m$ controls) that are dynamically
equivalent, it is interesting to ask: _What are the possible heights of a
dynamic equivalence?_ A particular instance is Theorem 3.2, which tells us
that the height suggests how control systems with $m=1$ and $m>1$ are
qualitatively different. The current section will present some new results in
this direction.

###  4.1 Some rank equalities and inequalities

Throughout this section, let $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$
be as above. Suppose that a dynamic equivalence between them has height
$(p,q)$ with $p,q>0$. Let $\mathcal{S}$ be an associated infinite permutation
matrix (equation (3.6)), obtained from a choice of coframes
$\big{(}\bar{\boldsymbol{\omega}}^{0},\bar{\boldsymbol{\omega}}^{1},\dots\big{)}$
and
$\big{(}\bar{\boldsymbol{\eta}}^{0},\bar{\boldsymbol{\eta}}^{1},\dots\big{)}$.
Let $\mathcal{R}(\mathcal{S})=\big{(}r^{i}_{j}\big{)}$ be the corresponding
rank matrix (Definition 3.6).

######  Proposition 4.1.

$r^{i}_{j}$ satisfy the following equalities

$\displaystyle\sum_{i\geq 0}r^{i}_{0}=n_{1},\qquad\sum_{j\geq
0}r^{0}_{j}=n_{2},\qquad\sum_{i\geq 0}r^{i}_{k}=\sum_{j\geq
0}r^{k}_{j}=m,\qquad k=1,2,\dots.$ (4.1)

###### Proof.

This is because the matrix $\mathcal{S}$ is an infinite permutation matrix.∎

######  Proposition 4.2.

$r^{i}_{j}$ satisfy the following inequalities:

  1. $(i)$

for $i,j\neq 0$, then

$\displaystyle
r^{i}_{j}\leq\min\left\\{\sum_{k=0}^{j+1}r^{i+1}_{k},\sum_{k=0}^{i+1}r^{k}_{j+1}\right\\};$

  2. $(ii)$

for $j\neq 0$,

$\displaystyle
r^{0}_{j}\leq\min\left\\{r^{0}_{j+1}+r^{1}_{j+1},(n_{2}-m)+\sum_{k=0}^{j+1}r^{1}_{k}\right\\};$

  3. $(iii)$

for $i\neq 0$,

$\displaystyle
r^{i}_{0}\leq\min\left\\{r^{i+1}_{0}+r^{i+1}_{1},(n_{1}-m)+\sum_{k=0}^{i+1}r^{k}_{1}\right\\};$

  4. $(iv)$

$r^{0}_{0}\leq\min\big{\\{}(n_{1}-m)+r^{0}_{1}+r^{1}_{1},(n_{2}-m)+r^{1}_{0}+r^{1}_{1}\big{\\}}$.

###### Proof.

To prove (i), the case when $r^{i}_{j}=0$ is trivial. Otherwise, suppose that
the submatrix $\bar{A}^{i}_{j}$ of $\mathcal{S}$ has $1$’s precisely at
positions $(a_{k},b_{k})$, $1\leq a_{k},b_{k}\leq m$, $1\leq k\leq r^{i}_{j}$.
Dropping pullback symbols, we have

$\displaystyle\bar{\eta}^{i}_{a_{k}}=\bar{\omega}^{j}_{b_{k}}.$

Condition C2 demands

$\displaystyle\mathrm{d}\bar{\eta}^{i}_{a_{k}}\equiv-\bar{\eta}^{i+1}_{a_{k}}\wedge\mathrm{d}t\mod{\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i}$
(4.2)

and

$\displaystyle\mathrm{d}\bar{\omega}^{j}_{b_{k}}\equiv-\bar{\omega}^{j+1}_{b_{k}}\wedge\mathrm{d}t\mod{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j}.$
(4.3)

At most $\sum\limits_{k=0}^{j}r^{i+1}_{k}$ congruences in (4.2) are reduced to
the following form once the congruence is taken modulo
${\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i},{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j}$:

$\displaystyle\mathrm{d}\bar{\eta}^{i}_{a_{k}}\equiv
0\mod{\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i},{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j};$

similarly, at most $\sum\limits_{k=0}^{i}r^{k}_{j+1}$ congruences in (4.3) are
reduced to the following form once the congruence is taken modulo
${\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j},{\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i}$:

$\displaystyle\mathrm{d}\bar{\omega}^{j}_{b_{k}}\equiv
0\mod{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j},{\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i}.$

The remaining congruences in (4.2) and (4.3), reduced modulo
${\boldsymbol{\eta}}^{0},\dots,{\boldsymbol{\eta}}^{i},{\boldsymbol{\omega}}^{0},\dots,{\boldsymbol{\omega}}^{j}$,
must match up as identical congruences; in particular, the corresponding
$\bar{\eta}^{i+1}_{a_{k}}$ and $\bar{\omega}^{j+1}_{b_{k}}$ must be equal.
Since the equalities $\bar{\eta}^{i+1}_{a_{k}}=\bar{\omega}^{j+1}_{b_{k}}$ are
at most $r^{i+1}_{j+1}$ in number, we obtain the inequalities

$\displaystyle r^{i}_{j}\leq\sum_{k=0}^{j+1}r^{i+1}_{k},\qquad
r^{i}_{j}\leq\sum_{k=0}^{i+1}r^{i}_{j+1},$

which justifies (i).

To prove (iv), suppose that $r^{0}_{0}>0$ and that the submatrix
$\bar{A}^{0}_{0}$ has $1$’s precisely at positions $(a_{k},b_{k})$, $1\leq
a_{k}\leq n_{2}$, $1\leq b_{k}\leq n_{1}$, $1\leq k\leq r^{0}_{0}$. We have

$\displaystyle\bar{\eta}^{0}_{a_{k}}=\bar{\omega}^{0}_{b_{k}}.$

There exist functions $C^{1}_{k},\dots,C^{s}_{k}$, $D^{1}_{k},\dots,D^{s}_{k}$
such that

$\displaystyle\mathrm{d}\bar{\eta}^{0}_{a_{k}}\equiv-\big{(}C^{1}_{k}\bar{\eta}^{1}_{1}+\cdots+C^{s}_{k}\bar{\eta}^{1}_{s}\big{)}\wedge\mathrm{d}t\mod{\boldsymbol{\eta}}^{0},$
(4.4)
$\displaystyle\mathrm{d}\bar{\omega}^{0}_{b_{k}}\equiv-\big{(}D^{1}_{k}\bar{\omega}^{1}_{1}+\cdots+D^{s}_{k}\bar{\omega}^{1}_{s}\big{)}\wedge\mathrm{d}t\mod{\boldsymbol{\omega}}^{0}.$
(4.5)

Since the rank of $\mathrm{d}{\boldsymbol{\eta}}^{0}$ (modulo
${\boldsymbol{\eta}}^{0}$) is $m$, we have

$\displaystyle\operatorname{rank}\big{(}C^{\alpha}_{k}\big{)}\geq
m-\big{(}n_{2}-r^{0}_{0}\big{)};$ (4.6)

similarly, we have

$\displaystyle\operatorname{rank}\big{(}D^{\alpha}_{k}\big{)}\geq
m-\big{(}n_{1}-r^{0}_{0}\big{)}.$ (4.7)

On the other hand, by taking the congruences (4.4) and (4.5) reducing modulo
both ${\boldsymbol{\omega}}^{0}$ and ${\boldsymbol{\eta}}^{0}$, it is not hard
to see that

$\displaystyle\operatorname{rank}\big{(}C^{\alpha}_{k}\big{)}\leq
r^{1}_{0}+r^{1}_{1},\qquad\operatorname{rank}\big{(}D^{\alpha}_{k}\big{)}\leq
r^{0}_{1}+r^{1}_{1}.$ (4.8)

Combining (4.6), (4.7) and (4.8), we obtain the inequalities in (iv).

The proofs of (ii) and (iii) are similar; we leave them to the reader.∎

###  4.2 Admissible heights

######  Theorem 4.3.

Let $\mathcal{E}$ be a dynamic equivalence between two control systems with
$n_{1}$, $n_{2}$ states, respectively, and $m$ controls. The height $(p,q)$ of
$\mathcal{E}$ must satisfy: if $p,q>0$, then

$\displaystyle\min\\{(p-1)\delta+r_{1}p+n_{1},(q-1)\delta+r_{2}q+n_{2}\\}\geq\max\\{r_{1}p+n_{1},r_{2}q+n_{2}\\},$
(4.9)

where $r_{1}=\operatorname{rank}(A_{\infty})$,
$r_{2}=\operatorname{rank}(B_{\infty})$, $\delta=m-r_{1}-r_{2}$.

###### Proof.

The rank matrix associated to $\mathcal{E}$ takes the form

$\displaystyle\mathcal{R}=\left(\begin{matrix}r^{0}_{0}&\cdots&r^{0}_{p}&{0}&\cdots\\\
\vdots&\ddots&\vdots&&\ddots\\\ r^{q}_{0}&\cdots&r^{q}_{p}\\\ {0}&&&\ddots\\\
\vdots&\ddots\end{matrix}\right),$

where $r^{k}_{p+k}=\operatorname{rank}(A_{\infty})=r_{1}$,
$r^{q+k}_{k}=\operatorname{rank}(B_{\infty})=r_{2}$ for all $k\geq 0$.

