\section{Introduction}
Hybrid systems exhibit continuous state evolution and discrete state jumps, and therefore can capture the behavior of complex dynamical systems. The introductory paper \cite{antsaklis1998hybrid} provides an overview and merits of hybrid systems. Stability of switched and hybrid systems has been studied extensively; for a detailed presentation of the solution concepts and the notion of stability for hybrid systems, the interested readers are referred to \cite{lygeros2004lecture,goebel2012hybrid}.

The survey paper \cite{teel2014stability} studies Lyapunov stability (LS), Lagrange stability and asymptotic stability (AS) for stochastic hybrid systems (SHS), and provides Lyapunov conditions for stability in probability. The paper also presents open problems on converse results on the stability in probability of SHS.
The review paper \cite{marty2015elements} discusses sufficient conditions for stability of three classes of hybrid systems, namely, (i) systems modeled on a time scale, (ii) impulsive hereditary systems, and (iii) weakly coupled systems defined on Banach spaces. In \cite{liu2016lyapunov}, the authors study hybrid systems exhibiting delay phenomena such as memory. They establish sufficient conditions for AS using Lyapunov-Razumikhin functions and Lyapunov-Krasovskii functionals. More recently, pointwise AS of hybrid systems is studied in \cite{goebel2016notions}, where the notion of set-valued Lyapunov functions is used to establish sufficient conditions for AS of a closed set. In \cite{teel2013lyapunov}, the authors impose an average dwell-time for the discrete jumps and devise Lyapunov-based sufficient conditions for exponential stability of closed sets; see \cite{goebel2016notions,teel2013lyapunov} for details on the notion of stability of closed sets.

In contrast to AS, which pertains to convergence as time goes to infinity, finite-time stability (FTS)\footnote{With slight abuse of notation, we use FTS to denote the phrase "finite-time stability" or "finite-time stable", depending on the context.} is a concept that requires convergence of the solution in finite time \cite{bhat2000finite}.
FTS of switched and hybrid systems has gained popularity in the last few years. The authors in \cite{liu2017finite} consider the problem of designing a controller for a linear switched system under delay and external disturbance with finite- and fixed-time convergence. In \cite{li2013robust}, the authors design a hybrid observer and show finite-time convergence in the presence of unknown, constant bias.  In \cite{nersesov2007finite}, the authors study FTS of nonlinear impulsive dynamical systems, and present sufficient conditions
to guarantee FTS. The work in \cite{li2013robust,nersesov2007finite} considers discrete jumps in the system states in a continuously evolving system, i.e., they consider one model for the continuous dynamics, and one model for the discrete dynamics. In this work, we consider the general case with $N_f$ continuous flows and $N_g$ discrete jump dynamics, where $N_f,N_g$ can be any positive integers. The authors in \cite{li2019finite} present conditions in terms of a common Lyapunov function for FTS of hybrid systems. They require the value of the Lyapunov function to be decreasing during the continuous flow and non-increasing at the discrete jumps. In \cite{zhangfinite}, the authors consider a switched system whose subsystems possess a homogeneous Lyapunov function and are of constant switching intervals. In \cite{orlov2004finite}, the authors introduce the concept of \textit{locally} homogeneous system and show FTS of switched systems with uniformly bounded uncertainties. More recently, \cite{zhang2018finite} studies FTS of homogeneous switched systems by introducing the concept of hybrid homogeneous degree, and relating negative homogeneity with FTS. In \cite{fu2015global}, the authors consider systems in strict-feedback form with positive powers and design a controller as well as a switching law so that the closed-loop system is FTS. In this work, we do not assume that the subsystems of the hybrid system are homogeneous or in strict feedback form, and present conditions for FTS in terms of multiple generalized Lyapunov functions.

In this paper, we consider a general class of hybrid systems under arbitrary switching, and develop sufficient conditions for FTS of its equilibrium. We develop sufficient conditions for FTS of the origin of the hybrid system in terms of multiple generalized Lyapunov functions. We relax the requirement in \cite{li2019finite,li2016results} that the Lyapunov function is non-increasing at the discrete jumps and strictly decreasing during the continuous flow. More specifically, we use generalized Lyapunov functions for each mode, and allow these functions to increase both during the continuous flow and at the discrete jumps, and only require that these increments are bounded. In other words, we allow the hybrid system to have unstable modes. We show that if the origin is stable and if there exists an FTS mode that is active for a sufficiently long cumulative time, then the origin is FTS under arbitrary switching. In our earlier work \cite{garg2019finite}, we have presented conditions for FTS of switched systems. The current paper extends these results to a general class of hybrid systems. To the best of authors' knowledge, this is a first work on FTS of hybrid systems using multiple generalized Lyapunov functions. We summarize in Remark \ref{remark: summary contri} how earlier results on FTS of continuous-time,  switched and hybrid systems are special cases of the results presented in this paper.

The paper is organized as follows: In Section \ref{FTS HS}, we present an overview of FTS followed by conditions for FTS of hybrid systems under arbitrary switching. Section \ref{Simulations} evaluates the performance of the proposed method via simulation results. Our conclusions and thoughts on future work are summarized in Section \ref{Conclusions}.

\section{FTS of Hybrid Systems}\label{FTS HS}

\subsection{Notations}

We denote by $\|\cdot\|$ the Euclidean norm of vector $(\cdot)$, $|\cdot|$ the absolute value if $(\cdot)$ is scalar and the length if $(\cdot)$ is a time interval. The set of non-negative reals is denoted by $\mathbb R_+ = [0,\infty)$, set of non-negative integers by $\mathbb Z_+$ and set of positive integers by $\mathbb N$. We denote by $\textrm{int}(S)$ the interior of the set $S$, and by $t^-$ and $t^+$ the time just before and after the time instant $t$, respectively.

\vspace{-10pt}
\subsection{Overview of FTS}

Consider the system:
\begin{equation}\begin{split}\label{ex sys}
\dot y(t) = f(y(t)),
\end{split}\end{equation}
where $y\in \mathbb R^n$, $f: D \rightarrow \mathbb R^n$ is continuous on an open neighborhood $D\subseteq \mathbb R^n$ of the origin and $f(0)=0$. The origin is said to be an FTS equilibrium of \eqref{ex sys} if it is Lyapunov stable and \textit{finite-time convergent}, i.e., for all $y(0) \in \mathcal N \setminus\{0\}$, where $\mathcal N$ is some open neighborhood of the origin, $\lim_{t\to T} y(t)=0$, where $T = T(y(0))<\infty$  \cite{bhat2000finite}.
The authors also presented Lyapunov conditions for FTS of \eqref{ex sys}:

\begin{Theorem}[\cite{bhat2000finite}]\label{FTS Bhat}
Suppose there exists a continuous function $V$: $D \rightarrow \mathbb{R}$ such that the following hold: \\
(i) $V$ is positive definite \\
(ii) There exist real numbers $c>0$ and $\alpha \in (0, 1)$ , and an open neighborhood $\mathcal{V}\subseteq D$ of the origin such that
\begin{equation}\begin{split} \label{FTS Lyap}
    \dot V(y) + cV(y)^\alpha \leq 0, \; y\in \mathcal{V}\setminus\{0\},
\end{split}\end{equation}
where $\dot V(y) \triangleq (D^+(V\circ\psi^x))(0)$ is the upper-right Dini derivative of $V(y)$. Then the origin is a FTS for \eqref{ex sys}.
\end{Theorem}

\subsection{Main result}

In this section, we consider the class of hybrid systems $\mathcal H = \{\mathcal F, \mathcal G , C, D\}$ described as
\begin{equation}\label{hybrid sys}
\begin{split}
    \dot x(t) & = f_{\sigma_f}(x(t)) \quad x(t)\in C,\\
    x(t^+) & = g_{\sigma_g}(x(t)) \quad x(t)\in D,
\end{split}\end{equation}
where $x\in \mathbb R^n$ is the state vector with $x(t_0) = x_0$, $f_{i}(x)\in \mathcal F \triangleq\{f_i\}$ for $i\in \Sigma_f \triangleq\{1,2,\cdots, N_f\}$ is the continuous flow for the system which are allowed on the subset of the state space $C\subset\mathbb R^n$ and $g_{j}(x) \in \mathcal G\triangleq\{g_j\}$ for $j\in \Sigma_g \triangleq\{1,2,\cdots, N_g\}$ defines the discrete behavior, which is allowed on the subset $D\subset\mathbb R^n$. Define $x^+(t) \triangleq x(t^+)$. The switching signals $\sigma_f$ and $\sigma_g$ are assumed to be piecewise constant, right-continuous signals, which can depend upon both state and time. Details about definitions and solution concepts of the hybrid system \eqref{hybrid sys} can be found in \cite{goebel2012hybrid}.

Denote by $T_{i_k} =  [t_{i_k},t_{i_k+1})$ the interval in which the flow $f_i$ is active for the $k-$th time for $i\in \Sigma_f$ and $k\in \mathbb N$, and $t^d_{j_m}$ as the time when discrete jump $x^+ = g_j(x)$ takes place for the $m-$th time for $j\in \Sigma_g$ and $m\in \mathbb N$. Define $J_{i} = \{t^d_{j_m}\; | t^d_{j_m}\in T_{i_k}, j\in \Sigma_g, m\in \mathbb N\}$ as the set of all time instances when a discrete jump takes place when the continuous flow $f_i$ is active. We assume that the solution $x(t)$ to \eqref{hybrid sys} exists and is maximal. For each interval $T_{i_k}$, define the largest connected sub-interval $\bar T_{i_k}\subset T_{i_k}$, such that there is no discrete jump in $\bar T_{i_k}$, i.e., $\textrm{int}(\bar T_{i_k})\bigcap J_{i_k} = \emptyset$. For example, if $T_{i_1} = [0,1)$ and $J_{i} = \{0.2, 0.4, 0.75\}$, then $\bar T_{i_1} = [0.4, 0.75)$. We make the following assumptions:

\begin{Assumption}\label{assum uniq equib}
The origin is the only equilibrium point for all the continuous flows and discrete jumps, i.e., $f_i(0) = g_j(0) = 0$ for all $f_i\in \mathcal F$ and $g_j\in \mathcal G$.
\end{Assumption}

In this work, per Assumption \ref{assum uniq equib}, we restrict our focus on the case when there is a unique equilibrium of the hybrid system \eqref{hybrid sys}. The case when there exists some $g_j\in \mathcal G$ and a set $\bar D\neq\{0\}$ such that $g_j(x) = 0$ for all $x\in \bar D\subset D$, can be treated by studying the stability property of set $\bar D$; see \cite{li2019finite,li2016results}.

\begin{Assumption}\label{assum bar T tm}
The time interval $\bar T_{F_k}$ for the FTS mode $F\in \Sigma$ is such that its length satisfies $|\bar T_{F_k}| \geq t_d>0$ for all $k\in \mathbb N$.
\end{Assumption}

Assumption \ref{assum bar T tm} implies that for the FTS flow $f_F$, in each interval $T_{F_k}$ when the system \eqref{hybrid sys} evolves along the flow $f_F$, there exists a sub-interval $\bar T_{F_k}$ of non-zero length $t_d$ such that there is no discrete jump in the system state during $\bar T_{F_k}$. Let $\bar T_{F_k} = [\bar t_{F_k},\bar t_{F_k+1})$ with $ \bar t_{F_k+1}-\bar t_{F_k}\geq t_d$, and $\{\bar V_{F_1}, \bar V_{F_2}, \cdots, \bar V_{F_p}\}$ and $\{\bar V_{F_1+1}, \bar V_{F_2+1}, \cdots, \bar V_{F_p+1}\}$ be the sequence of the values of the generalized Lyapunov function $V_F$ at the beginning and end of the intervals $\bar T_{i_k}$, respectively. Let $i^0, i^1, \cdots,i^l,\cdots\in\Sigma_f$ be the sequence of modes active on the intervals $[t_{0}, t_{1}), [t_{1},t_{2}), \cdots,[t_l,t_{l+1}),\cdots$, respectively. The conditions for FTS of the origin of \eqref{hybrid sys} in terms of a common Lyapunov function are already given in \cite{li2019finite,li2016results}. In this work, we direct our focus on conditions in terms of multiple generalized Lyapunov functions defined as follows.

\begin{Definition}\label{gen Lyap func}
(\textbf{Generalized Lyapunov Function}): A continuous, positive definite function $V_j:\mathbb R^n\rightarrow\mathbb R_+$ is called a generalized Lyapunov function if there exists a continuous function $\phi:\mathbb R_+\rightarrow \mathbb R_+$ satisfying $\phi(0) = 0$, such that $V_j(x(t)) \leq \phi(V_j(x(t_{j_i})))$ for all $t\in [t_{j_i}, t_{j_i+1})$.
\end{Definition}

\begin{Definition}\label{class GK}
(\textbf{Class-$\mathcal{GK}$ function}): A function $\alpha:\mathbb R_+\rightarrow\mathbb R_+$ is called a class-$\mathcal{GK}$ function if it is increasing, i.e., for all $x>y\geq 0$, $\alpha(x)>\alpha(y)$, and right continuous at the origin with $\alpha(0) = 0$.
\end{Definition}

\begin{Definition}\label{class GK infty}
(\textbf{Class-$\mathcal{GK}_\infty$ function}): A function $\alpha:\mathbb R_+\rightarrow\mathbb R_+$ is called a class-$\mathcal{GK}_\infty$ function if it is a class-$\mathcal{GK}$ function, and $\lim_{r\to\infty}\alpha(r) = \infty$.
\end{Definition}

Note that the class-$\mathcal{GK}$ functions have similar composition properties as those of class-$\mathcal K$ functions, i.e., for $\alpha_1, \alpha_2\in \mathcal{GK}$ and $\alpha\in \mathcal{K}$,
we have:
\begin{itemize}
    \item $\alpha_1\circ \alpha_2\in \mathcal{GK}$ and $\alpha_1+\alpha_2 \in \mathcal{GK}$;
    \item $\alpha\circ\alpha_1\in \mathcal{GK}$, $\alpha_1\circ\alpha\in \mathcal{GK}$ and $\alpha_1 + \alpha \in \mathcal{GK}$.
\end{itemize}

\begin{Lemma}\label{V diff lemma}
If a generalized Lyapunov function $V_F$ satisfies
\begin{equation}\begin{split}\label{v f equiv}
    \sum\limits_{k = 1}^p |V_F(x(t_{F_{k+1}})) - V_F(x(t_{F_k+1}))|\leq \alpha(\|x_0\|),
\end{split}\end{equation}
 for all $p \in \mathbb N$ where $\alpha \in \mathcal{GK}$, then there exists $\bar\alpha\in \mathcal{GK}$ such that the following holds for all $0\leq\beta<1$ and all $p\in \mathbb N$:
{\small
\begin{align}\label{v f actual}
\sum\limits_{k = 1}^p\Big(V_F(x(t_{F_{k+1}}))^{1-\beta} - V_F(x(t_{F_k+1}))^{1-\beta}\Big)\leq \bar\alpha(\|x_0\|).
\end{align}
}
\end{Lemma}
\begin{proof}
See Appendix \ref{app lemma 1 pf}.
\end{proof}

We now present our main result for FTS of hybrid systems.

\begin{Theorem}\label{FT Th 2}
The origin of \eqref{hybrid sys} is FTS if there exist generalized Lyapunov functions $V_i(x)$ for each $i\in \Sigma_f$ and the following hold:
\begin{itemize}
 \item[(i)] There exists $\alpha_1 \in \mathcal{GK}$, such that
\begin{equation}
\begin{split}
   \sum\limits_{k = 0}^{p}\Big(
   V_{i^{k+1}}(x(t_{{k+1}})) -&V_{i^k}(x(t_{{k+1}}))\Big)\leq \alpha_1(\|x_0\|), \label{Hyb cond 1}
\end{split}\end{equation}
holds for all $p\in \mathbb Z_+$;
\item[(ii)] There exists $\alpha_2\in \mathcal{GK}$ such that
\begin{equation}\begin{split}
   \sum\limits_{k = 0}^{p}\Big(
   V_{i^k}(x(t_{{k+1}})) -&V_{i^k}(x(t_{k}))\Big)\leq \alpha_2(\|x_0\|), \label{Hyb cond 2}
\end{split}\end{equation}
holds for all $p\geq 0$;
\item[(iii)] There exists $\alpha_3\in \mathcal{GK}$ such that for all $i\in \Sigma_f$,
\begin{equation}\begin{split}
    \sum\limits_{t\in J_{i}}\Big(V_i(x^+(t))-V_i(x(t))\Big)& \leq \alpha_3(\|x_0\|); \label{Hyb cond 3}
\end{split}\end{equation}
\item[(iv)] There exists $F\in \Sigma_f$ such that $\dot y = f_F(y)$ is FTS, and there exist a positive definite generalized Lyapunov function $V_F$ and constants $c>0$, $0<\beta<1$ such that
\begin{equation}\label{v dot cond}
\dot V_F \leq -cV_F^\beta,
\end{equation}
for all $t\in \bigcup [t_{F_k},t_{F_k+1})\setminus J_F$;
\item[(v)] There exists $\alpha_4\in \mathcal{GK}$ such that for all $p\geq 1$,
\begin{equation}\label{bar v f equiv}
\begin{split}
    \sum\limits_{k = 1}^p |\bar V_{F_{k+1}} - \bar V_{F_k+1}|\leq \alpha_4(\|x_0\|);
\end{split}\end{equation}
\item[(vi)] The accumulated duration of activation for mode $F$ without any discrete state jump satisfy $\bar T_F = \sum_i|\bar T_{i_k}| = \gamma(\|x_0\|)$ where $\gamma\in \mathcal{GK}$ where $|\bar T_{i_k}|$ is the length of the interval $T_{i_k}$.
\end{itemize}
Moreover, if all the conditions hold globally, the functions $V_i$ are radially unbounded for all $i\in \Sigma_f$, and $\alpha_l\in \mathcal{GK}_\infty$ for $l\in \{1,2,3,4\}$, then the origin of \eqref{hybrid sys} is globally FTS (GFTS).
\end{Theorem}

\begin{proof}
Let $x_0\in D$, where $D$ is some open neighborhood of the origin. For all $p\in \mathbb Z_+$, we have that
\begin{align*}
    V_{i^p}(x(t_p)) & = V_{i^0}(x(t_{0})) + \sum\limits_{k = 1}^{p}
   \Big(V_{i^k}(x(t_{k})) -V_{i^{k-1}}(x(t_{k}))\Big)\\
    & \quad + \sum\limits_{k = 0}^{p-1}
   \Big(V_{i^k}(x(t_{{k+1}})) -V_{i^k}(x(t_k))\Big)\\
   & \quad + \sum\limits_{k = 0}^{p}\sum_{t\in J_{k}\bigcap [t_k, t_{k+1})}
   \Big(V_{i^k}(x(t^+)) -V_{i^k}(x(t))\Big)\\
    & \overset{\eqref{Hyb cond 1},\eqref{Hyb cond 2},\eqref{Hyb cond 3}}{\leq} \alpha_0(\|x_0\|)+\alpha_1(\|x_0\|)\\
    & \quad \quad \quad +\alpha_2(\|x_0\|) + N\alpha_3(\|x_0\|),
\end{align*}
where $\alpha_0(r) = \max\limits_{i\in \Sigma, \|x\|\leq r}V_i(x)$. Since $V_i$ are positive definite, $\alpha_0$ is positive definite. Hence, there exists $\alpha\in \mathcal{GK}$ such that $\alpha_0(\|x_0\|)+\alpha_1(\|x_0\|)+\alpha_2(\|x_0\|) + N\alpha_3(\|x_0\|)\leq \alpha (\|x_0\|)$ for all $x_0\in D$. Consequently:
\begin{align}\label{v f a}
 V_F(x(t_{F_i})) \leq \alpha(\|x_0\|).
\end{align}

We also know that during $\bar T_{F_k}$, there is no discrete jump for all $k\in \mathbb N$. As per condition (iv), $\dot V_F \leq -cV_F^\beta$ for all $t\in \bigcup\bar T_{F_k}$. Using this and the Comparison lemma, we obtain:
\begin{equation*}\begin{split}
& |\bar T_{F_k}| \leq \frac{\bar V_{F_k}^{1-\beta}}{c((1-\beta)}-\frac{\bar V_{F_k+1}^{1-\beta}}{c(1-\beta)}\\
\implies \sum\limits_{k = 1}^M &|\bar T_{F_k}| \leq \sum\limits_{k = 1}^M\Big(\frac{\bar V_{F_k}^{1-\beta}}{c((1-\beta)}-\frac{\bar V_{F_k+1}^{1-\beta}}{c(1-\beta)}\Big)\\
& = \frac{\bar V_{F_1}^{1-\beta}}{c(1-\beta)} + \sum\limits_{i = 1}^
{M-1}\frac{\bar V_{F_{i+1}}^{1-\beta}-\bar V_{F_i+1}^{1-\beta}}{c(1-\beta)} -\frac{\bar V_{F_M+1}^{1-\beta}}{c(1-\beta)}.
\end{split}\end{equation*}
Using \eqref{v f a}, we obtain that $\frac{V_{F_1}^{1-\beta}}{c(1-\beta)} \leq \frac{\alpha(\|x_0\|)^{1-\beta}}{c(1-\beta)}$. Define $\gamma_1(\|x_0\|) \triangleq \frac{\alpha(\|x_0\|)^{1-\beta}}{c(1-\beta)}$ so that $\gamma_1\in \mathcal{GK}$. From condition (v) and Lemma \ref{V diff lemma}, there exists $\bar \alpha\in \mathcal{GK}$ such that:
\begin{align}\label{v f need}
\sum\limits_{i = 1}^{M-1}\frac{\bar V_{F_{i+1}}^{1-\beta}-\bar V_{F_i+1}^{1-\beta}}{c(1-\beta)} \leq \frac{\bar \alpha(\|x_0\|)}{c(1-\beta)}.
\end{align}
Define $\gamma(\|x_0\|) \triangleq \gamma_1(\|x_0\|) + \frac{\bar \alpha(\|x_0\|)}{c(1-\beta)}$ so that we obtain:
\begin{equation*}\begin{split}
   \bar T_F + \frac{\bar V_{F_M+1}^{1-\beta}}{c(1-\beta)} &\leq \frac{\bar V_{F_1}^{1-\beta}}{c(1-\beta)} + \sum\limits_{i = 1}^
{M-1}\frac{\bar V_{F_{i+1}}^{1-\beta}-\bar V_{F_i+1}^{1-\beta}}{c(1-\beta)}\\
& \leq \gamma(\|x_0\|).
\end{split}\end{equation*}
Clearly, $\gamma \in \mathcal{GK}$. Now, with $\bar T_F  = \gamma(\|x_0\|)$, we obtain
\begin{equation*}\begin{split}
     \bar T_F+\frac{\bar V_{F_M+1}^{1-\beta}}{c(1-\beta)}\leq \gamma(\|x_0\|) =  \bar T_F,
\end{split}\end{equation*}
which implies $\frac{\bar V_{F_M+1}^{1-\beta}}{c(1-\beta)} \leq 0$. But $\bar V_F\geq 0$, which implies $\bar V_{F_M+1} = 0$. Hence, if mode $F$ is active for the accumulated time $\bar T_F$ without any discrete jump in the system state, the value of the function $V_F$ converges to $0$ as $t\rightarrow \bar t_{F_M+1}$.

From Assumption \ref{assum bar T tm}, $|\bar T_{F_k}|\geq t_d$ for all $k\in \mathbb N$, and hence $Mt_d \leq \sum\limits_{i = 1}^M|\bar T_{F_k}| =  \gamma(\|x_0\|)$, which implies that $M\leq \frac{\gamma(\|x_0\|)}{t_d} <\infty$. Now we show that the time of convergence is finite, i.e., $\bar t_{F_M+1} <\infty$. From the above analysis, we have that if $\sum\limits_{i = 1}^M |\bar T_{F_i}| =  \bar T_F$, then there exists an interval $[\bar t_{F_M},\bar t_{F_M+1})$ such that $\bar V_{F_M+1} = 0$. Since $\bar V_{F_i} \leq \bar\alpha(\|x_0\|)<\infty$, we obtain that
\begin{align*}
  \bar t_{F_M+1}-\bar t_{F_M}& \leq \frac{\bar V_{F_M}^{1-\beta}-\bar V_{F_M+1}^{1-\beta}}{c(1-\beta)}\leq \frac{\bar V_{F_M}^{1-\beta}}{c(1-\beta)} <\infty
\end{align*}
for all $x_0\in D$. Now, there are two cases possible. If $\bar t_{F_M}<\infty$, then, we obtain that $\bar t_{F_M+1} \leq \bar t_{F_M} + \frac{\bar\alpha(\|x_0\|)^{1-\beta}}{c(1-\beta)}<\infty$ for all $x_0\in D$. If $\bar t_{F_M} = \infty$, we obtain that the time of activation for mode $F$ reads $\bar T_F = \sum\limits_{i = 1}^{M-1}|\bar T_i| <\gamma(\|x_0\|)$, which contradicts condition (vi). Hence, for condition (vi) to hold, it is required that $\bar t_{F_M}<\infty$ and hence, we obtain that $\bar t_{F_M+1}<\infty$. Hence, the trajectories of \eqref{hybrid sys} reach the origin with finite number of active intervals of the continuous flow $f_F$.