Let

$\displaystyle
C:=\sum_{i=0}^{q-1}\sum_{j=p}^{p+i}r^{i}_{j}-\sum_{i=0}^{q-1}r^{i}_{p+i},\qquad
D:=\sum_{j=0}^{p-1}\sum_{i=q}^{q+j}r^{i}_{j}-\sum_{j=0}^{p-1}r^{q+j}_{j}.$

By Proposition 4.1, we have

$\displaystyle 0\leq C\leq(q-1)\delta,\qquad 0\leq D\leq(p-1)\delta.$ (4.10)

Furthermore, we have

$\displaystyle
E:=(q-1)(m-r_{1})-\big{(}n_{1}-r^{0}_{0}-r_{2}\big{)}-C=(p-1)(m-r_{2})-\big{(}n_{2}-r^{0}_{0}-r_{1}\big{)}-D.$

Using (4.10), we obtain

$\displaystyle\max\\{r_{1}p+n_{1},r_{2}q+n_{2}\\}\leq E-r^{0}_{0}+n_{1}+n_{2}$
$\displaystyle\qquad{}\leq\min\\{(p-1)\delta+r_{1}p+n_{1},(q-1)\delta+r_{2}q+n_{2}\\}.$

The conclusion follows.∎

######  Corollary 4.4.

Let $\mathcal{E}$ be a dynamic equivalence between two control systems with
$n_{1}$, $n_{2}$ states, respectively, and $m$ controls. If
$\operatorname{rank}(A_{\infty})+\operatorname{rank}(B_{\infty})=m$, then the
height $(p,q)$ of $\mathcal{E}$ must satisfy: when $p,q>0$, we have

$\displaystyle n_{1}+\operatorname{rank}(A_{\infty})\cdot
p=n_{2}+\operatorname{rank}(B_{\infty})\cdot q.$ (4.11)

###### Proof.

This is an immediate consequence of setting $\delta=0$ in (4.9).∎

######  Theorem 4.5.

The height $(p,q)$ of a dynamic equivalence between two control systems with
$n_{1}$ and $n_{2}$ states, respectively, and $2$ controls must satisfy
$n_{1}+p=n_{2}+q$.

###### Proof.

Let $(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ be the two control systems
in question with a dynamic equivalence given by submersions $\Phi\colon
M^{(p)}\rightarrow N$ and $\Psi\colon N^{(q)}\rightarrow M$.

First consider the case when $n_{1}=n_{2}$. By Proposition 2.8, we have either
$p=q=0$ or $p,q>0$. In the former case, there is nothing to prove. In the
latter case, when $m=2$, Lemma 3.4 implies that the only possibility is
$\operatorname{rank}(A_{\infty})=\operatorname{rank}(B_{\infty})=1$; then
apply Corollary 4.4.

It remains to consider the case when $n_{1}\neq n_{2}$. Assume that
$n_{1}>n_{2}$. The case when $p,q>0$ follows from a similar argument as the
above. Thus, it suffices to consider the case when $p=0$. Suppose that, under
a choice of coordinates $(t,{\bf y},{\bf v})$ on $N$, $\Psi$ depends
nontrivially on $v_{\alpha}^{(q)}$ for some $\alpha$. Fixing an index
$\beta\neq\alpha$, we can construct a partial prolongation
$\big{(}\bar{N},\bar{\mathcal{C}}_{N}\big{)}$ by letting
$\bar{N}=N\times\mathbb{R}^{n_{1}-n_{2}}$ with coordinates $\big{(}t,{\bf
y},{\bf v},v_{\beta}^{(1)},\dots,v_{\beta}^{(n_{1}-n_{2})}\big{)}$ and
$\bar{\mathcal{C}}_{N}$ being generated by

$\displaystyle\mathcal{C}_{N},\,\mathrm{d}v_{\beta}-v_{\beta}^{(1)}\mathrm{d}t,\,\dots,\,\mathrm{d}v_{\beta}^{(n_{1}-n_{2}-1)}-v_{\beta}^{(n_{1}-n_{2})}\mathrm{d}t.$

Let $\bar{\Phi}\colon M^{(n_{1}-n_{2})}\rightarrow\bar{N}$ (indicated by the
dashed arrow in the diagram below) denote the submersion induced by $\Phi$.
Let $\bar{\pi}^{(q)}\colon\bar{N}^{(q)}\rightarrow N^{(q)}$ be the submersion
induced by the natural projection $\bar{\pi}\colon\bar{N}\rightarrow N$

${M^{(n_{1}-n_{2})}}$${\bar{N}^{(q)}}$${N^{(q)}}$${M}$${\bar{N}}$${N.}$$\scriptstyle{\bar{\pi}^{(q)}}$$\scriptstyle{\Psi}$$\scriptstyle{\pi}$$\scriptstyle{\Phi}$$\scriptstyle{\bar{\pi}}$

It is easy to see that the pair of submersions $\bar{\Phi}$ and
$\Psi\circ\bar{\pi}^{(q)}$ establishes a dynamic equivalence between
$(M,\mathcal{C}_{M})$ and $\big{(}\bar{N},\bar{\mathcal{C}}_{N}\big{)}$ of
height $(n_{1}-n_{2},q)$. Since $M$ and $\bar{N}$ have the same dimension, we
conclude that $n_{1}-n_{2}=q$. This completes the proof. ∎

######  Remark 4.6.

Theorem 4.5 suggests a qualitative distinction between control systems with
$m=2$ and those with $m>2$. In fact, when $m>2$, it may be, for a dynamic
equivalence, that
$\operatorname{rank}(A_{\infty})+\operatorname{rank}(B_{\infty})<m$, and
(4.11) does not need to hold. For details, see Example 4.7 below.

###  4.3 Examples

######  Example 4.7.

Let $(M,\mathcal{C}_{M})$ be a control system with 4 states and 3 controls,
where $M$ has coordinates $(t,{\bf x},{\bf u})$ and $\mathcal{C}_{M}$ is
generated by

$\displaystyle\mathrm{d}x_{1}-u_{1}\mathrm{d}t,\qquad\mathrm{d}x_{2}-x_{1}\mathrm{d}t,\qquad\mathrm{d}x_{3}-u_{2}\mathrm{d}t,\qquad\mathrm{d}x_{4}-u_{3}\mathrm{d}t.$
(4.12)

A partial prolongation $(M,\mathcal{C}_{M})$ can be obtained by adjoining
equations of the form $\dot{u}_{\alpha}^{(k)}=u_{\alpha}^{(k+1)}$ to the
original system. For example, consider $N_{1}=M\times\mathbb{R}^{3}$ with the
coordinates

$\displaystyle\big{(}t,\big{(}{\bf
x},u_{1},u_{2},u_{2}^{(1)}\big{)},\big{(}u_{1}^{(1)},u_{2}^{(2)},u_{3}\big{)}\big{)},$

where $u_{1}^{(1)}$, $u_{2}^{(1)}$ and $u_{2}^{(2)}$ are coordinates on the
$\mathbb{R}^{3}$ component; let $\mathcal{C}_{1}$ be the Pfaffian system
generated by (4.12) and the $1$-forms

$\displaystyle\mathrm{d}u_{1}-u_{1}^{(1)}\mathrm{d}t,\qquad\mathrm{d}u_{2}-u_{2}^{(1)}\mathrm{d}t,\qquad\mathrm{d}u_{2}^{(1)}-u_{2}^{(2)}\mathrm{d}t.$

Alternatively, consider $N_{2}=M\times\mathbb{R}^{3}$ with the coordinates

$\displaystyle\big{(}t,\big{(}{\bf
x},u_{3},u_{3}^{(1)},u_{3}^{(2)}\big{)},\big{(}u_{1},u_{2},u_{3}^{(3)}\big{)}\big{)},$

where $u_{3}^{(1)}$, $u_{3}^{(2)}$ and $u_{3}^{(3)}$ are coordinates on the
$\mathbb{R}^{3}$ component; let $\mathcal{C}_{2}$ be the Pfaffian system
generated by (4.12) and the $1$-forms

$\displaystyle\mathrm{d}u_{3}-u_{3}^{(1)}\mathrm{d}t,\qquad\mathrm{d}u_{3}^{(1)}-u_{3}^{(2)}\mathrm{d}t,\qquad\mathrm{d}u_{3}^{(2)}-u_{3}^{(3)}\mathrm{d}t.$

The standard submersions $\Phi\colon N_{1}^{(3)}\rightarrow N_{2}$,
$\Psi\colon N_{2}^{(2)}\rightarrow N_{1}$ give rise to a dynamic equivalence
between $(N_{1},\mathcal{C}_{1})$ and $(N_{1},\mathcal{C}_{2})$ (both having
$7$ states and $3$ controls) with height $(p,q)=(3,2)$. The associated rank
matrix is

$\displaystyle(r^{i}_{j})=\left(\begin{matrix}4&1&1&1&&&&&\\\ 2&0&0&0&1&&&&\\\
1&1&0&0&0&1&&&\\\ &1&1&0&0&0&1&&\\\ &&1&1&0&0&0&1&\\\
\hphantom{\ddots}&\hphantom{\ddots}&\hphantom{\ddots}&{\ddots}&\hphantom{\ddots}&\hphantom{\ddots}&\hphantom{\ddots}&\hphantom{\ddots}&\ddots\end{matrix}\right).$

In this example, (4.9) becomes an equality.