If all the conditions (i)-(vi) hold globally and the functions $V_i$ are radially unbounded, we obtain that $\alpha_0$ is also radially unbounded. Using the fact that $\alpha_1, \alpha_2, \alpha_3,\alpha_4\in \mathcal{GK}_\infty$, we have  $\bar\alpha(\|x_0\|) <\infty$ for all $\|x_0\|<\infty$, and hence, we obtain $t_{F_M+1} \leq t_{F_M} + \frac{\bar\alpha(\|x_0\|)^{1-\beta}}{c(1-\beta)}<\infty$ for all $x_0$, which implies GFTS.
\end{proof}

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.92\columnwidth,clip]{V_inc_hybrid2.eps}
	\caption{Conditions (i), (ii) and (iii) of Theorem \ref{FT Th 2}: change in the values of the generalized Lyapunov functions. The increment shown by blue, red and black double-arrows pertain to condition (i), (ii) and (iii), respectively. }
	\label{fig:V i incr}
\end{figure}

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.92\columnwidth,clip]{V_F_inc.eps}
	\caption{Increment in the value of $V_F$ during the switched-off period (condition (v) of Theorem \ref{FT Th 2}).}
	\label{fig:V F incr}
\end{figure}

\noindent Condition (i) means that at switching instants, the cumulative value of the differences between the consecutive Lyapunov functions is bounded by a class-$\mathcal{GK}$ function; condition (ii) means that the cumulative increment in the value of the individual Lyapunov functions when the respective modes are active, is bounded by a class-$\mathcal{GK}$ function (see Figure \ref{fig:V i incr}). Some authors use the time derivative condition, i.e., $\dot V_i\leq \lambda V_i$ with $\lambda>0$, in place of condition (ii) to allow growth of $V_i$, hence, requiring the function to be continuously differentiable (e.g. \textnormal{\cite{wang2018conditions}}). Our condition allows the use of non-differentiable generalized Lyapunov function; condition (iii) means that the cumulative increment in the value of the generalized Lyapunov function $V_i$ is bounded by a class-$\mathcal{GK}$ function during the discrete jumps; condition (iv) means that there exists an FTS mode $F\in \Sigma$; condition (v) means that the accumulated increment in the value of the Lyapunov function $V_F$ during the "switched-off" periods is bounded (see Figure \ref{fig:V F incr}); and condition (vi) means that the FTS mode $F$ is active for a minimum accumulated time.

\begin{Remark}
In general, the inequality \eqref{v f need} can be difficult to obtain directly. Consider the case when we only know that the mode $F$ is homogeneous, with negative degree of homogeneity. From \cite[Theorem 7.2]{bhat2005geometric}, we know that in this case, condition (iii) of Theorem \ref{FT Th 2} holds for some $\beta$, but its exact value might not be known. In this case, it is not possible to bound the left-hand side (LHS) of \eqref{v f need}. Lemma \ref{V diff lemma} allows one to bound this LHS with a class-$\mathcal{GK}$ function without explicitly knowing the value of $\beta$. \end{Remark}

Compared to \cite{li2019finite,li2016results}, our method is less conservative since we allow generalized Lyapunov functions to increase during the continuous flow (per \eqref{Hyb cond 2}) as well as at the discrete jump (per \eqref{Hyb cond 3}). Also, during the continuous flow the generalized Lyapunov functions are allowed to grow when switching from one continuous flow to another (per \eqref{Hyb cond 1}), whereas the aforementioned work imposes that the common Lyapunov function is always non-increasing.

We discuss the relation of Assumption \ref{assum bar T tm} with the average dwell-time (ADT) for discrete jumps method that is often used in the literature \cite{teel2013lyapunov}. Discrete jumps under ADT means that in any given interval $[t_1, t_2]$, the number of discrete jumps of the states $N([t_1,t_2])$ satisfies $N([t_1,t_2])\leq N_0 + \delta(t_2-t_1)$, where $\delta\geq 0$ and $N_0\geq 1$. (see \cite{teel2013lyapunov,cai2008smooth}). We note that if the conditions of the ADT method hold for the mode $F$, instead of the minimum dwell-time condition of Assumption \ref{assum bar T tm}, then the FTS result still holds since the parameter $M$ used in the proof of Theorem \textnormal{\ref{FT Th 2}} can be defined as $M = N_0 + \frac{T_F}{\tau_a}$, where $T_F$ is the total time of activation of mode $F$.

So far we have considered the cases when only one of the modes of the system \eqref{hybrid sys} is FTS. Next, we present a result for the case when all the continuous modes of the hybrid system are FTS.

\begin{Theorem}
The origin of \eqref{hybrid sys} is FTS if the origin of $\dot x = f_i(x)$ is FTS for all $i\in \Sigma_f$ and there exist generalized Lyapunov functions $V_{i}$ for each $i\in \Sigma_f$ such that conditions (i), (iii) and (v) of Theorem \ref{FT Th 2} are satisfied.
\end{Theorem}

\begin{proof}
Let $T_i$ denote the total time duration for which mode $i\in \Sigma$ is active without any discrete jump in the state. As the generalized
Lyapunov function $V_{i}$ satisfies condition (iv) of Theorem \eqref{FT Th 2}, it satisfies the condition (ii) also with $\alpha_2 = 0$. Let $T_F(\|x_0\|) = \max\limits_i \gamma_i(\|x_0\|)$. In any time interval $[0, T)$, at least one mode $i$ satisfies $T_i \geq \frac{T}{N}$. Hence, for $T\geq T_M \triangleq NT_F(\|x_0\|)$, we obtain that there exists at least one mode $i$
satisfying $T_i \geq \frac{T}{N} \geq \frac{T_M}{N} = T_F(\|x_0\|)\geq \gamma_i(\|x_0\|)$. This implies that all the conditions of Theorem \ref{FT Th 2} are satisfied. Therefore, we obtain that the origin of \eqref{hybrid sys} is FTS.
\end{proof}

\begin{Remark}\label{remark: summary contri}
For switched systems, i.e., systems with no discrete jump, Theorem \ref{FT Th 2} guarantees FTS of the origin with conditions (i), (ii), (iv), (v) and (vi), since with $D = \emptyset$, condition (iii) is not needed. Hence, the results presented in this paper are applicable to switched systems as well. Furthermore, if $N_f = 1$, i.e., if the system \eqref{hybrid sys} reduces to a continuous-time dynamical system, Theorem \ref{FT Th 2} reduces to Theorem \ref{FTS Bhat}. Thus, the continuous-time result for FTS is a special case of Theorem \ref{FT Th 2}. Finally, some of the results presented in \cite{li2019finite} are special cases of Theorem \ref{FT Th 2} with $V = V_F$ and $\alpha_3 = 0$.
\end{Remark}

\vspace{-10pt}

\section{Simulations}\label{Simulations}
We present a numerical example with number of modes $N = 5$, where the fifth mode is FTS, i.e., $F = 5$. The  system modes are chosen as
\begin{align*}
    &f_1 = \begin{bmatrix}0.01x_1^2+x_2\\-0.01x_1^3+x_2\end{bmatrix},\quad     f_2 = \begin{bmatrix}0.01x_1-x_2\\-x_1^2+0.01x_2\end{bmatrix},\\
    &f_3 = \begin{bmatrix}-x_1-x_2\\x_1-x_2\end{bmatrix},\quad
    f_4 = \begin{bmatrix}0.01x_1^2+0.01x_1x_2\\-0.01x_1^3+x_2^2\end{bmatrix},\\
    &\qquad \quad f_5 = \begin{bmatrix} x_2-20\textrm{sign}(x_1)|x_1|^\alpha\\-10\textrm{sign}(x_1)|x_1|^{2-2\alpha}\end{bmatrix}.
\end{align*}
The generalized Lyapunov functions are chosen as $V_i(x) = x^TP_ix$ where matrices $P_i$ are chosen as $P_1 = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix},  P_2 = \begin{bmatrix}5 & 2\\ 2 & 4\end{bmatrix}, P_3 = \begin{bmatrix}1 & 0\\ 0 & 3\end{bmatrix}, P_4 = \begin{bmatrix}6 & 1\\ 2 & 3\end{bmatrix}$, and $V_5(x) = \frac{k_2}{2\alpha}|x_1|^{2\alpha} + \frac{1}{2}|x_2|^2$. The discrete jumps are defined by $g = \begin{bmatrix}-1.1x_1 \\-1.1x_2\end{bmatrix}$ so that the states $x_1$ and $x_2$ change their signs and increase in the magnitude at the discrete jumps. Note that this example is more general than the examples considered in \cite{li2019finite}, as we allow the dynamics to have unstable modes. In this case, we allowed the switches in the continuous flows after $0.2$ sec, i.e., $|T_{i_k}| = 0.2$ sec and discrete jumps after $0.1$ sec, so that $t_d = 0.1$ sec, i.e., $|\bar T_{i_k}| = 0.1$, for all $i\in \{1, 2,\cdots, 5\}$ and $k\in \mathbb Z_+$ (see Assumption \ref{assum bar T tm}). Figure \ref{fig:sim hyb mode} shows the switching signal $\sigma_f(t)$.  Figure \ref{fig:sim hyb states} shows the states $x_1(t)$ and $x_2(t)$ for the first 10 seconds of the simulation. The states change sign at the discrete jumps. It can be seen in the figure that the system moves away from the origin while operating in the unstable modes. Figure \ref{fig:sim hyb norm} shows the norm of the state vector $x(t)$ on log scale. It can be seen that the system states reach the origin in finite time. Figure \ref{fig:sim hyb V} shows the evolution of generalized Lyapunov functions $V_i$ with respect to time for first 10 seconds of the simulation. As can be seen from the figure, the generalized Lyapunov functions increase at the time of discrete jumps, the switching instants, as well as during the continuous flows along the unstable modes.

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.92\columnwidth,clip]{hyb_mode.eps}
	\caption{Switching signal $\sigma_f(t)$ for the considered hybrid system.}
	\label{fig:sim hyb mode}

	\centering
	\includegraphics[width=0.92\columnwidth,clip]{hyb_states.eps}
	\caption{System states $x_1(t)$ and $x_2(t)$ for $t\in [0,10]$ sec. }
	\label{fig:sim hyb states}

	\centering
	\includegraphics[width=0.92\columnwidth,clip]{hyb_norm_x.eps}
	\caption{The norm of the state vector $x(t)$ for the considered hybrid system.}
	\label{fig:sim hyb norm}
\end{figure}

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.92\columnwidth,clip]{hyb_V.eps}
	\caption{Generalized Lyapunov functions $V_i(t)$ for $t\in [0, 10]$ sec for the considered hybrid system.}
	\label{fig:sim hyb V}
\end{figure}

The provided example validates that the system can achieve FTS even if one or more modes are unstable, if the FTS mode is active for long enough.

\section{Conclusions and Future Work}\label{Conclusions}
In this paper, we studied FTS of hybrid systems. We showed that under some mild conditions on the bounds on the difference of the values of generalized Lyapunov functions, if the FTS mode is active for a minimum required time, then FTS can be achieved. Our proposed method allows the individual generalized Lyapunov functions to increase both during the continuous flow as well as at the discrete jumps, i.e., it allows for the hybrid system to have unstable modes.

Traditionally, optimization-based formulations are used to find control inputs that satisfy the Lyapunov conditions. Our ongoing research focuses on incorporating input and state constraints in the hybrid framework for specifications involving both spatial and temporal requirements. More specifically, we are investigating how to incorporate prescribed-time, rather than merely finite-time, convergence for the system modes, so that the overall framework can be used in the synthesis and analysis of controllers with spatiotemporal specifications.

\vspace{-10pt}
\appendices

\section{Proof of Lemma 1}
\begin{proof}\label{app lemma 1 pf}
Let the function $V_F$ satisfy \eqref{v f equiv} for all $k\mathbb N$, where $\alpha\in \mathcal{GK}$. Lemma 3.3, 3.4 of \cite{zuo2016distributed} establish the following set of inequalities for $z_i\geq 0$ and $0<r\leq 1$
\begin{align}
    \left(\sum_{i = 1}^M z_i \right)^r \leq \sum_{i = 1}^M z_i^r \leq M^{1-r} \left(\sum_{i = 1}^M z_i \right)^r.
\end{align}
Hence, we have that for $a\geq b\geq 0$ and $0<r\leq 1$, $ a^r = (b + (a-b))^r \leq b^r  + (a-b)^r$, or equivalently,
\begin{equation}\begin{split}
    a^r-b^r \leq (a-b)^r.
\end{split}\end{equation}
Define $a_i \triangleq V_F(x(t_{F_{i+1}}))$ and $b_i \triangleq V_F(x(t_{F_i+1}))$ so that whenever $a_i\geq b_i$, or, $V_F(x(t_{F_{i+1}})) \geq V_F(x(t_{F_i+1}))$, we have that $a_i^r-b_i^r \leq (a_i-b_i)^r$. Denote $i \in I_1 \triangleq \{i_1,i_2,\cdots, i_m\}$ for some $m\leq k$, for which $a_i\geq b_i$ and $i\in I_2\triangleq \{i_{m+1}, i_{m+2},\cdots, i_k\}$ for which $a_i\leq b_i$ (or equivalently, $a_i^r-b_i^r\leq 0$). Hence, we have that for any $0<r\leq 1$,
\begin{equation*}\begin{split}
    \sum_{i = 1}^k (a_i^r-b_i^r) & \leq \sum_{i\in I_1}(a_i^r-b_i^r) \leq \sum_{i\in I_1}(a_i-b_i)^r\\
    & \leq  m^{1-r}\left(\sum_{i\in I_1}(|a_i-b_i|)\right)^r.
\end{split}\end{equation*}

Using this, we obtain that

\begin{equation*}\begin{split}
  & \sum_{i = 1}^k ( V_F(x(t_{F_{i+1}}))^r-V_F(x(t_{F_i+1}))^r)\\
  \leq & m^{1-r}\left(\sum_{i \in I_1} (| V_F(x(t_{F_{i+1}}))-V_F(x(t_{F_i+1}))|)\right)^r\\
  \leq & m^{1-r}\left(\sum_{i = 1}^k | V_F(x(t_{F_{i+1}}))-V_F(x(t_{F_i+1}))|\right)^r\\
  \leq & m^{1-r}\alpha(\|x(0)\|)^r \triangleq \bar \alpha(\|x(0)\|)
\end{split}\end{equation*}

where $\bar \alpha$ is clearly a class $\mathcal{GK}$ function. Hence, we obtain that \eqref{v f actual} holds for all $0<\beta = 1-r<1$.
\end{proof}

\bibliographystyle{IEEEtran}


\section{Introduction}

\subfile{1introduction}

\section{The Topology on A-mod}

\subfile{2topology}

\section{Cohomology on A-mod}\label{deformations}

\subfile{3deformations}

\section{Extensions and Cohomology}\label{extandcohom}

\subfile{4extandcohom}

\section{Illusie's Exact Sequence}\label{illusiecomparison}

\subfile{5illusieexactseq}

\subfile{zappendix}

\bibliographystyle{unsrt

\section{The Grothendieck Spectral Sequence}

In this appendix, we describe a map belonging to the 5-term exact sequence induced from the Grothendieck Spectral Sequence in the special case of a morphism of topoi.

We also specify our sign convention for torsors and gerbes:

\begin{definition}\label{torsorgerbesignconvention}
Given a short exact sequence
\[0 \rightarrow A \rightarrow B \overset{g}{\rightarrow} C \rightarrow 0\]
\noindent of abelian groups in $E$, consider the corresponding long exact sequence in sheaf cohomology:
\[\begin{tikzcd}
\cdots \arrow{r}      &H^0(B) \arrow{r}{g_*}     &H^0(C) \arrow{r}{\partial^0}     &H^1(A) \arrow{r}     &\cdots\\
\cdots \arrow{r}      &H^1(B) \arrow{r}{g_*}     &H^1(C) \arrow{r}{\partial^1}     &H^2(A) \arrow{r}     &\cdots
\end{tikzcd}\]

We agree that
\begin{itemize}
\item The boundary map $\partial^0$ sends $\gamma \in H^0(C)$ to the $A$-torsor $P$ whose sections over $U \in E$ are
\[P(U) := \{\beta \in H^0(U, B) \:\:|\:\: g_*\beta = \gamma|_U\}\]
\item The boundary map $\partial^1$ sends the class of a $C$-torsor $P$ to the $A$-gerbe whose sections over $U \in E$ are $B|_U$-torsors $Q$ on $E/U$ with a $g$-equivariant map $Q \rightarrow P|_U$. The arrows are $B$-maps.
\end{itemize}

\end{definition}

Another convention is to choose the boundary maps $-\partial^i$ instead.

Now consider $\pi : X \rightarrow Y$ and $\rho : Y \rightarrow Z$ two functors between abelian categories with enough injectives and suppose $\pi$ sends injective objects to $\rho$-acyclic ones. Given $G \in X$, we construct the Grothendieck Spectral Sequence as follows (see the diagrams below):
\begin{itemize}

\item Resolve $G$ by injectives, $\{J^p\}$.

\item Apply $\pi$.

\item Find a Cartan-Eilenberg resolution $\{I^{p, q}\}$ of the resulting complex and apply $\rho$ to it.

\item Take horizontal ($p$) and then vertical ($q$) cohomology to get $E_2^{p, q} = R^q \rho R^p \pi G$.

\end{itemize}

Remark 13.21.4 of \cite[015G]{sta} observes the naturality of Cartan-Eilenberg resolutions with respect to maps of chain complexes $J^\bullet$, and any choice of injective resolutions is natural in $G$. The Grothendieck Spectral Sequence is then natural in $G$ via the usual functoriality of the pair of right-derived functors.

We use the convention of \cite[012X]{sta} that the Total Complex should have differentials $d_\rightarrow^{p,q} + (-1)^p d_\uparrow^{p,q}$, where the horizontal differential $d_\rightarrow$ has degree $(+1, 0)$ and the vertical $d_\uparrow$ degree $(0, +1)$.

Our case of interest is when $X$ and $Y$ are ringed topoi, $\pi = f_*$ for a morphism $f$ of topoi, and $\rho = \Gamma$. We construct this spectral sequence.

\[\begin{tikzcd}
I^{03}      &       &       &       \\
I^{02} \arrow{u} \arrow{r}    &I^{12}     &       &       \\
I^{01} \arrow{u} \arrow{r}    &I^{11} \arrow{u}{d_\uparrow} \arrow{r}{d_\rightarrow}    &I^{21}     &       \\
I^{00} \arrow{u} \arrow{r}      &I^{10} \arrow{u} \arrow{r}    &I^{20}  \arrow{u} \arrow{r}   &I^{30}
\end{tikzcd}
E_1 :
\left[\begin{tikzcd}
h_\rightarrow^{03}      &       &       &       \\
h_\rightarrow^{02} \arrow{u}     &h_\rightarrow^{12}     &       &       \\
h_\rightarrow^{01} \arrow{u}     &h_\rightarrow^{11} \arrow{u}{d_\uparrow}    &h_\rightarrow^{21}     &       \\
h_\rightarrow^{00} \arrow{u}       &h_\rightarrow^{10} \arrow{u}     &h_\rightarrow^{20}  \arrow{u}    &h_\rightarrow^{30}
\end{tikzcd}\right.\]

\[E_2 :
\left[\begin{tikzcd}
H^3(f_*(G))      &       &       &       \\
H^2(f_*(G))      &H^2(R^1f_*(G))     &       &       \\
H^1(f_*(G))      &H^1(R^1f_*(G)) \arrow{ulu}{d_2}    &H^1(R^2f_*(G))     &       \\
\Gamma(f_*(G))        &\Gamma(R^1f_*(G)) \arrow{ulu}     &\Gamma(R^2f_*(G))  \arrow{uul}    &\Gamma(R^3f_*(G))
\end{tikzcd}\right.\]

A section of the derived pushforward $\Gamma(R^1 f_* G)$ may be thought of as an element $\beta \in \Gamma(I^{10})$ such that $d_\rightarrow \beta = 0$ and $d_\uparrow \beta$ is in the image of $d_\rightarrow$. Choose a lift $\gamma \in \Gamma(I^{01})$ of $d_\uparrow \beta$.

Suppose another section $\gamma'$ maps to $d_\uparrow \beta$. Then $\gamma-\gamma'$ lies in $h^{01}_\rightarrow$. Since $H^2(f_* G)$ is a quotient by the image of $h^{01}_\rightarrow$, the image $d_\uparrow \gamma = d_\uparrow \gamma'$ is a well-defined cohomology class.

Consider the two subsheaves of $I^{01}$:

\[P(U) := \{s \in \Gamma(U, I^{01}) \: | \: d_\rightarrow s = d_\uparrow \beta \text{ and } d_\uparrow s = 0\}\]
and
\[Q(U) := \{t \in \Gamma(U, I^{01}) \: | \: d_\uparrow t = d_\uparrow \gamma \text{ and } d_\rightarrow t = 0\}\]

Let $L$ denote the kernel of both differentials $d_\uparrow$ and $d_\rightarrow$ in $I^{01}$. Then $P$ and $Q$ are naturally $L$-torsors. The sum of elements from $P$ and $Q$ is the $L$-torsor of local sections which map to $d_\rightarrow \beta$ horizontally and $d_\uparrow \gamma$ vertically; $\gamma$ trivializes this torsor.

\[\begin{tikzcd}
L \oplus L \arrow{r}{+} \arrow[symbol=\circlearrowleft]{d}      &L \arrow[symbol=\circlearrowleft]{d}      \\
P \oplus Q \arrow{r}{+} \arrow[hook]{d}      &L \arrow[hook]{d}      \\
I^{01} \oplus I^{01} \arrow{r}{+}        &I^{01}
\end{tikzcd}\]

This diagram witnesses that the sum of the torsors $P$ and $Q$ is zero, hence that they are inverses. Since we have an exact sequence
\[0 \rightarrow f_*G \rightarrow I^{00} \rightarrow L \rightarrow 0\]
with middle term injective, we may identify $L$-torsors with $f_*G$-gerbes via the boundary map.

Remark that, by definition, $P$ was the $f_*G$-gerbe of lifts of $\beta$ to a $G$-torsor on $Y$. Likewise, $Q$ was the image of $\beta$ under the $E_2$-page differential. That is, the map $\Gamma(R^1f_*G) \rightarrow H^2(f_* G)$ in the spectral sequence and the corresponding 5-term exact sequence sends a global section to the inverse of the gerbe of its lifts to a torsor on $Y$.

We package the observations made in this section into a lemma.

\begin{lemma}\label{grothssinterpretation}

The Grothendieck Spectral Sequence is natural in $G$. The map $\Gamma(R^1 f_*G) \rightarrow H^2(f_*G)$ sends a global section to the inverse of the class of its gerbe of lifts to a $G$-torsor on $Y$.

\end{lemma}

\section{Butterflies}\label{butterfliesappendix}

Several essential properties of the 2-groupoid of 2-extensions are transcribed from \cite{butterflies} or \cite{sga7i} to our context. In the process, we fix many of the conventions requisite for working with them.

\begin{definition}\label{butterfly2categorydefinitions}

We define and begin our study of $\underline{\text{Ext}}_A^2(\underline{\enspace \thinspace}, K) \rightarrow {A \text{-mod}}$.

The 2-groupoid $\underline{\text{Ext}}_A^2(\underline{\enspace \thinspace}, K) \rightarrow {A \text{-mod}}/M$ has:

\begin{itemize}
\item Sections: 2-extensions
\[0 \rightarrow K \rightarrow X \rightarrow Y \rightarrow T \rightarrow 0\]
\noindent over $T \rightarrow M$.

\item Morphisms: Butterflies inducing the identity on $K$:
\begin{equation}\label{butterflydiagram}
\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &T  \arrow{dd}\arrow{r}    &0      \\
        &       &       &Q \arrow{dr} \arrow{ur}      &       &       &       \\
0 \arrow{r}      &K \arrow{r}     &X'\arrow{rr} \arrow{ur}     &      &Y' \arrow{r}      &T'  \arrow{r}    &0
\end{tikzcd}\end{equation}
\noindent lie over the morphism $T \rightarrow T'$.

\item 2-Isomorphisms: Maps between the extensions defining the two butterflies.
\end{itemize}
\end{definition}

\paragraph{Induced Butterfly}\label{inducedbutterfly}
Given a map of chain complexes that are 2-extensions:

$$\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{d}     &X\arrow{d} \arrow{r}           &Y \arrow{r} \arrow{d}       &T  \arrow{d}\arrow{r}    &0      \\
0 \arrow{r}      &K \arrow{r}     &X'\arrow{r}           &Y' \arrow{r}      &T'  \arrow{r}    &0
\end{tikzcd}$$

\noindent we get a butterfly:

$$\begin{tikzcd}
0 \arrow{r}      &K \arrow{r}\arrow[equals]{dd}     &X \arrow{rr} \arrow{dr}      &&Y \arrow{r}     &T \arrow{r}\arrow{dd}     &0      \\
        &       &       &X' \oplus Y \arrow{dr} \arrow{ur}       &       &       &\\
0 \arrow{r}      &K \arrow{r}     &X' \arrow{rr} \arrow{ur}     &&Y' \arrow{r}     &T' \arrow{r}     &0
\end{tikzcd}$$

The map $X \rightarrow X' \oplus Y$ is the sum of the two maps $X \rightarrow X'$ and $X \rightarrow Y$; the map $X' \oplus Y \rightarrow Y'$ is $X' \rightarrow Y'$ \textit{minus} $Y \rightarrow Y'$. Define the identity to be the induced butterfly from the identity map on chain complexes.