######  Example 4.8.

Let $(M,\mathcal{C}_{M})$ ($\dot{\bf x}={\bf f}({\bf x},{\bf u})$) and
$(N,\mathcal{C}_{N})$ $(\dot{\bf y}={\bf f}({\bf y},{\bf v}))$ be two copies
of a same control system with $3$ states and $2$ controls:

$\displaystyle\begin{cases}\dot{x}_{1}=u_{1},\\\ \dot{x}_{2}=u_{2},\\\
\dot{x}_{3}=f(x_{2},x_{3},u_{2}),\end{cases}\qquad\begin{cases}\dot{y}_{1}=v_{1},\\\
\dot{y}_{2}=v_{2},\\\ \dot{y}_{3}=f(y_{2},y_{3},v_{2}).\end{cases}$

For any $p>1$, the following pair of submersions $\Phi\colon
M^{(p)}\rightarrow N$ and $\Psi\colon N^{(p)}\rightarrow M$ define a dynamic
equivalence with height $(p,p)$ between $(M,\mathcal{C}_{M})$ and
$(N,\mathcal{C}_{N})$:

$\displaystyle({\bf y},{\bf v})=\Phi\big{(}{\bf x},{\bf u},\dots,{\bf
u}^{(p)}\big{)}=\big{(}u_{2}^{(p-1)}-x_{1},x_{2},x_{3};u_{2}^{(p)}-u_{1},u_{2}\big{)},$
$\displaystyle({\bf x},{\bf u})=\Psi\big{(}{\bf y},{\bf v},\dots,{\bf
v}^{(p)}\big{)}=\big{(}v_{2}^{(p-1)}-y_{1},y_{2},y_{3};v_{2}^{(p)}-v_{1},v_{2}\big{)}.$

This shows that there is no _a priori_ upper bound of dynamic equivalences
relating a fixed pair of control systems; it would only be meaningful to ask
whether a ‘minimum height’ exists among all possible dynamic equivalences
between such a fixed pair.

######  Example 4.9.

Assume $n_{1}>n_{2}$ in Theorem 4.5. This theorem tells us that even though
$\dim N^{(q)}\geq\dim M$ (in order for $N^{(q)}$ to submerse onto $M$) as long
as

$\displaystyle q\geq\frac{n_{1}-n_{2}}{2},$

an actual dynamic equivalence could only exist with $q\geq n_{1}-n_{2}$. One
may compare this fact with Theorem 52 in [13].

As an example, we consider the PVTOL system (see [10]).

Using the coordinates
$(t,(x,z,\theta,\dot{x},\dot{z},\dot{\theta}),(u_{1},u_{2}))$ on $M$, let
$\mathcal{C}_{M}$ be generated by the six $1$-forms:

$\displaystyle\mathrm{d}x-\dot{x}\mathrm{d}t,$
$\displaystyle\mathrm{d}z-\dot{z}\mathrm{d}t,$
$\displaystyle\mathrm{d}\theta-\dot{\theta}\mathrm{d}t,$
$\displaystyle\mathrm{d}\dot{x}-(-u_{1}\sin\theta+\epsilon
u_{2}\cos\theta)\mathrm{d}t,$
$\displaystyle\mathrm{d}\dot{z}-(u_{1}\cos\theta+\epsilon
u_{2}\sin\theta-1)\mathrm{d}t,$
$\displaystyle\mathrm{d}\dot{\theta}-u_{2}\mathrm{d}t,$

where $\epsilon$ is a constant.

Using the coordinates $(t,(y_{1},y_{2}),(\dot{y}_{1},\dot{y}_{2}))$ on $N$,
let $\mathcal{C}_{N}$ be the trivial system generated by the two $1$-forms:

$\displaystyle\mathrm{d}y_{1}-\dot{y}_{1}\mathrm{d}t,$
$\displaystyle\mathrm{d}y_{2}-\dot{y}_{2}\mathrm{d}t.$

By the argument above, if there exists a dynamic equivalence between
$(M,\mathcal{C}_{M})$ and $(N,\mathcal{C}_{N})$ (aka. $(M,\mathcal{C}_{M})$
being ‘differentially flat’), then such a dynamic equivalence must satisfy
$q\geq n_{1}-n_{2}=6-2=4$.

In fact, one can verify that the relation

$\displaystyle(y_{1},y_{2})=(x-\epsilon\sin\theta,z+\epsilon\cos\theta)$

induces a dynamic equivalence between $(M,\mathcal{C}_{M})$ and
$(N,\mathcal{C}_{N})$ of height $(p,q)=(0,4)$. For more details, see [10].

### Acknowledgements

We would like to thank our referees for their careful reading of the
manuscript and helpful comments that led to considerable improvement of this
article. The first and third authors were supported in part by NSF grant
DMS-1206272. The first author was supported in part by a Collaboration Grant
for Mathematicians from the Simons Foundation.

## References

  * [1] Anderson R.L., Ibragimov N.H., Lie–Bäcklund transformations in applications, SIAM Studies in Applied Mathematics, Vol. 1, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979.
  * [2] Aranda-Bricaire E., Moog C.H., Pomet J.-B., Infinitesimal Brunovský form for nonlinear systems with applications to dynamic linearization, in Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), Banach Center Publ., Vol. 32, Polish Acad. Sci. Inst. Math., Warsaw, 1995, 19–33.
  * [3] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  * [4] Cartan E., Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France 42 (1914), 12–48.
  * [5] Chetverikov V.N., Invertible linear ordinary differential operators, J. Geom. Phys. 113 (2017), 10–27.
  * [6] Fliess M., Lévine J., Martin P., Rouchon P., A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control 44 (1999), 922–937.
  * [7] Jakubczyk B., Equivalence of differential equations and differential algebras, Tatra Mt. Math. Publ. 4 (1994), 125–130.
  * [8] Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  * [9] Lévine J., On necessary and sufficient conditions for differential flatness, Appl. Algebra Engrg. Comm. Comput. 22 (2011), 47–90, arXiv:math.OC/0605405.
  * [10] Martin P., Murray R.M., Rouchon P., Flat systems, equivalence and feedback, in Advances in the Control of Nonlinear Systems (Murcia, 2000), Lect. Notes Control Inf. Sci., Vol. 264, Springer, London, 2001, 5–32.
  * [11] Nicolau F., Respondek W., Flatness of multi-input control-affine systems linearizable via one-fold prolongation, SIAM J. Control Optim. 55 (2017), 3171–3203.
  * [12] Pomet J.-B., A differential geometric setting for dynamic equivalence and dynamic linearization, in Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), Banach Center Publ., Vol. 32, Polish Acad. Sci. Inst. Math., Warsaw, 1995, 319–339.
  * [13] Sluis W.M., Absolute equivalence and its applications to control theory, Ph.D. Thesis, University of Waterloo, 1992.
  * [14] Stackpole M.W., Dynamic equivalence of control systems via infinite prolongation, Asian J. Math. 17 (2013), 653–688, arXiv:1106.5437.


# Updated Photometry of the Yellow Supergiant Progenitor and Late-time
Observations of the Type IIb Supernova 2016gkg

Charles D. Kilpatrick Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA  David
A. Coulter Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA 95064, USA  Ryan J. Foley Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA  Anthony L.
Piro The Observatories of the Carnegie Institution for Science, 813 Santa
Barbara St., Pasadena, CA 91101, USA  Armin Rest Department of Physics and
Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD
21218, USA  Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  César Rojas-Bravo Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA  Matthew R.
Siebert  Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA 95064, USA

###### Abstract

We present Hubble Space Telescope (HST) observations of the type IIb supernova
(SN) 2016gkg at 652, 1698, and 1795 days from explosion with the Advanced
Camera for Surveys (ACS) and Wide Field Camera 3 (WFC3). Comparing to pre-
explosion imaging from 2001 obtained with the Wide Field Planetary Camera 2,
we demonstrate that SN 2016gkg is now fainter than its candidate counterpart
in the latest WFC3 imaging, implying that the counterpart has disappeared and
confirming that it was the SN progenitor star. We show the latest light curve
and Keck spectroscopy of SN 2016gkg, which implies that SN 2016gkg is
declining more slowly than the expected rate for 56Co decay during its nebular
phase. We find that this emission is too luminous to be powered by other
radioisotopes, thus we infer that SN 2016gkg is entering a new phase in its
evolution where it is powered primarily by interaction with circumstellar
matter. Finally, we re-analyze the progenitor star spectral energy
distribution and late-time limits in the context of binary evolution models
and including emission from a potential companion star and find that all
companion stars would be fainter than our limiting magnitudes.

stars: evolution — supernovae: general — supernovae: individual (SN 2016gkg)

††facilities: HST (WFPC2, ACS, WFC3), Keck (LRIS), Swope (Direct)††software:
dolphot (Dolphin, 2016), DoPhot (Schechter et al., 1993), hst123 (Kilpatrick,
2021), photpipe (Rest et al., 2005), SWarp (Bertin, 2010)

fixltx2e firstofone

##  1 INTRODUCTION

Supernova (SN) 2016gkg is a type IIb SN (SN IIb, with broad, transient lines
of hydrogen; Filippenko, 1997) that exploded in NGC 613 on 20 September 2016
(Kilpatrick et al., 2017). Similar to other nearby SNe IIb with deep pre-
explosion imaging (SNe 1993J, 2008ax, 2011dh, and 2013df; Aldering et al.,
1994; Crockett et al., 2008; Maund et al., 2011; Van Dyk et al., 2014), its
counterpart was identified in archival Hubble Space Telescope (HST) data as a
yellow supergiant (Kilpatrick et al., 2017; Tartaglia et al., 2017; Bersten et
al., 2018). This stellar classification is consistent with the expectation
that SNe IIb have depleted hydrogen envelopes and thus hotter photospheric
temperatures than the red supergiant (RSG) progenitor stars of type II-P SNe
(Falk & Arnett, 1973; Falk, 1978; Smartt, 2009). However, the pathway through
which these yellow supergiants shed their hydrogen envelopes remains
ambiguous; the two most common explanations are enhanced radiative or eruptive
mass loss (e.g., Langer et al., 1994; Maeder & Meynet, 2000; Heger et al.,
2003; Smith, 2014) or interaction with and Roche-Lobe overflow onto a close
binary companion (Woosley et al., 1994; Izzard et al., 2004; Yoon, 2017).
Additional constraints on the pre-explosion evolution, circumstellar
environment, or possible companion star of SN 2016gkg can therefore help to
disentangle the pre-explosion evolution for SN IIb progenitor stars generally.

SN 2016gkg was also characterized by a rapid rise and luminous, ultraviolet-
bright early peak in its light curve (Kilpatrick et al., 2017; Arcavi et al.,
2017; Tartaglia et al., 2017; Bersten et al., 2018), which has been
interpreted as shock cooling emission from the hydrogen envelope of its
progenitor star. This interpretation is consistent with observations of other
well-observed SNe IIb with distinct early light curve peaks (e.g., SN 2011dh;
Arcavi et al., 2011; Bersten et al., 2012). When compared with models that
describe the rapid evolution as cooling in the outer envelope (Rabinak &
Waxman, 2011; Nakar & Piro, 2014; Sapir & Waxman, 2017; Piro et al., 2017,
2021), photometric data can be used to independently constrain the size and
mass of the hydrogen envelope. For SN 2016gkg, this radius was constrained as
30–150 $R_{\odot}$ (see, e.g., fig. 4 in Arcavi et al., 2017), depending
primarily on the shock cooling model and envelope structure assumed.

This indirect constraint on the progenitor star mass can be directly compared
to estimates from direct detection of the pre-explosion counterpart in Hubble
Space Telescope (HST) Wide Field Planetary Camera 2 (WFPC2) data. In
Kilpatrick et al. (2017), comparison between the pre-explosion photometry and
binary star models from BPASS (Eldridge et al., 2017) implied a progenitor
star mass of around $15~{}M_{\odot}$. Subsequent analyses of the same data
yielded estimates of 15–20 $M_{\odot}$ (Tartaglia et al., 2017) and 19.5
$M_{\odot}$ (Bersten et al., 2018), with both estimates invoking close binary
models to explain the photometry and low hydrogen envelope mass in the
progenitor star. However, subsequent analyses using Modules for Experiments in
Stellar Astrophysics (MESA) models suggest that for certain mass loss
prescriptions there are both single-star and binary systems that can reproduce
the pre-explosion photometry (Sravan et al., 2018; Gilkis & Arcavi, 2021).

Follow-up imaging of SN 2016gkg can reveal whether the pre-explosion
counterpart has disappeared, implying that it was indeed dominated by the
progenitor star (similar to the SN 2011dh progenitor; Van Dyk et al., 2013),
or if there is residual flux consistent with the pre-explosion source,
implying that SN 2016gkg exploded from a fainter star. In addition, deep
imaging, especially in bluer optical bands or the ultraviolet, may reveal the
presence of a companion star at the site of SN 2016gkg, similar to the
putative companion of the type IIb SN 1993J (Maund et al., 2004; Fox et al.,
2014) and potentially 2011dh (Folatelli et al., 2014; Maund et al., 2015).
This is critical for analysis of the pre-explosion WFPC2 data, which have
inferior resolution compared with later Advanced Camera for Surveys (ACS) and
Wide Field Camera 3 (WFC3) imaging and may reveal blended sources near the SN
2016gkg counterpart suggested in previous analyses (e.g., Tartaglia et al.,
2017).

Here we present follow-up HST and Swope imaging of SN 2016gkg, spanning
32–1795 days from explosion, as well as late-time spectroscopy. We present new
photometry of the pre-explosion source using the improved relative astrometry
of SN 2016gkg provided by HST imaging at 652 and 1795 days post-explosion and
updated analysis of that progenitor source. We also analyze the light curve of
SN 2016gkg using the new late-time photometry as well as constraints on the
progenitor system of SN 2016gkg in the context of new photometry and limits on
a companion star to the SN 2016gkg progenitor.

As in Kilpatrick et al. (2017), we adopt a Tully-Fisher distance to NGC 613 of
$D_{L}=26.4\pm 5.3$ Mpc (Nasonova et al., 2011). We assume a line-of-sight
Milky Way reddening of $E(B-V)=0.017$ mag (Schlafly & Finkbeiner, 2011). We
further assume that there is additional line-of-sight extinction to SN 2016gkg
based on the relatively weak Na i D line observed in its spectra at the
redshift of NGC 613 (Arcavi et al., 2017; Tartaglia et al., 2017), which
combined with the conversions of Poznanski et al. (2012) suggests emission
from SN 2016gkg was reddened by $E(B-V)=0.09\begin{subarray}{c}+0.06\\\
-0.03\end{subarray}$ mag.

##  2 Observations

###  2.1 HST Imaging

We analyzed all Hubble Space Telescope (HST) covering the site of SN 2016gkg,
including the pre-explosion WFPC2 imaging from 2001, ACS imaging from 2018,
and WFC3 imaging from 2021. Following methods described in Kilpatrick et al.
(2021), we downloaded all frames and aligned and drizzled them using the
python-based code hst123 (Kilpatrick, 2021). Using the drizzled ACS F606W
frame as a reference, we performed photometry in each epoch using dolphot. We
used standard dolphot parameters for each epoch as described in hst123. All
individual epochs and bands were drizzled separately, and we show all of the
F450W, F435W, and F438W (roughly $B$-band) and F606W (roughly $V$-band) in
Figure 1 and Table 1.

SN 2016gkg is clearly detectable in the post-explosion F606W frames at the
site of the pre-explosion counterpart, and appears to fade significantly from
2018 to 2021 as expected for an evolved SN IIb. By performing forced
photometry at the site of this source in the remaining images, we are able to
update and improve the photometric uncertainties on the counterpart detected
in the WFPC2 data compared with Kilpatrick et al. (2017), which is also
presented in Table 1.

Figure 1: HST WFPC2, ACS, and WFC3 imaging of the site of SN 2016gkg. We highlight the location of SN 2016gkg as determined from the ACS/WFC F606W image (bottom middle) from 2018, at which time the SN exhibited $m_{\rm F606W}=22.79\pm 0.01$ mag (AB). The counterpart appears fainter in WFC3 F438W and F606W imaging from 2021 (left top and bottom) than coincident pre-explosion emission from 2001 as observed in WFPC2 F450W and F606W (right top and bottom). Epoch | MJD | Instrument | Filter | Magnitude | Uncertainty
---|---|---|---|---|---
(days) |  |  |  | (mag) | (mag)
$-$5482.29 | 52142.01 | WFPC2 | F450W | 24.37 | 0.22
$-$5482.28 | 52142.02 | WFPC2 | F606W | 24.40 | 0.18
$-$5482.27 | 52142.03 | WFPC2 | F814W | 24.43 | 0.20
648.55 | 58302.89 | ACS | F435W | 24.04 | 0.07
648.62 | 58302.96 | ACS | F606W | 22.59 | 0.01
648.69 | 58303.03 | ACS | F814W | 22.89 | 0.01
1689.42 | 59348.86 | WFC3 | F275W |  $>$26.04 | —
1785.90 | 59445.82 | WFC3 | F606W | 25.10 | 0.07
1785.91 | 59445.83 | WFC3 | F438W | 26.61 | 0.27
Table 1: Pre- and Post-explosion HST photometry of the counterpart to SN
2016gkg. Epoch is given in rest-frame days relative to the explosion date of
2016 Sep 20.165 (Kilpatrick et al., 2017). All magnitudes are on the AB system
(Oke & Gunn, 1983).

###  2.2 Swope Imaging

We observed SN 2016gkg in $uBVgri$ bands from 22 October 2016 to 8 November
2017 with the Swope 1m telescope at Las Campanas Observatory, Chile. Following
standard procedures in the photpipe imaging and photometry package (Rest et
al., 2005) and as described in Kilpatrick et al. (2018), all Swope data were
corrected for bias and flat-fielded using calibration frames obtained in the
same instrumental configuration as the SN 2016gkg imaging. We astrometrically
aligned each Swope image to a common frame using 2MASS astrometric standards
in the same field as SN 2016gkg (Skrutskie et al., 2006). Finally, we
regridded each image to correct for geometric distortion using SWarp (Bertin,
2010) and performed photometry in the individual frames with DoPhot (Schechter
et al., 1993). The $BVgri$ data were photometrically calibrated using Pan-
STARRS DR2 photometric standards in the same field as SN 2016gkg and
calibrated to the Swope natural system using SuperCal (Scolnic et al., 2015).
The $u$-band data were calibrated using SkyMapper $u$-band standards (Onken et
al., 2019). As SN 2016gkg is relatively isolated and much brighter than its
HST counterpart in all Swope data, we did not perform image subtraction on
these images but instead use the unsubtracted frames to obtain final
photometry, which is shown in Table 2 and Figure 2.

Figure 2: The combined $uBVgri$ Swope (squares) and F435W, F438W, F606W, and
F814W HST (stars) light curve of SN 2016gkg comprising photometry from +32 to
+1795 days from explosion (32–1786 rest-frame days). We compare F435W and
F438W to $B$-band, F606W to $V$-band, and F814W to $i$-band. For comparison,
we show a simulated light curve powered by 56Co$\rightarrow{}^{56}$Fe decay
(dashed line) to compare to the photometry at $>$250 rest-frame days. However,
in the 1795 day epoch, there appears to be excess emission (solid line) beyond
the predicted emission from this model, which we model here using a
circumstellar interaction model assuming a wind with $\dot{M}=$3–4$\times
10^{-7}~{}M_{\odot}~{}\text{yr}$ and $v_{w}=$150–200 km s-1.