\paragraph{Composition of Butterflies}
Define composition as follows:

$$\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &T  \arrow{dd}\arrow{r}    &0      \\
        &       &       &Q \arrow{dr} \arrow{ur}      &       &       &       \\
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X'\arrow{dr} \arrow{ur} \arrow{rr}     &      &Y' \arrow{r}       &T'  \arrow{dd}\arrow{r}    &0      \\
        &       &       &Q' \arrow{dr} \arrow{ur}      &       &       &       \\
0 \arrow{r}      &K \arrow{r}     &X''\arrow{rr} \arrow{ur}     &      &Y'' \arrow{r}      &T''  \arrow{r}    &0
\end{tikzcd}$$

\noindent compose to the butterfly:
\[\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &T  \arrow{dd}\arrow{r}    &0      \\
        &       &       &Q\oplus_{Y'}^{X'}Q' \arrow{dr} \arrow{ur}      &       &       &       \\
0 \arrow{r}      &K \arrow{r}     &X''\arrow{rr} \arrow{ur}     &      &Y'' \arrow{r}      &T''  \arrow{r}    &0
\end{tikzcd}\]

Here, the term $Q \oplus_{Y'}^{X'} Q'$ refers to the cokernel of $X' \rightarrow Q \times_{Y'} Q'$. The SW-NE diagonal sequence is the cokernel of the map of extensions
\[\begin{tikzcd}
        &0  \arrow{r}     &X'\arrow[equals]{r} \arrow{d}    &X'\arrow{r} \arrow{d}    &0      \\
0 \arrow{r}      &X'' \arrow{r}       &Q \times_{Y'} Q'\arrow{r}       &Q \arrow{r}     &0
\end{tikzcd}\]

It is exact by the snake lemma. Exactness of the NW-SE diagonal follows analogously from the following lemma.

\begin{lemma}
The module $Q \oplus_{Y'}^{X'} Q'$ is the kernel of $Q \amalg_{X'} Q' \rightarrow Y'$.
\end{lemma}

\begin{proof}
Consider the pair of diagrams:

\[\begin{tikzcd}
0 \arrow{r}   &X'' \arrow{r} \arrow[equals]{d}    &Q \times_{Y'} Q' \arrow{r} \arrow{d} \arrow[phantom, very near start, "\ulcorner"]{dr}      &Q\arrow{r} \arrow{d}     &0      \\
0 \arrow{r}  &X''\arrow{r}     &Q'\arrow{r}     &Y' \arrow{r}       &0
\end{tikzcd}\quad
\begin{tikzcd}
0 \arrow{r}   &X' \arrow{r} \arrow{d}    &Q \arrow{r} \arrow{d}       &Y\arrow{r} \arrow[equals]{d}     &0      \\
0 \arrow{r}  &Q'\arrow{r}     &Q \amalg_{X'} Q'\arrow{r}  \arrow[phantom, very near start, "\lrcorner"]{ul}   &Y \arrow{r}       &0
\end{tikzcd}.\]

Let $\pi_i$ denote the projections of $Q \times_{Y'} Q'$ and $\iota_i$ denote the inclusions of $Q \amalg_{X'} Q'$, $i = 1,2$. We get a map from the pullback to the pushout above:
\[\begin{tikzcd}
0 \arrow{r}      &X'' \arrow{r} \arrow{d} \arrow[phantom, near start, "\circ"]{dr}    &Q \times_{Y'} Q' \arrow{dl}{\pi_2} \arrow{r}{\pi_1} \arrow[phantom, near end, "\circ"]{dr} &Q \arrow{r} \arrow{d} \arrow{dl}[swap]{\iota_1}    &0      \\
0 \arrow{r}      &Q' \arrow{r}[swap]{-\iota_2}    &Q \amalg_{X'} Q' \arrow{r}      &Y \arrow{r}     &0
\end{tikzcd}\]

The left and right vertical maps are structure maps coming from the butterfly. The left and right triangles commute because they are the left and right squares of the pullback and pushout diagrams above, respectively. We don't claim the inner parallelogram is commutative.

The arrows $\pi_2$ and $\iota_1$ define a nullhomotopy of a morphism of chain complexes. We place the nullhomotopic morphism $\psi = \iota_1 \circ \pi_1 - \iota_2 \circ \pi_2$ in the middle of a diagram:
\[\begin{tikzcd}
       &0 \arrow{r}      &X' \arrow[equals]{r} \arrow{d}      &X' \arrow{r} \arrow{d}       &0       \\
0 \arrow{r}      &X'' \arrow{r} \arrow{d}     &Q \times_{Y'} Q'  \arrow{r} \arrow{d}{\psi}  &Q \arrow{r} \arrow{d}     &0      \\
0 \arrow{r}      &Q' \arrow{r} \arrow{d}    &Q \amalg_{X'} Q' \arrow{r} \arrow{d}      &Y \arrow{r}     &0      \\
0 \arrow{r}       &Y' \arrow[equals]{r}     &Y' \arrow{r}     &0
\end{tikzcd}\]

Then either of the vertical composites of chain maps in the above diagram is zero, and we get a map from the cokernel of the first chain map to the kernel of the last:

\[\begin{tikzcd}
0 \arrow{r}       &X'' \arrow[equals]{d} \arrow{r}     &Q \oplus_{Y'}^{X'} Q' \arrow{r} \arrow{d}      &Y \arrow{r} \arrow[equals]{d}     &0       \\
0 \arrow{r}      &X''  \arrow{r}      &\ker \arrow{r}      &Y\arrow{r}      &0
\end{tikzcd}\]

Here $\ker = \ker (Q \amalg_{X'} Q' \rightarrow Y')$. The careful reader will notice that the structure map $X'' \rightarrow \ker$ is the negative of the usual map because we used $-\iota_2$, but we still have an isomorphism $Q \oplus_{Y'}^{X'} Q' \simeq \ker$.

\end{proof}

\begin{lemma}
Suppose $T \rightarrow T'$ is an isomorphism in Diagram \eqref{butterflydiagram}. The butterfly given by flipping the diagram upside-down is its inverse (up to 2-isomorphism).
\end{lemma}

\begin{proof}
The composite is

\[\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &T  \arrow{d}\arrow{r}    &0      \\
        &       &       &Q\oplus_{Y'}^{X'}Q \arrow{dr} \arrow{ur}      &       &T' \arrow{d}       &       \\
0 \arrow{r}      &K \arrow{r}     &X \arrow{rr} \arrow{ur}     &      &Y \arrow{r}      &T  \arrow{r}    &0
\end{tikzcd}\]

The diagonal $Q \rightarrow Q \times_{Y'} Q$ gives a section of the middle row of the diagram constructing $Q \oplus_{Y'}^{X'} Q$:
\[\begin{tikzcd}
        &0 \arrow{r}     &X' \arrow[equals]{r} \arrow{d}    &X'\arrow{r} \arrow{d}    &0      \\
0 \arrow{r}      &X \arrow{r} \arrow{d}    &Q \times_{Y'} Q \arrow{r} \arrow{d}      &Q \arrow{r} \arrow{d} \arrow[bend right, dashed]{l} \arrow[dashed]{dl}    &0      \\
0 \arrow{r}      &X \arrow{r}     &Q \oplus_{Y'}^{X'} Q \arrow{r}        &Y \arrow{r} \arrow[bend right, dashed]{l}     &0
\end{tikzcd}\]

Compose to get the map $Q \dashrightarrow Q \oplus_{Y'}^{X'} Q$. The map $X' \rightarrow Q \times_{Y'} Q$ factors through the diagonal, since the maps $X' \rightarrow Q$ are the same.

Then the map $X' \rightarrow Q \dashrightarrow Q \oplus_{Y'}^{X'} Q$ factors as $X' \rightarrow Q \times_{Y'} Q \rightarrow Q \oplus_{Y'}^{X'} Q$, which is the zero map. Hence $Q \dashrightarrow Q \oplus_{Y'}^{X'} Q$ factors through the cokernel, inducing a section $Y \dashrightarrow Q \oplus_{Y'}^{X'} Q$.

The exact sequence defining the composite butterfly splits, so it's isomorphic to the identity butterfly with $X \oplus Y$ in the center.

\end{proof}

\begin{lemma}

Up to 2-isomorphism, butterflies $\xi \simeq \eta$ over $N \rightarrow M \in {A \text{-mod}}$ are the same as butterflies $\xi \simeq \eta|_N$ over $id_N$.

\end{lemma}

\begin{proof}

Let $\xi|_N$ denote the 2-extension
\[0 \rightarrow K \rightarrow X \rightarrow Y|_N \rightarrow N \rightarrow 0\]
and so forth.

Given a butterfly

\[\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &N  \arrow{dd}\arrow{r}    &0      \\
        &       &       &Q \arrow{dr} \arrow{ur}      &       &        &       \\
0 \arrow{r}      &K \arrow{r}     &X' \arrow{rr} \arrow{ur}     &      &Y' \arrow{r}      &M  \arrow{r}    &0
\end{tikzcd}\]

\noindent over $N \rightarrow M \in {A \text{-mod}}$,

\[\begin{tikzcd}
K \arrow{r} \arrow{d}       &X \arrow{r} \arrow{d}      &P \arrow{d}     \\
X' \arrow{r} \arrow{d}     &Q \arrow{r} \arrow[dashed]{d}     &Y \arrow{d}     \\
P' \arrow{r} \arrow[equals]{d}     &Y'|_N \arrow{r} \arrow{d} \arrow[phantom, very near start, "\ulcorner"]{dr}     &N  \arrow{d}    \\
P' \arrow{r}     &Y' \arrow{r}     &M
\end{tikzcd}\]

The dashed arrow comes from the rest of the diagram. Discard the bottom row to get a 3x3 grid of modules whose rows are known to be exact. The 3x3 Lemma ensures that the middle column is exact. We can rearrange to obtain a butterfly $\xi \simeq \eta|_N$.

The composite butterfly $\xi \simeq \eta|_N \simeq \eta$ is then

\[\begin{tikzcd}
0 \arrow{r}      &K \arrow{r} \arrow[equals]{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &N  \arrow{dd}\arrow{r}    &0      \\
        &       &       &(Q \oplus X')/X' \arrow{dr} \arrow{ur}      &       &       &       \\
0 \arrow{r}      &K \arrow{r}     &X' \arrow{rr} \arrow{ur}     &      &Y' \arrow{r}      &M  \arrow{r}    &0
\end{tikzcd}\]

\noindent where $(Q \oplus X')/X'$ includes $X'$ via the sum of the two natural maps. Unwinding definitions carefully, we see that the diagonal arrows are:
\begin{itemize}
\item $X' \rightarrow (Q \oplus X')/X'$ includes the second summand.
\item $X \rightarrow (Q \oplus X')/X'$ is the structure map $X \rightarrow Q$.
\item $Q \oplus X' \rightarrow Y'|_N \oplus X' \rightarrow Y'$, where the first arrow is the pair of natural maps and the second is $X' \rightarrow Y'$ \textit{minus} $Y'|_N \rightarrow Y'$.
\item $Q \oplus X' \rightarrow Q \rightarrow Y$.
\end{itemize}
Precomposing the latter two maps by the sum of the natural maps $X' \rightarrow Q \oplus X'$ gives zero, yielding the factorization through $(Q \oplus X')/X'$.

The reader may check commutativity of the following diagrams.

\[\begin{tikzcd}[column sep=small, row sep=small]
        &(Q \oplus X')/X' \arrow{dd} \arrow{dr}       &       \\
X' \arrow{ur} \arrow{dr}      &       &Y      \\
        &Q \arrow{ur}
\end{tikzcd}
\begin{tikzcd}[column sep=small, row sep=small]
        &(Q \oplus X')/X' \arrow{dd} \arrow{dr}       &       \\
X \arrow{ur} \arrow{dr}      &       &Y'      \\
        &Q \arrow{ur}
\end{tikzcd}\]

The map $Q \oplus X' \rightarrow Q$ is the map $X' \rightarrow Q$ \textit{minus} $id_Q$. The 3-Lemma says this arrow is an isomorphism, and the two diagrams together build a 2-isomorphism between the two butterflies.

\end{proof}

\begin{remark}\label{splitpushoutbutterfly}

By a symmetric proof, butterflies $\xi \simeq \eta$

\[\begin{tikzcd}
\xi:        &0 \arrow{r}      &K \arrow{r} \arrow{dd}     &X\arrow{dr} \arrow{rr}     &      &Y \arrow{r}       &M  \arrow[equals]{dd}\arrow{r}    &0      \\
&        &       &       &Q \arrow{dr} \arrow{ur}      &       &        &       \\
\eta :      &0 \arrow{r}      &L \arrow{r}     &X' \arrow{rr} \arrow{ur}     &      &Y' \arrow{r}      &M  \arrow{r}    &0
\end{tikzcd}\]

\noindent which allow non-identity left vertical arrows are the same as butterflies $(K \rightarrow L) \smile \xi \simeq \eta$ with left vertical arrow $id_L$.

\end{remark}

We provide a reassurance that the group $\text{Ext}_A^2(M, K)$ of connected components of $\underline{\text{Ext}}_A^2(M, K)$ is the same, whether the maps are butterflies or maps of chain complexes.

\begin{lemma}

The connected components of $\underline{\text{Ext}}_A^2(M, K)$ under butterflies are the same as the connected components of 2-extensions under morphisms of complexes.