###  2.3 Keck/LRIS Spectroscopy

We observed SN 2016gkg with the Low Resolution Imaging Spectrometer on the
Keck-I 10m telescope on 2017 September 15, 2018 November 16, and 2018 January
16 UT. Each epoch was observed with the B600/4000 grism and R400/8500 grating
in conjunction with the D560 dichroic and the 1.0${}^{\,\prime\prime}$ wide
slit. We reduced these data using standard IRAF111IRAF is distributed by the
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation. and python routines to correct for bias,
flat-fielding, flux calibration, and telluric absorption. All spectra are
shown in Figure 3. As in Kuncarayakti et al. (2020), we detect emission lines
of [Mg i] $\lambda$4571, [O i] $\lambda\lambda$6300, 6364, H$\alpha$, and [Ca
ii] $\lambda\lambda$7292, 7324. For comparison, we also plot the spectrum of
SN 2016gkg from Gemini-S+GMOS and obtained on 2018 November 10 presented in
Kuncarayakti et al. (2020). This final spectrum exhibits stronger emission in
H$\alpha$, highlighting that SN 2016gkg was likely entering a phase of
enhanced interaction with circumstellar matter (CSM).

Figure 3: Our nebular spectra of SN 2016gkg from Keck/LRIS as described in
Section 2.3. We highlight lines of [Mg i] $\lambda$4571, [O i]
$\lambda\lambda$6300, 6364, H$\alpha$, and [Ca ii] $\lambda\lambda$7292, 7324.
For comparison, we show the latest Gemini-S/GMOS spectrum from Kuncarayakti et
al. (2020), which shows that H$\alpha$ emission is enhanced as the SN evolves.

##  3 Analysis

###  3.1 HST Photometry of SN 2016gkg and Its Progenitor Star

We note that SN 2016gkg declines significantly in F435W/F438W and F606W from
2018 to 2021. Indeed, SN 2016gkg is fainter than its pre-explosion counterpart
in F438W (comparing to F450W) and F606W. We infer from this decline that the
SN 2016gkg pre-explosion counterpart has disappeared and is in fact the
progenitor star.

Using the updated progenitor star photometry from Table 1, we use a blackbody
model and stellar spectral energy distributions (SEDs) from Pickles (1998) to
infer the effective temperature and luminosity of that star. Following methods
in Kilpatrick et al. (2021), we fit the pre-explosion photometry using a Monte
Carlo Markov Chain approach and find that the progenitor star SED is best fit
by a blackbody with $T_{\rm eff}=10800\pm 1000$ K and
$\log(L/L_{\odot})=4.7\pm 0.5$ as shown in Figure 4. The uncertainty in the
latter quantity is dominated by the uncertainty in the distance modulus. Such
a blackbody photosphere implies a radius of 60$\pm$30 $R_{\odot}$.

Figure 4: Spectral energy distribution (SED) of the SN 2016gkg pre-explosion
counterpart from F450W, F606W, and F814W (red points) with vertical error bars
including the measurement uncertainty and horizontal error bars indicating the
effective filter width. We overplot two models used to describe the overall
SED; a blackbody with $T_{\rm eff}=10,800$ K and a stellar SED for a A3I star
derived from Pickles & Depagne (2010) models as described in Section 3.1.

From our fits using Pickles (1998) SEDs, we find the best-fitting model is a
A3 star with $T_{\rm eff}=8900\begin{subarray}{c}+1100\\\ -700\end{subarray}$
K and $\log(L/L_{\odot})=4.5\pm 0.5$. Given the lower temperature, this source
would have a marginally more extended photospheric radius of $70\pm
30~{}R_{\odot}$. The difference in effective temperature from the blackbody
models arises because we are probing the Rayleigh-Jeans tail of the SED in
both cases and the stellar SED models provide a more realistic, absorbed
quasi-photosphere and UV suppression at $>$4000Å due to metal lines (Figure 4)
that also matches the progenitor photometry at cooler temperatures. Therefore,
we consider the A3 star temperature and implied luminosity to be a better
match to the progenitor detections. Taking the endpoint luminosity as an
indicator for the initial mass of the progenitor star, this would place SN
2016gkg at 10$\begin{subarray}{c}+2\\\ -1\end{subarray}$ $M_{\odot}$, in
agreement with the expectations for source B in Tartaglia et al. (2017).

###  3.2 Late-time Photometric Evolution of SN 2016gkg and Its Spectral Energy
Distribution

SN 2016gkg appears to decline following the radioactive decay rate of 56Co as
shown in Figure 2. As in Prentice et al. (2019), we model this phase of the
light curve assuming that SN 2016gkg initially produced 0.085 $M_{\odot}$ of
56Ni and assuming its light curve is powered primarily by radioactive decay
from 250–652 days post-explosion. This assumption is supported by the overall
light curve shape at this phase as well as the lack of strong spectroscopic
features due to interaction with CSM at $<$500 days (Figure 3 and Kuncarayakti
et al., 2020).

From 652 to 1795 days post-explosion, there is significant flattening in the
light curve compared with predictions from 56Co decay. We infer that some
other source of emission dominated SN 2016gkg in the most recent epoch. At the
same time, the color $m_{\rm F435W}-m_{\rm F606W}=1.45\pm 0.07$ mag at 652
days is consistent with the color $m_{\rm F438W}-m_{\rm F606W}=1.51\pm 0.28$
mag at 1795 days. This finding is consistent with expectations for a nebular
SN, where the emission line dominated spectrum undergoes a “freeze out” at
late times ($>$1 year as in Maeda et al., 2007; Jerkstrand et al., 2011, 2014)
as cooling due to emission lines is balanced by the heating rate due to the
shock and radioactive species.

In order to test whether the late-time photometry is consistent with coming
from SN 2016gkg, we show the latest F438W and F606W HST photometry in Figure 5
with comparisons to a blackbody spectral energy distribution similar to those
used for the pre-explosion photometry in Section 3.1 and our +483 day
Keck/LRIS spectrum, which covers $>$90% of the enclosed energy for both HST
bands. If the underlying spectral source is a blackbody, such as from a
coincident star, the implied temperature and luminosity are $T_{\rm eff}=3990$
K and $\log(L/L_{\odot})=4.3$. On the other hand, the photometry is not well
fit to the Keck/LRIS spectrum, with the spectrum suggesting $m_{\rm
F438W}-m_{\rm F606W}=0.54$ mag (AB) compared with 1.51 mag from the
photometry. This implies that there is either some other source or significant
spectral evolution at +1795 days compared with $\sim 1300$ days earlier. Based
solely on the spectral evolution observed in Section 2.3, we infer that there
may be significantly more CSM interaction in the latest photometric
observations. Thus the H$\alpha$ feature, which is observable in F606W,
dominates the emission profile. However, we cannot rule out the possibility
that some blended source or another emission mechanism contributing to the
photometry.

Figure 5: The HST photometry from SN 2016gkg at +1786 rest-frame days (red) as
shown in Figure 2 compared with a scaled Keck/LRIS spectrum of SN 2016gkg from
+483 days (black) and the best-fitting blackbody spectrum with $T_{\rm
eff}=3990$ K (green). The scaled Keck spectrum corresponds to a much bluer
color than the late-time HST photometry, implying that there is significant
spectral evolution out to this late phase, such as H$\alpha$ emission as in
Figure 3, or there is some other source of emission dominating SN 2016gkg.

To account for the excess emission above that of 56Co-decay, we first consider
other radioactive species such as 57Co, 55Fe, and 44Ti. Integrating the
emission in F438W and F606W at 1795 days and accounting for extinction and the
distance $26.4$ Mpc to SN 2016gkg, we find the total pseudo-bolometric
$B+V$-band emission is $2.6\pm 0.5\times 10^{38}$ erg s-1 (including the
uncertainty for distance). Under the assumption that all of this emission is
due to various radioisotopes with 100% thermalization efficiency, we can place
a lower limit on the total mass of that radioisotopes assuming it accounts for
all of the SN luminosity. Again assuming 100% thermalization and a rest-frame
time $t$, a radioisotope with atomic mass $m_{A}$, decay constant
$\lambda_{A}$, mean energy per decay $q_{A}$, and initially ($t=0$)
synthesized with a total mass $M_{A}$ will produce a luminosity

$L(t)=\frac{\lambda_{A}M_{A}q_{A}}{m_{A}}e^{-\lambda_{A}t}.$ (1)

We consider 57Co ($\lambda=2.56\times 10^{-3}$ day-1, $m=56.9$ u $q=836.0$
keV), 55Fe ($\lambda=6.93\times 10^{-4}$ day-1, $m=54.9$ u, $q=231.4$ keV),
and 44Ti ($\lambda=3.16\times 10^{-5}$ day-1, $m=44.0$ u, $q=3920.8$ keV) to
compare to Equation 1 (Firestone et al., 1999). For 44Ti, we have assumed that
the daughter isotope 44Sc, with a half-life of 3.9 hr, decays effectively
instantaneously and added its energy per decay with that of its parent.
Assuming SN 2016gkg emitted $2.6\pm 0.5\times 10^{38}$ erg s-1 at a rest-frame
time of 1785.9 days, we find minimum initial masses of 0.03 $M_{\odot}$
(57Co), 0.01 $M_{\odot}$ (55Fe), and 0.004 $M_{\odot}$ (44Ti). For the former
two radioisotopes, these would imply extremely large mass ratios to the
primary iron-peak isotope 56Ni that are not favored by nucleosynthesis in
core-collapse SNe (e.g., in calculations by Curtis et al., 2019; Andrews et
al., 2020). Even for 44Ti, which has been observed in the Galactic supernova
remnant Cassiopeia A with an implied initial mass $1.5\times
10^{-4}~{}M_{\odot}$ (Grefenstette et al., 2014) and in SN 1987A with an
initial mass $1.5\times 10^{-4}~{}M_{\odot}$ (Boggs et al., 2015), SN 2016gkg
would require at least $25\times$ more of this radioisotope than has been
observed for other core-collapse SNe. We therefore find it more likely that
there is some other energy source powering the late-time emission.