\end{lemma}

\begin{proof}

If we have a morphism of complexes between two 2-extensions, we get an induced butterfly as above. Conversely, suppose we have a butterfly between two 2-extensions. Choose a cover $\{N_i \rightarrow M\}$ which trivializes the butterfly, available due to Lemma \ref{isomisagerbe}. Let $T := \bigoplus N_i$. Then the butterfly pulls back under $T \rightarrow M$ to the split butterfly. Butterflies come from morphisms of chain complexes if and only if they're split, so we have morphisms of chain complexes connecting the two 2-extensions which shared a butterfly.

\[\begin{tikzcd}
0  \arrow{r}     &K \arrow{r}       &X \arrow{r}      &Y \arrow{r}      &M  \arrow{r}     &0      \\
0 \arrow{r}      &K \arrow{r} \arrow[equals]{d} \arrow[equals]{u}    &X \arrow{r} \arrow[equals]{u} \arrow{d}     &Y|_T  \arrow{u}\arrow{r} \arrow{d}     &T  \arrow{u}\arrow{r} \arrow[equals]{d}     &0      \\
0  \arrow{r}     &K \arrow{r} \arrow[equals]{d}     &X'\arrow{r} \arrow[equals]{d}     &Y'|_T  \arrow{r} \arrow{d}    &T \arrow{r} \arrow{d}     &0      \\
0 \arrow{r}      &K \arrow{r}     &X' \arrow{r}    &Y' \arrow{r}    &M \arrow{r}     &0
\end{tikzcd}\]

\end{proof}


\section{Semiclassics}

Here we include back-scattering processes in the derivation of Eq.~(16) in the main text for the local density of states at large frequency, $\omega\gg\omega_\star$; we find that it only modifies the expression~(17) for $\omega_\text{cr}$ (the factor $\sqrt{3}$ would be replaced with $\sqrt{2}$ if only forward-scattering processes were present). Namely, at $\omega\gg\omega_\star$, we use the analogy with the one-dimensional Schr\"odinger equation for a particle in a Gaussian disorder potential to evaluate the averaged local plasmon density of states that appears in Eq.~(12) in the main text. For this, we employ the Fokker-Planck method reviewed in \cite{Lifshitz1988}.

We may express the Green function that solves Eq.~(10) in the main text, together with boundary conditions, as
\begin{equation}
G(x,x';\omega)=\frac {\pi } {vW} \left[\Psi_-(x)\Psi_+(x')\Theta(x'-x)+\Psi_-(x')\Psi_+(x)\Theta(x-x')\right]\,.
\label{eq:GFclassical}
\end{equation}
Here $\Psi_+$ and $\Psi_-$ are two solutions of the differential equation
\begin{equation}
\label{eq:diff}
\omega^2 \Psi_\pm(x)=-v^2\partial_x^2 \Psi_\pm(x)+V(x)\Psi_\pm(x)
\end{equation}
with boundary conditions $\partial_x \Psi_-(0)=0$ and $\Psi_-(0)=1$, and $\partial_x \Psi_+(d)=1$ and $\Psi_+(d)=0$, respectively,
$W$ is a spatially-independent Wronskian,
\begin{equation}
W=\Psi_-(x)\partial_x\Psi_+(x)-\partial_x\Psi_-(x)\Psi_+(x)\,,
\end{equation}
and $\Theta(x)$ is the Heaviside function [with $\Theta(0)=1/2$].
Using the Wronskian and the boundary conditions, we find from Eq.~\eqref{eq:GFclassical}
\begin{equation}
\label{eq:G00}
G(0,0;\omega)=\frac{\pi }v\frac{\Psi_+(0)}{\partial_x\Psi_+(0)}\,.
\end{equation}
We further define four functions $\rho_\pm$ and $\phi_\pm$ such that
\begin{equation}
\label{eq:parametrize}
\Psi_\pm(x)=\rho_\pm(x)\sin\phi_\pm(x) \quad\text{and}\quad
\partial_x\Psi_\pm(x)=(\omega/v) \rho_\pm(x)\cos\phi_\pm(x)\,.
\end{equation}
Equation \eqref{eq:diff} and its boundary conditions then read equivalently
\begin{equation}
\label{eq:neq-eq}
\partial_x \rho_\pm=-\frac{V}{2v\omega}\sin2\phi_\pm \quad\mathrm{and}\quad
\partial_x \phi_\pm=\frac\omega v-\frac{V}{v\omega}\sin^2\phi_\pm
\end{equation}
with boundary conditions $\rho_+(d)=v/\omega$ and $\phi_+(d)=0$, and $\rho_-(0)=1$ and $\phi_-(0)=\pi/2$, respectively.
Inserting Eq.~\eqref{eq:parametrize} into Eq.~\eqref{eq:G00}, we relate
the statistics of
\begin{equation}
\label{eq:G00-param}
G(0,0;\omega)=\frac{\pi }\omega\tan\phi_+(0)
\end{equation}
with that of the solution $\phi_+$ of the relevant pair of Eqs.~\eqref{eq:neq-eq}.

In general, $G(0,0;\omega)$ is complex. Its real part readily follows from Eq.~\eqref{eq:G00-param}. In order to derive its imaginary part, one should include a small imaginary part to the frequency (in the upper complex plane), $\omega\to \omega+i0^+$, which in turn adds a small imaginary part with the same sign to $\phi_+$. Using Eq.~(13) in the main text, we can then relate the statistical properties of the local density of states to those of $\phi_+(0)$,
\begin{equation}
\label{eq:res-rvdG3}
\nu(x=0,\omega)
 =-\frac 2 v \sum_m\delta\left(\phi_+(0)-(m+\frac12)\pi\right)\,.
\end{equation}

To proceed further, we use the assumption of large frequency to treat the term $\propto V$ in Eq.~(\ref{eq:neq-eq}) as a perturbation,
and decompose $\phi_+(x)=\omega (x-d)/v+\zeta(x)$ , where $\zeta(x)$ is a slowly varying function on the scale of $v/\omega$.
Furthermore, the disorder potential contains three relevant terms for the spatial variation of $\zeta(x)$,
\begin{equation}
\label{eq:semiclassical-disorder}
V(x)=V_0(x)+V_1(x)e^{2i\omega (x-d)/v}+V^*_1(x)e^{-2i\omega(x-d)/v}\,,
\end{equation}
where the slowly varying (on the scale of $v/\omega$) components $V_0(x)$ and $V_1(x)$ describe forward- and back-scattering, and have Gaussian averages,
\begin{equation}
\label{eq:correl-reduced-2}
\langle V_0(x)V_0(x')\rangle=\langle V_1(x)V^*_1(x')\rangle= 8 \pi^2R_\star\omega_\star^4\delta(x-x')\,.
\end{equation}
Equation~\eqref{eq:correl-reduced-2} reproduces the correlator for $V= (4\pi K\lambda v/\hbar a)\cos(2\bar\theta+\chi)$, in which the spatial variations of $\bar\theta$ are safely ignored at $\omega\gg\omega_\star$ (see discussion in main text) and the correlator for $\cos\chi$ is given by Eq.~(4) in the main text. Then retaining only the slowly varying components in Eq.~(\ref{eq:neq-eq}), we find
\begin{equation}
\label{eq:neq-eq3}
\partial_x \zeta=-\frac 1{2v\omega}V_0+ \frac 1{4v\omega}\left[V_1^*e^{2i\zeta}+V_1e^{-2i\zeta}\right]
\end{equation}
with the boundary condition $\zeta(d)=0$. Equation~\eqref{eq:neq-eq3} is a Wiener process whose Fokker-Planck equation reads~\cite{Lifshitz1988}
\begin{equation}
\label{eq:const-diff}
\frac{\partial P}{\partial x}={\cal D}\frac{\partial ^2 P}{\partial\zeta^2}\qquad\mathrm{with}\qquad
{\cal D}=\frac {3\pi^2} 2 \frac{R_\star\omega_\star^4}{v^2\omega^2}=\frac{3\pi^2}{2\ell(\omega)}\,;
\end{equation}
its boundary condition is $P(x=d,\zeta)=\delta(\zeta)$. The corresponding solution of Eq.~(\ref{eq:const-diff}) is
\begin{equation}
\label{eq:gaussian-distrib}
P(x=0,\zeta)=\frac1{2\sqrt{\pi {\cal D} d}}e^{-\zeta^2/(4{\cal D}d)}\,.
\end{equation}
The derivation of Eq.~(\ref{eq:gaussian-distrib}) required $\omega\gg\omega_\star$. Once this condition is satisfied, the obtained distribution function is valid at any ratio $d/\ell(\omega)$. [In particular, forward and backward scattering processes contribute additively (in a 2/3-1/3 ratio) to the ``diffusion constant'' $\cal D$.] This is quite remarkable, as the waves' localization length and mean free path are of the same order in a one-dimensional system.

Averaging Eq.~\eqref{eq:res-rvdG3} over the distribution \eqref{eq:gaussian-distrib} then yields its disorder-averaged value at any $d$, see Eq.~(14) in the main text. Subsequently, one gets the disorder-averaged real part of the reflection amplitude through Eq.~(12) in the main text.

Using Kramers-Kronig relations, we get similarly the disorder-averaged real part of the local Green function,
\begin{equation}
\label{eq:r-im}
\frac{2\omega}{\pi^2 v}\text{Re}\langle G(0,0;\omega)\rangle = 2\nu_0
\sum_m\frac{ \sqrt{ 2}\Delta}{\pi \delta(\omega)} D\left(\frac{\omega-(m+1/2)\Delta}{\sqrt{2}\delta(\omega)}\right)\,,
\end{equation}
where $D(x)=e^{-x^2}\int_0^xdt e^{t^2}$ is the Dawson function, with asymptotes $D(x)\approx x$ at $|x|\ll 1$ and $D(x)\approx 1/(2x)$ at $|x|\gg 1$. In the frequency range $\omega_\star\ll\omega\ll \omega_\text{cr}$, the Poisson summation formula applied to Eq.~\eqref{eq:r-im} yields
\begin{equation}
\label{eq:r-im-2}
\frac{2\omega}{\pi^2 v} \text{Re}\langle G(0,0;\omega)\rangle \rangle = 4\nu_0\sin\left({2\pi \omega}/{\Delta}\right)e^{-2(\pi \omega_{\rm cr}/{\omega})^2}\,.
\end{equation}
Equations \eqref{eq:r-im} and \eqref{eq:r-im-2} determine the disorder-averaged imaginary part of the reflection amplitude,
\begin{equation}
\label{eq:local-dos}
\langle r''(\omega)\rangle =\frac{K}{K_0}\frac{2\omega}{\pi} \text{Re}\langle G(0,0;\omega)\rangle \,.
\end{equation}

\section{Soft modes}

Here we calculate the disordered-averaged local density of states at low frequency $\omega\ll\omega_\star$ in the strong-pinning regime. We confirm the scaling captured by Eq.~(17) in the main text, and we obtain the numerical factor in front of it.

Infinitely-strong pinning on separate impurities reduces the system to a sequence of independent segments. Averaging the local density of state in the segment nearby the interface at $x=0$, and whose length is given by the Poisson distribution, $P(L)=ce^{-cL}$, yields
\begin{equation}
\label{eq:DOS-Gorkov}
\langle \nu(x=0,\omega)\rangle
\approx 2c/\omega e^{-\pi v c/(2\omega)} \quad\text{at}\quad \omega\ll\pi v c\,.
\end{equation}

Finite-strength impurities allow for special configurations carrying soft excitations with arbitrarily-low frequency, as was noticed in~\cite{Aleiner1994}. Closely following their work, we consider three impurities at positions $x_1=L_1,x_2=x_1+L,x_3=x_2+L_2>0$ nearby $x=0$, and such that $L\ll 1/(\gamma c) \ll L_1,L_2$, where $\gamma=K\lambda/(\hbar v\sqrt{ac^3})$ is the large parameter in the strong pinning regime. The sum of elastic and potential energies associated with a static charge density such that $\partial_x\bar\theta(0)=0$ and $\bar\theta(x_j)=\theta_j$,
\begin{equation}
E=\frac{\hbar v}{2\pi K}
\left[\frac{(\theta_2-\theta_1)^2}{L}+\frac{(\theta_3-\theta_2)^2}{L_2}\right]-\frac\lambda{\sqrt{ac}}\sum_{j=1}^3\cos(2\theta_j+\chi_j)\,,
\end{equation}
is minimized with $\theta_3=-\chi_3/2$
and $\theta_1-\theta_2=-2\pi  \gamma c L \sin[(\chi_1-\chi_2)/2]\cos \Phi \ll 1$ with $\Phi=\theta_1+\theta_2+(\chi_1+\chi_2)/2$. Then, if $\chi_1-\chi_2=\pi+2\varepsilon$ with $0<\varepsilon\ll 1$, the equilibrium solution for $\Phi$ (with $|\Phi|\ll 1$) is obtained from the minimization of a quartic potential,
\begin{equation}
\label{eq:E-quartic}
U=-h\Phi+A\Phi^2+B\Phi^4\,,
\end{equation}
where $h=\hbar v[\theta_3+(\chi_1+\chi_2)/4]/(2\pi K L_2)$, $A=\hbar v/(8\pi K L_2)-\varepsilon \lambda/\sqrt{ac}+ 4\pi K\lambda^2 L/(\hbar vac)$, and $B=\varepsilon \lambda/(12\sqrt{ac})-4\pi K\lambda^2 L/(3\hbar vac)$.

Using a Born-Oppenheimer approximation, we find the oscillation frequency $\omega_0$ in the minimum of the potential~\eqref{eq:E-quartic} using the Lagrangian $L=T-U$ where $T=M\dot\Phi^2/2$ with an effective ``mass'' $M=\hbar (L_1+L_2/3)/(\pi  vK)$ for the coordinate $\Phi$. Namely,
\begin{equation}
\label{eq:freq2}
\omega_0^2=\frac{4}M (h^2B)^{1/3} F\left( A/{(h^2B)^{1/3}}\right),
\end{equation}
where $F(\eta)$ is defined implicitly by $F(\eta)=\eta+6\xi^2$ with $-1+2\eta\xi+4\xi^3=0$. Function $F(\eta)$ reaches a minimum of order $1$ at $\eta=3/2^{5/3}$.

Using Eq.~\eqref{eq:freq2} and the properties of $F(\eta)$, we find that the phase space $\Gamma$ that allows for small oscillation frequencies $\omega_0\ll \omega$ is bounded by critical values for $|h|$ and $|A|$ scaling like $\omega^3$ and $\omega^2$, respectively. Thus, $\Gamma\propto \omega^5$, yielding a density of state $\nu\sim d\Gamma/d\omega\propto \omega^4$. Furthermore, the conditions $B>0$ and $A\approx 0$, which are necessary to find soft modes, yield two constraints: $\varepsilon<1/(6\pi L_2c\gamma)$ and $L<\varepsilon/(16\pi c\gamma)$, where $L_2\sim 1/c$. They yield an additional smallness in the suppression of the local density of states,
\begin{equation}
\label{eq:dos-universal}
\langle \nu(x=0,\omega)\rangle \propto \nu_0 \gamma^{-3} (\omega/vc)^4
\,.
\end{equation}
 The power-law \eqref{eq:dos-universal} dominates at low frequencies, while the exponential suppression \eqref{eq:DOS-Gorkov} takes over in an intermediate range of frequencies, $vc/\ln\gamma\ll \omega\ll vc$.

The missing numerical factor in Eq.~(\ref{eq:dos-universal}) is found by an explicit calculation of the average $\langle\nu(x=0,\omega)\rangle$ at $\omega\ll\omega_\star$,

\begin{eqnarray}
\label{eq:dos-interm}
\langle \nu(x=0,\omega)\rangle=\int_0^\infty dL_1P(L_1) \int_0^\infty dL_2P(L_2) \int_0^{L_0} dL P(L)
\int_0^{2\pi} \frac{d\chi_1}{2\pi}\int_0^{2\pi} \frac{d\chi_2}{2\pi}\int_0^{2\pi} \frac{d\chi_3}{2\pi}\nonumber\\
\times \int_{-\infty}^\infty dh' \int_{-\infty}^\infty  dA'\delta(h-h')\delta(A-A')
\frac 1{L_1+L_2/3}\delta(\omega-\omega_0(A',h'))\,,
\end{eqnarray}
where the factor $1/(L_1+L_2/3)$ comes from the normalisation of the wavefunction associated with small oscillations. Here we used the Poisson distribution for the segments' lengths, and we assumed a uniform distribution for phases $\chi_1,\chi_2,\chi_3$ in the impurity potential. Furthermore, the upper limit $L_0=1/(96 \pi^2 L_2 c^2\gamma^2)$ in the integral over $L$ is set by the validity of expansion (\ref{eq:E-quartic}) leading to Eq.~(\ref{eq:freq2}). Now, setting $h'\approx A'\approx 0$ in the Dirac distributions (which is required to obtain low-frequency modes) allows performing the integrations over $h'$ and $A'$,
\begin{equation}
\int\!\!
dh' \int\!\!
dA'
\delta(\omega-\omega_0(A',h'))=2\omega \int \!\!d\eta \int \!\!dh (h^2B)^{1/3}\delta\left(\omega^2-\frac{4(h^2B)^{1/3}}{M}F(\eta)\right)
=\frac 3{16}\frac{M^{5/2}\omega^4}{B^{1/2}}\int\!\!
\frac{d\eta}{F(\eta)^{5/2}}
=\frac 1{8}\frac{M^{5/2}\omega^4}{B^{1/2}}\,.
\end{equation}
The remaining integrations in Eq.~\eqref{eq:dos-interm} can be performed to yield
\begin{eqnarray}
\langle \nu(x=0,\omega)\rangle&=&\frac 1\pi \nu_0\frac{1}{\gamma^2}\left(\frac\omega{vc}\right)^4c^5
\int_0^\infty dL_1\int_0^\infty dL_2 \int_0^{L_0} dL \frac{(L_1+L_2/3)^{3/2}L_2}{\sqrt{L_0-L}} e^{-c(L_1+L_2+L)}
\nonumber
\\
\nonumber\\
&=&\frac 1{2\sqrt{6}\pi^2}\nu_0\frac{1}{\gamma^3}\left(\frac\omega{vc}\right)^4 \int_0^\infty dx_1 dx_2 \sqrt{x_2(x_1+x_2/3)^3}e^{-x_1-x_2}
\nonumber\\
&\approx& 0.12\, \nu_0\frac{1}{\gamma^3}\left(\frac\omega{vc}\right)^4\,,
\end{eqnarray}
where we used that the ranges of integration giving the dominant contributions are $L\ll 1/c$ and $L_1,L_2\sim 1/c$. The scaling is the same as in~\cite{Aleiner1994} for the global density of states, but the numerical prefactor is different.

At $\gamma \sim 1$, a crossover to the weak pinning occurs. That leaves no room for the intermediate asymptote \eqref{eq:DOS-Gorkov} and broadens the region for the $\omega^4$-dependence to $\omega\lesssim\omega_\star$, leading to
\begin{equation}
\label{eq:Dos-w4}
\langle \nu(x=0,\omega)\rangle \propto \nu_0 (\omega/\omega_\star)^4
\,.
\end{equation}
The missing here, but appearing in Eq.~(17) of the main text, proportionality coefficient is found within a numerical simulation outlined in the next Section.

\section{Numerics}
\label{sec:numerics}

Here we provide details on the numerical computation of the local density of state at arbitrary frequency, which allowed us plotting Fig.~2 in the main text.

For this, we start by formulating the problem in dimensionless variables. Namely, by rescaling the spatial dimension $x=\xi y$ with length $\xi=R_\star/(2\pi^2)^{1/3}$,
we find that the static field $\bar\theta(y)$ setting the charge density in a given charge disorder configuration in the disconnected Josephson junction chain should admit boundary conditions $\partial_y\bar\theta(0)=0$ and $\bar\theta(d/\xi)=0$, and minimize the energy
\begin{equation}
{\cal E}[\bar\theta]=\frac{\hbar v}{2\pi K\xi}\int_0^{d/\xi} dy\left[(\partial_y\bar\theta)^2-{\cal V}'\cos2\bar\theta+{\cal V}''\sin2\bar\theta\right]\,.
\end{equation}
Here ${\cal V}'(y)$ and ${\cal V}''(y)$ are two independent real random fields characterizing the disorder configuration, such that
\begin{equation}
\label{eq:correl-dimless}
\langle {\cal V}'(y){\cal V}'(y')\rangle=\langle {\cal V}''(y){\cal V}''(y')\rangle=\delta(y-y'')\,.
\end{equation}
The spectrum of small oscillations in the potential set by the static field $\bar\theta$ is also found as a dimensionless eigenproblem,
\begin{equation}
\label{eq:eigen-pb}
\Omega_n^2\psi_n=-\partial_y^2\psi_n+2({\cal V}'\cos2\bar\theta-{\cal V}''\sin2\bar\theta)\psi_n
\end{equation}
with eigenfrequency $\Omega_n=\omega_n \xi/v$ and (real) eigenfunction $\psi_n$ normalized such that ${\int_0^{d/\xi} dy\psi^2_n(y)}=1$.
The local density of states then reads as
\begin{equation}
\nu(x=0,\omega)=\pi\nu_0\sum_n {\psi^2_n(0)}\delta(\Omega-\Omega_n)\,.
\end{equation}

Next we discretize the above equations by introducing a small spacing $\epsilon$, such that the (dimensionless) length corresponds to $M=(d/\xi)/\epsilon$ sites. The random fields with correlators \eqref{eq:correl-dimless} are replaced with discrete fields ${\cal V}'_{m}$ and ${\cal V}''_{m}$ at sites $y_m=m\epsilon$ ($0\leq m<M$), which are uncorrelated from site to site, and which are drawn with flat probability in the interval $-\sqrt{3/\epsilon}<{\cal V}'_{m},{\cal V}''_{m}<\sqrt{3/\epsilon}$.

The random static field $\bar \theta$ is obtained by minimization of the energy functional
\begin{equation}
{\cal E}[\{\bar\theta_m\}]=\frac{\hbar v\epsilon}{2\pi K\xi}
\left\{
\sum_{m=1}^{M-1}\left[
\frac{(\bar\theta_{m-1}-\bar\theta_{m})^2}{\epsilon^2}
-({\cal V}'_{m}\cos2\bar\theta_m-{\cal V}''_{m}\sin2\bar\theta_m)\right]
+\frac{\bar\theta^2_{M-1}}{\epsilon^2}-\frac 12
({\cal V}'_{0}\cos2\bar\theta_0-{\cal V}''_{0}\sin2\bar\theta_0)
\right\}\,.
\end{equation}
The two last terms are such that the boundary conditions for $\bar \theta$ are satisfied in the continuum limit, $\epsilon\to 0$. (In particular, in order to reproduce the boundary condition at $y=0$ and get this result, one may think of $\bar\theta$ as the even solution of a twice longer chain with symmetric disorder on either side of the central node $m=0$.) We then use the algorithm described in Ref.~\cite{Gurarie2003} to obtain the variables $\{\bar\theta_m\}$ [taking discrete values $2\pi p/P$ ($0\leq p\leq P-1$ with integer $P\gg 1$)] that realize the absolute minimum of the energy $\cal E$.

Once the static configuration field has been determined, we can obtain the local density of states,
\begin{equation}
{\nu(x=0,\omega)}=-2{\nu_0}\Omega\text{Im}G_{0,0}(\Omega)\,,
\end{equation}
in terms of the retarded Green function $G_{m,m'}(\Omega)$ associated with Eq.~\eqref{eq:eigen-pb}. The boundary condition at site $m=0$ is obtained by considering a Green function $\tilde G_{m,m'}(\Omega)$ in a twice longer space,
\begin{equation}
G_{m,m'}=\tilde G_{m,m'}+\tilde G_{-m,m'}\,,
\end{equation}
such that $G_{0,0}=2\tilde G_{0,0}$. The later Green function is obtained as a continued fraction using the recursive equation
\begin{equation}
(\Omega^2-{\cal W}_m+\frac 2{\epsilon^2})\tilde G_{m,m'}(\Omega)
-\frac 1{\epsilon^2}\left[\tilde G_{m+1,m'}(\Omega)+\tilde G_{m-1,m'}(\Omega)\right]=\delta_{m,m'}
\end{equation}
with ${\cal W}_m=2({\cal V}'_m\cos2\bar\theta_m-{\cal V}''_m\sin2\bar\theta_m)$.