We next consider that the counterpart may be a light echo arising from the SN
light scattered by a local dust sheet into our line of sight. Light echoes
have been observed for a few extragalactic SNe (e.g., SN 2006X and 2014J; Wang
et al., 2008; Crotts, 2015) in addition to Galactic SN remnants such as
Cassiopeia A and Tycho (Rest et al., 2008). For unresolved light echoes from
sufficiently thick dust sheets, the light echo SED resembles the luminosity-
weighted average of their complete time-varying SED, which will be dominated
by the SN SED around peak light. As an approximation, we estimate that the
$m_{\rm F435W}-m_{\rm F606W}$ color of a SN 2016gkg light echo would be its
color around the second $V$-band maximum, which was about $B-V=0.4$ mag (AB;
Tartaglia et al., 2017). Including emission from the initial but shorter first
peak, this color would be even bluer. Thus the redder $m_{\rm F435W}-m_{\rm
F606W}=1.51$ mag counterpart we see is unlikely to be from a light echo.

Motivated by observations of possible CSM interaction in other SN IIb light
curves (e.g., SNe 1993J, 2011dh, and 2013df in Maeda et al., 2015; Smith et
al., 2017; Kundu et al., 2019; Maund, 2019), we model the excess emission
using a CSM interaction model for a $\rho\propto r^{-2}$ wind profile due to a
constant mass loss rate $\dot{M}$ and wind speed $v_{w}$. We use a relatively
simple prescription for this emission with the bulk luminosity following the
analytical model in Smith et al. (2010, their equation 1, implying luminosity
$\propto t^{-3}$) and no color evolution from the previous epochs. Assuming a
wind speed of $\approx$150–200 km s-1, corresponding to the escape speed of a
star with a terminal mass 7–10 $M_{\odot}$ and radius 90 $R_{\odot}$, the
excess corresponds to a mass-loss rate of $\dot{M}=$3–4$\times
10^{-7}~{}M_{\odot}~{}\text{yr}$. This is a relatively low mass-loss rate for
a $10~{}M_{\odot}$ star (e.g., compared with Smith et al., 2004; Vink, 2011;
Smith, 2014), but it is a plausible scenario considering the systematic
uncertainties in our light curve model. Detailed modeling of the complete
radio, X-ray, and optical light curve of SN 2016gkg can provide a more
complete picture of that system’s pre-explosion mass-loss history and what
progenitor scenarios it matches. However, the excess need not arise entirely
from CSM interaction but may be partly from a static source coincident with SN
2016gkg. We investigate the excess optical emission in the context of possible
companion models below.

###  3.3 The Spectral Energy Distribution of SN 2016gkg at +1786 days and
Limits on a Companion Star to the SN 2016gkg Progenitor

In Kilpatrick et al. (2017), the authors used publicly available binary
stellar evolution models from the Binary Population and Spectral Synthesis
code (BPASS; Eldridge et al., 2017) to consider scenarios in which one star in
a binary system could produce SN 2016gkg. We repeat that analysis here but
including the additional constraints on the presence of a companion star
either assuming that the excess emission observed from SN 2016gkg represents a
companion or an upper limit on a companion star if it is still dominated by
the SN itself. We also include upper limits in the ultraviolet filter F275W
observed at 1689 days in Table 1.

Among the BPASS models in v2.2, the best-fitting binary stellar endpoint for
our data involves a 10+3 $M_{\odot}$ binary where the primary terminates with
a final mass $M_{f}=2.2~{}M_{\odot}$ and a envelope hydrogen mass of
$0.07~{}M_{\odot}$. At the time the primary star explodes, the secondary star
is still a B-type main sequence star with $m>30$ mag at the distance of NGC
613 in optical and ultraviolet bands. This would imply that the late-time
counterpart does not correspond to the companion star and our limits are
unconstraining for any binary scenarios.

Even in the most extreme cases allowed by our photometry of the SN 2016gkg
progenitor star, the secondary star can have at most an initial mass of
$7~{}M_{\odot}$ with $m_{V}\approx 30.7$ mag at the time the primary explodes.
This reinforces our finding above that the spectral energy distribution of the
late-time emission is more consistent with SN 2016gkg itself.

##  4 Discussion & Conclusions

The improved astrometric alignment between SN 2016gkg and its pre-explosion
counterpart over ground-based adaptive optics imaging (Kilpatrick et al.,
2017; Tartaglia et al., 2017) and short-exposure HST imaging in which the SN
is nearly saturated (Bersten et al., 2018) enables a novel analysis of its
progenitor system. It is clear from the superior resolution of the late-time
HST imaging that the WFPC2 counterpart is indeed split into multiple
components as hypothesized by Tartaglia et al. (2017), and the counterpart
corresponds to their source B. Our revised photometry is in good agreement
with their original analysis, although we have improved the photometric
precision by using the position of SN 2016gkg provided by the late-time ACS
and WFC3 imaging. Our updated estimates of the luminosity of this source
further suggest that SN 2016gkg exploded from a relatively compact
($\approx$70 $R_{\odot}$) yellow supergiant star with an initial mass of
10$\begin{subarray}{c}+2\\\ -1\end{subarray}$ $M_{\odot}$. These results agree
with most of the early-time shock cooling modeling (Arcavi et al., 2017;
Kilpatrick et al., 2017; Piro et al., 2017; Tartaglia et al., 2017; Piro et
al., 2021), especially those using the Piro (2015) and Sapir & Waxman (2017)
that tend to lead to smaller radii.

Investigating the late-time evolution of SN 2016gkg, the light curve initially
follows the 56Co decay rate, which is also suggested by the lack of H$\alpha$
emission in the nebular spectra of Kuncarayakti et al. (2020). The relative
brightnesses of SN 2016gkg in F435W, F606W, and F814W (Table 1) are consistent
with expectations that the SED of SN 2016gkg is dominated by [Mg i]
$\lambda$4571, [O i] $\lambda\lambda$6300, 6364, and [Ca ii]
$\lambda\lambda$7292, 7324 at around 652 days. It is therefore unlikely that
there is any significant contribution from CSM interaction at these phases,
although detailed radio and X-ray observations could reveal an underlying
shock interaction with a surrounding wind or other medium.

However, at 1786 rest-frame days, there appears to be a flattening in the
light curve implying that it is no longer 56Co-powered. The emission is too
bright to be any other radioisotope typically found in core-collapse SNe such
as 44Ti. At this very late phase, it is possible that CSM interaction in the
shock begins to dominate the optical radiation provided by radioactive decay,
especially as the latest optical spectrum of SN 2016gkg showed an enhancement
in H$\alpha$ emission, likely due to an interaction with H-rich material.
While it is possible that there is some static source contributing to the SN
2016gkg counterpart at this phase, the spectral energy distribution of that
emission is consistent with earlier spectra of SN 2016gkg, and so we find it
more likely that the SN still dominates the HST photometry. Moreover, our
initial search of BPASS models that are consistent with the counterpart do not
reveal any possible companion stars that can simultaneously reproduce the 1786
day counterpart and the SN 2016gkg progenitor source. Additional follow up of
SN 2016gkg is needed reveal if the source continues to decline in brightness
consistent with CSM interaction or if there remains a persistent source of
emission.

Table 2: Swope Photometry of SN 2016gkg

MJD | Filter | Magnitude | Uncertainty
---|---|---|---
|  | (mag) | (mag)
57686.31 | $B$ | 17.08 | 0.01
57687.33 | $B$ | 17.15 | 0.01
57688.34 | $B$ | 17.19 | 0.01
57691.20 | $B$ | 17.32 | 0.01
57694.33 | $B$ | 17.47 | 0.01
57697.29 | $B$ | 17.52 | 0.04
57697.30 | $B$ | 17.51 | 0.01
57698.30 | $B$ | 17.57 | 0.03
57698.31 | $B$ | 17.55 | 0.01
57715.25 | $B$ | 17.81 | 0.05
57715.25 | $B$ | 17.79 | 0.01
57744.18 | $B$ | 18.11 | 0.02
57750.08 | $B$ | 18.13 | 0.02
57752.14 | $B$ | 18.19 | 0.01
57753.14 | $B$ | 18.20 | 0.05
57753.14 | $B$ | 18.19 | 0.01
57757.14 | $B$ | 18.22 | 0.01
57683.31 | $g$ | 16.24 | 0.01
57686.30 | $g$ | 16.47 | 0.01
57687.32 | $g$ | 16.56 | 0.01
57688.33 | $g$ | 16.59 | 0.01
57691.19 | $g$ | 16.69 | 0.01
57694.32 | $g$ | 16.84 | 0.01
57697.31 | $g$ | 16.96 | 0.01
57698.32 | $g$ | 16.99 | 0.01
57715.27 | $g$ | 17.25 | 0.01
57744.19 | $g$ | 17.62 | 0.01
57750.07 | $g$ | 17.63 | 0.01
57752.13 | $g$ | 17.65 | 0.01
57753.16 | $g$ | 17.73 | 0.01
57757.12 | $g$ | 17.75 | 0.01
57774.13 | $g$ | 18.02 | 0.02
57775.12 | $g$ | 18.00 | 0.02
57782.09 | $g$ | 18.14 | 0.01
57785.06 | $g$ | 18.13 | 0.01
57801.03 | $g$ | 18.39 | 0.02
57803.04 | $g$ | 18.38 | 0.02
57908.41 | $g$ | 19.83 | 0.02
57946.37 | $g$ | 20.59 | 0.08
57954.43 | $g$ | 20.63 | 0.03
58010.32 | $g$ | 20.96 | 0.05
58019.38 | $g$ | 21.30 | 0.05
57683.31 | $i$ | 15.35 | 0.01
57686.30 | $i$ | 15.45 | 0.01
57687.32 | $i$ | 15.49 | 0.01
57688.33 | $i$ | 15.48 | 0.01
57691.19 | $i$ | 15.54 | 0.01
57694.32 | $i$ | 15.68 | 0.01
57697.31 | $i$ | 15.77 | 0.01
57698.32 | $i$ | 15.82 | 0.01
57715.27 | $i$ | 16.01 | 0.01
57744.19 | $i$ | 16.76 | 0.01
57750.07 | $i$ | 16.75 | 0.01
57752.13 | $i$ | 16.80 | 0.01
57753.16 | $i$ | 16.92 | 0.01
57757.12 | $i$ | 16.92 | 0.01
57774.13 | $i$ | 17.26 | 0.02
57782.09 | $i$ | 17.39 | 0.01
57785.06 | $i$ | 17.41 | 0.01
57801.03 | $i$ | 17.64 | 0.02
57803.04 | $i$ | 17.62 | 0.02
57908.41 | $i$ | 18.96 | 0.02
57916.43 | $i$ | 19.04 | 0.03
57946.36 | $i$ | 19.43 | 0.03
57954.42 | $i$ | 19.50 | 0.03
58001.25 | $i$ | 20.15 | 0.05
58010.31 | $i$ | 20.28 | 0.04
58019.37 | $i$ | 20.32 | 0.04
58065.23 | $i$ | 20.77 | 0.09
57683.30 | $r$ | 15.41 | 0.02
57683.31 | $r$ | 15.41 | 0.01
57685.20 | $r$ | 15.49 | 0.02
57686.29 | $r$ | 15.54 | 0.02
57686.29 | $r$ | 15.53 | 0.01
57686.29 | $r$ | 15.58 | 0.01
57687.32 | $r$ | 15.61 | 0.02
57688.32 | $r$ | 15.63 | 0.02
57688.32 | $r$ | 15.61 | 0.01
57691.19 | $r$ | 15.68 | 0.01
57694.32 | $r$ | 15.83 | 0.01
57697.32 | $r$ | 15.95 | 0.01
57698.32 | $r$ | 15.98 | 0.01
57715.27 | $r$ | 16.36 | 0.01
57744.19 | $r$ | 16.90 | 0.01
57750.06 | $r$ | 16.99 | 0.03
57750.07 | $r$ | 16.96 | 0.01
57752.12 | $r$ | 17.02 | 0.01
57753.16 | $r$ | 17.07 | 0.01
57757.12 | $r$ | 17.10 | 0.04
57757.12 | $r$ | 17.13 | 0.01
57774.13 | $r$ | 17.43 | 0.01
57775.11 | $r$ | 17.41 | 0.05
57775.12 | $r$ | 17.41 | 0.05
57782.08 | $r$ | 17.48 | 0.05
57782.09 | $r$ | 17.58 | 0.01
57785.06 | $r$ | 17.59 | 0.01
57801.02 | $r$ | 17.84 | 0.02
57803.04 | $r$ | 17.81 | 0.01
57908.40 | $r$ | 19.18 | 0.02
57916.43 | $r$ | 19.20 | 0.02
57946.36 | $r$ | 19.57 | 0.03
57954.41 | $r$ | 19.68 | 0.03
58001.24 | $r$ | 20.26 | 0.06
58010.30 | $r$ | 20.25 | 0.04
58019.36 | $r$ | 20.35 | 0.03
58034.24 | $r$ | 20.41 | 0.09
58065.22 | $r$ | 20.79 | 0.08
57683.32 | $u$ | 18.26 | 0.03
57686.30 | $u$ | 18.57 | 0.03
57687.32 | $u$ | 18.66 | 0.03
57688.33 | $u$ | 18.69 | 0.03
57691.19 | $u$ | 18.84 | 0.03
57694.33 | $u$ | 18.86 | 0.04
57697.30 | $u$ | 18.94 | 0.03
57698.31 | $u$ | 18.97 | 0.03
57715.26 | $u$ | 18.99 | 0.04
57744.18 | $u$ | 19.23 | 0.06
57750.07 | $u$ | 19.27 | 0.03
57752.13 | $u$ | 19.18 | 0.03
57753.15 | $u$ | 19.26 | 0.04
57757.13 | $u$ | 19.34 | 0.04
57686.31 | $V$ | 15.95 | 0.01
57687.33 | $V$ | 16.02 | 0.01
57688.34 | $V$ | 16.05 | 0.01
57691.20 | $V$ | 16.14 | 0.01
57694.33 | $V$ | 16.30 | 0.01
57697.30 | $V$ | 16.37 | 0.01
57698.31 | $V$ | 16.44 | 0.01
57715.26 | $V$ | 16.73 | 0.01
57744.18 | $V$ | 17.23 | 0.01
57750.08 | $V$ | 17.27 | 0.01
57752.14 | $V$ | 17.30 | 0.01
57753.15 | $V$ | 17.30 | 0.01
57757.14 | $V$ | 17.41 | 0.01

Note. — All photometry is on the AB magnitude system.

ACKNOWLEDGMENTS

We would like to thank I. Arcavi and A. Gilkis for helpful comments related to
this manuscript. This work includes data obtained with the Swope Telescope at
Las Campanas Observatory, Chile, as part of the Swope Time Domain Key Project
(PI Piro; Co-Is Burns, Cowperthwaite, Dimitriadis, Drout, Foley, French,
Holoien, Hsiao, Kilpatrick, Madore, Phillips, and Rojas-Bravo). The Hubble
Space Telescope imaging presented in this manuscript come from programs
GO-9042 (PI Smartt), GO-15272 (PI Folatelli), GO-16287 (PI Lyman), and
SNAP-16179 (PI Filippenko). Based on observations made with the NASA/ESA
Hubble Space Telescope, obtained from the data archive at the Space Telescope
Science Institute. STScI is operated by the Association of Universities for
Research in Astronomy, Inc. under NASA contract NAS 5-26555. Some of the data
presented herein were obtained at the W. M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California, and NASA. The Observatory was made
possible by the generous financial support of the W. M. Keck Foundation. The
authors wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Maunakea has always had within the indigenous
Hawaiian community. We are most fortunate to have the opportunity to conduct
observations from this mountain.

## References

  * Aldering et al. (1994) Aldering, G., Humphreys, R. M., & Richmond, M. 1994, AJ, 107, 662, doi: 10.1086/116886
  * Andrews et al. (2020) Andrews, S., Fryer, C., Even, W., Jones, S., & Pignatari, M. 2020, ApJ, 890, 35, doi: 10.3847/1538-4357/ab64f8
  * Arcavi et al. (2011) Arcavi, I., Gal-Yam, A., Yaron, O., et al. 2011, ApJ, 742, L18, doi: 10.1088/2041-8205/742/2/L18
  * Arcavi et al. (2017) Arcavi, I., Hosseinzadeh, G., Brown, P. J., et al. 2017, ApJ, 837, L2, doi: 10.3847/2041-8213/aa5be1
  * Bersten et al. (2012) Bersten, M. C., Benvenuto, O. G., Nomoto, K., et al. 2012, ApJ, 757, 31, doi: 10.1088/0004-637X/757/1/31
  * Bersten et al. (2018) Bersten, M. C., Folatelli, G., García, F., et al. 2018, Nature, 554, 497, doi: 10.1038/nature25151
  * Bertin (2010) Bertin, E. 2010, SWarp: Resampling and Co-adding FITS Images Together.  http://ascl.net/1010.068
  * Boggs et al. (2015) Boggs, S. E., Harrison, F. A., Miyasaka, H., et al. 2015, Science, 348, 670, doi: 10.1126/science.aaa2259
  * Crockett et al. (2008) Crockett, R. M., Eldridge, J. J., Smartt, S. J., et al. 2008, MNRAS, 391, L5, doi: 10.1111/j.1745-3933.2008.00540.x
  * Crotts (2015) Crotts, A. P. S. 2015, ApJ, 804, L37, doi: 10.1088/2041-8205/804/2/L37
  * Curtis et al. (2019) Curtis, S., Ebinger, K., Fröhlich, C., et al. 2019, ApJ, 870, 2, doi: 10.3847/1538-4357/aae7d2
  * Dolphin (2016) Dolphin, A. 2016, DOLPHOT: Stellar photometry.  http://ascl.net/1608.013
  * Eldridge et al. (2017) Eldridge, J. J., Stanway, E. R., Xiao, L., et al. 2017, PASA, 34, e058, doi: 10.1017/pasa.2017.51
  * Falk (1978) Falk, S. W. 1978, in BAAS, Vol. 10, Bulletin of the American Astronomical Society, 425
  * Falk & Arnett (1973) Falk, S. W., & Arnett, W. D. 1973, ApJ, 180, L65, doi: 10.1086/181154
  * Filippenko (1997) Filippenko, A. V. 1997, ARA&A, 35, 309, doi: 10.1146/annurev.astro.35.1.309
  * Firestone et al. (1999) Firestone, R. B., Ekstrom, L. P., & Chu, S. Y. F. 1999, in APS Division of Nuclear Physics Meeting Abstracts, APS Meeting Abstracts, CE.13
  * Folatelli et al. (2014) Folatelli, G., Bersten, M. C., Benvenuto, O. G., et al. 2014, ApJ, 793, L22, doi: 10.1088/2041-8205/793/2/L22
  * Fox et al. (2014) Fox, O. D., Azalee Bostroem, K., Van Dyk, S. D., et al. 2014, ApJ, 790, 17, doi: 10.1088/0004-637X/790/1/17
  * Gilkis & Arcavi (2021) Gilkis, A., & Arcavi, I. 2021, arXiv e-prints, arXiv:2111.04432.  https://arxiv.org/abs/2111.04432
  * Grefenstette et al. (2014) Grefenstette, B. W., Harrison, F. A., Boggs, S. E., et al. 2014, Nature, 506, 339, doi: 10.1038/nature12997
  * Heger et al. (2003) Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., & Hartmann, D. H. 2003, ApJ, 591, 288, doi: 10.1086/375341
  * Izzard et al. (2004) Izzard, R. G., Ramirez-Ruiz, E., & Tout, C. A. 2004, MNRAS, 348, 1215, doi: 10.1111/j.1365-2966.2004.07436.x
  * Jerkstrand et al. (2011) Jerkstrand, A., Fransson, C., & Kozma, C. 2011, A&A, 530, A45, doi: 10.1051/0004-6361/201015937
  * Jerkstrand et al. (2014) Jerkstrand, A., Smartt, S. J., Fraser, M., et al. 2014, MNRAS, 439, 3694, doi: 10.1093/mnras/stu221
  * Kilpatrick (2021) Kilpatrick, C. D. 2021, charliekilpatrick/hst123: hst123, v1.0.0, Zenodo, doi: 10.5281/zenodo.5573941
  * Kilpatrick et al. (2017) Kilpatrick, C. D., Foley, R. J., Abramson, L. E., et al. 2017, MNRAS, 465, 4650, doi: 10.1093/mnras/stw3082
  * Kilpatrick et al. (2018) Kilpatrick, C. D., Foley, R. J., Drout, M. R., et al. 2018, MNRAS, 473, 4805, doi: 10.1093/mnras/stx2675
  * Kilpatrick et al. (2021) Kilpatrick, C. D., Drout, M. R., Auchettl, K., et al. 2021, MNRAS, 504, 2073, doi: 10.1093/mnras/stab838
  * Kuncarayakti et al. (2020) Kuncarayakti, H., Folatelli, G., Maeda, K., et al. 2020, ApJ, 902, 139, doi: 10.3847/1538-4357/abb4e7
  * Kundu et al. (2019) Kundu, E., Lundqvist, P., Sorokina, E., et al. 2019, ApJ, 875, 17, doi: 10.3847/1538-4357/ab0d81
  * Langer et al. (1994) Langer, N., Hamann, W. R., Lennon, M., et al. 1994, A&A, 290, 819
  * Maeda et al. (2007) Maeda, K., Tanaka, M., Nomoto, K., et al. 2007, ApJ, 666, 1069, doi: 10.1086/520054
  * Maeda et al. (2015) Maeda, K., Hattori, T., Milisavljevic, D., et al. 2015, ApJ, 807, 35, doi: 10.1088/0004-637X/807/1/35
  * Maeder & Meynet (2000) Maeder, A., & Meynet, G. 2000, A&A, 361, 159.  https://arxiv.org/abs/astro-ph/0006405
  * Maund (2019) Maund, J. R. 2019, ApJ, 883, 86, doi: 10.3847/1538-4357/ab2386
  * Maund et al. (2004) Maund, J. R., Smartt, S. J., Kudritzki, R. P., Podsiadlowski, P., & Gilmore, G. F. 2004, Nature, 427, 129
  * Maund et al. (2011) Maund, J. R., Fraser, M., Ergon, M., et al. 2011, ApJ, 739, L37, doi: 10.1088/2041-8205/739/2/L37
  * Maund et al. (2015) Maund, J. R., Arcavi, I., Ergon, M., et al. 2015, MNRAS, 454, 2580, doi: 10.1093/mnras/stv2098
  * Nakar & Piro (2014) Nakar, E., & Piro, A. L. 2014, ApJ, 788, 193, doi: 10.1088/0004-637X/788/2/193
  * Nasonova et al. (2011) Nasonova, O. G., de Freitas Pacheco, J. A., & Karachentsev, I. D. 2011, A&A, 532, A104, doi: 10.1051/0004-6361/201016004
  * Oke & Gunn (1983) Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713, doi: 10.1086/160817
  * Onken et al. (2019) Onken, C. A., Wolf, C., Bessell, M. S., et al. 2019, PASA, 36, e033, doi: 10.1017/pasa.2019.27
  * Pickles & Depagne (2010) Pickles, A., & Depagne, É. 2010, PASP, 122, 1437, doi: 10.1086/657947
  * Pickles (1998) Pickles, A. J. 1998, PASP, 110, 863, doi: 10.1086/316197
  * Piro (2015) Piro, A. L. 2015, ApJ, 808, L51, doi: 10.1088/2041-8205/808/2/L51
  * Piro et al. (2021) Piro, A. L., Haynie, A., & Yao, Y. 2021, ApJ, 909, 209, doi: 10.3847/1538-4357/abe2b1
  * Piro et al. (2017) Piro, A. L., Muhleisen, M., Arcavi, I., et al. 2017, ApJ, 846, 94, doi: 10.3847/1538-4357/aa8595
  * Poznanski et al. (2012) Poznanski, D., Prochaska, J. X., & Bloom, J. S. 2012, MNRAS, 426, 1465, doi: 10.1111/j.1365-2966.2012.21796.x
  * Prentice et al. (2019) Prentice, S. J., Ashall, C., James, P. A., et al. 2019, MNRAS, 485, 1559, doi: 10.1093/mnras/sty3399
  * Rabinak & Waxman (2011) Rabinak, I., & Waxman, E. 2011, ApJ, 728, 63, doi: 10.1088/0004-637X/728/1/63
  * Rest et al. (2005) Rest, A., Stubbs, C., Becker, A. C., et al. 2005, ApJ, 634, 1103, doi: 10.1086/497060
  * Rest et al. (2008) Rest, A., Welch, D. L., Suntzeff, N. B., et al. 2008, ApJ, 681, L81, doi: 10.1086/590427
  * Sapir & Waxman (2017) Sapir, N., & Waxman, E. 2017, ApJ, 838, 130, doi: 10.3847/1538-4357/aa64df
  * Schechter et al. (1993) Schechter, P. L., Mateo, M., & Saha, A. 1993, PASP, 105, 1342, doi: 10.1086/133316
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Scolnic et al. (2015) Scolnic, D., Casertano, S., Riess, A., et al. 2015, ApJ, 815, 117, doi: 10.1088/0004-637X/815/2/117
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163, doi: 10.1086/498708
  * Smartt (2009) Smartt, S. J. 2009, ARA&A, 47, 63, doi: 10.1146/annurev-astro-082708-101737
  * Smith (2014) Smith, N. 2014, ARA&A, 52, 487, doi: 10.1146/annurev-astro-081913-040025
  * Smith et al. (2010) Smith, N., Chornock, R., Silverman, J. M., Filippenko, A. V., & Foley, R. J. 2010, ApJ, 709, 856, doi: 10.1088/0004-637X/709/2/856
  * Smith et al. (2004) Smith, N., Vink, J. S., & de Koter, A. 2004, ApJ, 615, 475, doi: 10.1086/424030
  * Smith et al. (2017) Smith, N., Kilpatrick, C. D., Mauerhan, J. C., et al. 2017, MNRAS, 466, 3021, doi: 10.1093/mnras/stw3204
  * Sravan et al. (2018) Sravan, N., Marchant, P., Kalogera, V., & Margutti, R. 2018, ApJ, 852, L17, doi: 10.3847/2041-8213/aaa2f5
  * Tartaglia et al. (2017) Tartaglia, L., Fraser, M., Sand, D. J., et al. 2017, ApJ, 836, L12, doi: 10.3847/2041-8213/aa5c7f
  * Van Dyk et al. (2013) Van Dyk, S. D., Zheng, W., Clubb, K. I., et al. 2013, ApJ, 772, L32, doi: 10.1088/2041-8205/772/2/L32
  * Van Dyk et al. (2014) Van Dyk, S. D., Zheng, W., Fox, O. D., et al. 2014, AJ, 147, 37, doi: 10.1088/0004-6256/147/2/37
  * Vink (2011) Vink, J. S. 2011, Ap&SS, 336, 163, doi: 10.1007/s10509-011-0636-7
  * Wang et al. (2008) Wang, X., Li, W., Filippenko, A. V., et al. 2008, ApJ, 677, 1060, doi: 10.1086/529070
  * Woosley et al. (1994) Woosley, S. E., Eastman, R. G., Weaver, T. A., & Pinto, P. A. 1994, ApJ, 429, 300, doi: 10.1086/174319
  * Yoon (2017) Yoon, S.-C. 2017, MNRAS, 470, 3970, doi: 10.1093/mnras/stx1496