In the plots shown here and in Fig.~2 in the main text, we choose a number of sites $M=200,400,\,\text{and}\,800$, and a mesh $\epsilon=0.05$, corresponding to $d/R_\star=M\epsilon /(2 \pi^{2})^{1/3}\approx 3.7,7.4,\,\text{and}\,14.8$, $\omega_\star/\Delta=n\epsilon /(2 \pi^{5})^{1/3}\approx 1.2,2.4,\,\text{and}\,4.7$, and $\omega_{\text{cr}} =\sqrt{3/2}(n\epsilon)^{3/2} /\pi^2\approx 3.9,11.1,\,\text{and}\,31.4$, respectively. We also introduce an imaginary broadening $\Omega\to\Omega+i\tilde \gamma$ with $\tilde \gamma=0.02 \Delta$.

\begin{figure}
(a)\includegraphics[width=0.35\columnwidth]{Fig-20-angle.pdf}
(b)\includegraphics[width=0.34\columnwidth]{Fig-20-1config.pdf}
\caption{\label{Fig:single-dis} (a) Spatial profile of the static configuration $\bar\theta(x)$ for a given disorder configuration in a chain with length $d=7.4 R_\star$, corresponding to $\omega_\star\approx 2.4 \Delta$ and $\omega_{\text{cr}} \approx 11.1 \Delta$. (b) Local density of states (in units of $\nu_0$) as a function of the frequency (in units of $\Delta$) for that disorder configuration.}
\end{figure}

In Fig.~\ref{Fig:single-dis}, we plot the spatial profile of the static configuration along the chain for a single disorder configuration (left panel). We also show the local density of states (normalized by the bulk value $\nu_0$) as a function of the frequency (in units of $\Delta$) for that disorder configuration. Typically, the distant levels do contribute to the local density of states at low energies, but the contributions are exponentially small, $\exp(-d/R_\star)$. Thus one sees a few ``representative'' levels contributing to the local density of states at $\omega\lesssim\omega_\star$. By contrast, a (roughly) regular spectrum of equidistant states is visible at large frequencies  $\omega\gtrsim\omega_\text{cr}$.

\begin{figure}
(a)\includegraphics[width=0.35\columnwidth]{Fig-10.pdf}
(b)\includegraphics[width=0.35\columnwidth]{Fig-20.pdf}
\\
(c)\includegraphics[width=0.35\columnwidth]{Fig-40.pdf}
(d)\includegraphics[width=0.35\columnwidth]{Fig-All.pdf}
\caption{\label{Fig:av-dis} (a-c) Local density of states (in units of $\nu_0$) as a function of the frequency (in units of $\Delta$) averaged over 5000 disorder configurations, and comparison with the asymptotic predictions at frequencies below and above $\omega_\star$, for chains with length $d=3.7,7.4,\,\text{and}\,14.8 R_\star$, respectively (Fig.~\ref{Fig:av-dis}(b) is same as Fig.~2 in the main text). (d) Disorder-averaged local density of states as a function of the frequency (in units of $\omega_\star$) for the three chain's lengths.}
\end{figure}

In Figs.~\ref{Fig:av-dis} and 2 in the main text, we plot the local density of states averaged over a large number of disorder configurations for three different lengths. We compare the result with the asymptotic formula
\begin{equation}
\langle \nu(0,\omega)\rangle /\nu_0= C(\omega/\omega_\star)^4 \quad \text{with}\quad C\sim 0.032\,,
\end{equation}
which was speculated to hold at low frequency $\omega\lesssim\omega_\star$, and with the prediction
\begin{equation}
\langle \nu(0,\omega)\rangle/\nu_0 =
\sqrt{\frac 2 \pi}\frac{\omega_{\text{cr}}}{\omega}\sum_{n=0}^\infty \exp\left[-\frac{\omega^2}{2\omega_{\text{cr}}^2}\left(\frac \omega \Delta-n-\frac 12\right)^2\right]\,,
\end{equation}
which holds at frequencies $\omega\gg\omega_\star$, yielding well resolved peaks at $\omega\gg\omega_\text{cr}$.
\begin{figure}
\includegraphics[width=0.35\columnwidth]{Fig-All-local-global.pdf}
\caption{\label{Fig:all} Local and global disorder-averaged density of states as a function of the frequency, for the same parameters as in Fig.~\ref{Fig:av-dis}.}
\end{figure}
In particular, Fig.~\ref{Fig:av-dis}(d) shows the scaling of the disorder-averaged local density of states as a function of $\omega/\omega_\star$ for three different lengths of the chain.

In addition, Fig.~\ref{Fig:all} shows both the local and global densities of states; it illustrates the relation $\langle \nu(x=0,\omega)\rangle\approx 2 \langle \nu(\omega)\rangle$, which approximately holds at low frequencies, $\omega\lesssim\omega_\star$.

\section{Inelastic scattering}

Here we use perturbation theory in the phase slip term in Hamiltonian~(3) of the main text to provide an alternative derivation of the frequency shift, yielding Eq.~(16) in the main text in long chains, $d\gg R_\star$, at large frequencies, $\omega\gg\omega_\star$. Then we use the same approach to argue that inelastic scattering processes give negligible corrections to the result.\\

\subsection{Frequency shift}
Hamiltonian (2) in the main text, with boundary conditions $\partial_x\theta(x=0)=0$ and $\theta(x=d)=0$ in a finite-length chain, is diagonalized as
\begin{equation}
H_0= \sum_{n=0}^\infty (n+1/2)\Delta a^\dagger_na_n\,,
\end{equation}
where $a_n$ is an annihilation operator of a plasmon, such that
\begin{equation}
\label{eq:theta}
\theta(x)= \sum_{n=0}^\infty \sqrt{\frac{K}{n+1/2}}(a_n+ a_n^\dagger ) \cos (q_n x)
\end{equation}
with $q_n=\pi (n+1/2)/d$.

The dominant effect of a small but finite $\lambda$ is to shift the resonance frequencies, $\omega_n=(n+1/2)\Delta +\delta_n$. In first order in $\lambda$, the frequency shift of mode $n$ is the energy difference between no plasmon in the chain and the singly-occupied $n$-plasmon mode, $\delta_n=1/\hbar\left(\langle 1_n|H_1|1_n\rangle-\langle 0|H_1|0\rangle\right)$, where $|0\rangle$ is the vacuum state, $|1_n\rangle=a^\dagger_n|0\rangle$, and
\begin{equation}
\label{eq:phase-slip-term}
H_1=-\frac\lambda a\int dx\cos(2\theta+\chi)
\end{equation}
is the phase slip term. Inserting Eq.~\eqref{eq:theta} in it, we get
\begin{eqnarray}
\delta_n
&=&-\frac\lambda {\hbar a} \mathrm{Re}\int dx\langle 0|\prod_m
\left( a_ne^{2i\sqrt{\frac{ K}{m+1/2}}( a_m+ a^\dagger_m)\cos (q_m x)} a_n^\dagger-
e^{2i\sqrt{\frac{ K}{m+1/2}}( a_m+ a^\dagger_m)\cos (q_m x)}\right)|0\rangle e^{i\chi(x)}
 \nonumber\\
&\approx&\frac{4K \lambda }{\hbar a(n+1/2)}\int _0^d dx\sin^2(q_n x)\exp\left(-{2 K}\sum_{m}\frac{\sin^2(q_m x)}{m+1/2}\right)\cos\chi(x)\,,
\end{eqnarray}
where we used $n\gg 1$ to expand the factor with $m=n$ up to lowest order in $K$, and to add back the (small) term with $m=n$ in the exponential. In long arrays, $d\gg R_\star$ (which can only happen if $K<3/2$, according to Eq.~(7) in the main text), the exponential factor can be evaluated by taking the continuum limit, $\sum_m\approx (d/\pi ) \int dq$, replacing $\sin^2(q_mx)\approx 1/2$, and using  $1/\ell_{\mathrm{sc}}$ and $1/R_\star (\gg 1/d)$ as the high and low momenta cutoffs of the log-divergent sum (this takes into account the fact that plasmons with frequency $\omega<\omega_\star$ are localized, and do not contribute to the sum). It yields
\begin{equation}
\label{eq:dn}
\delta_n
=\frac{4 K\lambda (R_\star/\ell_{\rm sc})^{-K} }{\hbar a(n+1/2)}\int _0^d dx\sin^2(q_n x)\cos\chi(x)\,.
\end{equation}
On average, $\langle\delta_n\rangle=0$. Using the correlator (4) in the main text, we find that its variance can be presented as
\begin{equation}
\label{eq:var}
\frac{\langle\delta^2_{n}\rangle}{\Delta^2}
=\frac{3}{\pi^2 (n+1/2)^2}\left(\frac{d}{R_\star}\right)^3\,,
\end{equation}
in agreement with Eq.~(15) in the main text upon identification $\delta\left(\omega=(n+1/2)\Delta\right)\equiv{\langle\delta^2_{n}\rangle}^{1/2}$.

\subsection{Internal dissipation}
Considered in the harmonic approximation, the plasmon excitation spectrum of the Josephson-junction chain is broadened by radiative decay. At strong impedance mismatch with the waveguide, $K/K_0\ll 1$, the corresponding level width $\Gamma=(2v/d)K/K_0$  is small [for discussion of $\Gamma$, see the main text above Eq.~(10)]. Interaction between the plasmons may provide additional channels of ``internal dissipation''. In the absence of phase slips, charge disorder does not affect the dynamics of phases $\varphi_n$ in Eq.~(1) of the main text. The anharmonicity stemming from the expansion of $\cos(\varphi_n-\varphi_{n-1})$ beyond the second order in $\varphi_n-\varphi_{n-1}$, results in a momentum-conserving interaction between the plasmons. Because the plasmon spectrum $\omega(q)$ is convex, decay of a plasmon occurs only in collisions with other plasmons present in the system (a spontaneous decay is not allowed by the energy and momentum conservation laws), see, {\sl e.g.}, Ref.~\cite{Lin2013}. On the contrary, interaction stemming from the phase slip term \eqref{eq:phase-slip-term} allows a plasmon to decay into a number of plasmons of lower frequency and may provide an additional channel of ``internal dissipation''. Leaving aside a complex question regarding the possibility of many-body localization~\cite{Aleiner2006} of excitations of an isolated system at $\Gamma\to 0$, here we provide a crude estimate of the contribution of the internal dissipation to the plasmon line width.

The decay rate of a state $|i\rangle$ of Hamiltonian $H_0$ due to a perturbation $H_1$ is given by the Fermi's Golden Rule,
\begin{equation}
\Gamma_i=2\pi\sum_f\left|\langle f|H_1|i\rangle\right|^2\delta(\varepsilon_i-\varepsilon_f)
={\rm Re}\int_{-\infty}^{\infty} dt e^{i0t} \langle i|e^{iH_0t}H_1e^{-iH_0t}H_1|i\rangle\,.
\end{equation}
Here $\varepsilon_i$ and $\varepsilon_f$ are, respectively, the energies of the initial state $|i\rangle$ and final states $|f\rangle$. We take a plasmon state $|i\rangle=a^\dagger _n|0\rangle$ with the (unperturbed) value of energy $\varepsilon_i= (n+1/2)\Delta$ as an approximate eigenstate of $H_0$. Inserting Eq.~\eqref{eq:theta} into the phase slip term \eqref{eq:phase-slip-term}, we then identify contributions leading to its decay into $p$ plasmons,
\begin{equation}
H_1=\sum_{p \geq 2}\sum_{n_1\dots n_p}V_{n,n_1\dots n_p}  a_{n_1}^\dagger\dots a_{n_p}^\dagger a_n
+\text{H.c.}+\text{other terms}
\end{equation}
with
\begin{equation}
V_{n,n_1\dots n_p}
=\frac{(2i)^{p+1}}{p!}\frac{\lambda(R_\star/\ell_{\rm sc})^{-K}}{a}\frac{K^{(p+1)/2}}{\sqrt{nn_1\dots n_pn}}\int_0^d dx\frac {e^{i\chi(x)}+(-1)^pe^{-i\chi(x)}}{2}\cos(q_{n_1} x)\dots \cos(q_{n_p} x) \cos(q_{n} x)\,.
\end{equation}
Here the factor $(R_\star/\ell_{\rm sc})^{-K}$ has the same origin as the one that appears in Eq.~\eqref{eq:dn}; furthermore we used $n_i+1/2\approx n_i$ at $n_i\gg 1$.
Then, we evaluate the inelastic decay rate induced by $H_1$ using Wick's theorem,
\begin{eqnarray}
\Gamma_n&\approx&
\frac{\lambda^2}{a^2}
\left(\frac{R_\star}{\ell_{\rm sc}}\right)^{-2K}
\sum_p \frac{(4K)^{p+1}}{p!}
\sum_{n_1\dots n_p}
\frac 1{n_1\dots n_p n}
\left(\frac{da}{2^{p+2}}\right)
\int dt \langle 0|a_na^\dagger_n(t)a_{n_1}(t)\dots a_{n_{p}}(t)c_{n_{p}}^\dagger \dots a_{n_1}^\dagger a_{n}a_{n}^\dagger |0\rangle
\nonumber\\
&=&\frac{\lambda^2}{a^2}
\left(\frac{R_\star}{\ell_{\rm sc}}\right)^{-2K}
\sum_p \frac{(4K)^{p+1}}{p!}
\sum_{n_1\dots n_p}
\frac 1{n_1\dots n_p n}
\left(\frac{da}{2^{p+2}}\right)
\int dt iG_n^>(-t)iG_{n_1}^>(t)\dots iG_{n_p}^>(t)
\label{Gamman1}
\end{eqnarray}
with $iG_l^>(t)= \langle a_l(t)a^\dagger_l(0)\rangle$.
For estimates, we replaced here a factor which depends on the disorder configuration by a respective average:
\begin{equation}
\left(\frac{da}{2^{p+2}}\right)
=
\int_0^d dx \int_0^ddy \frac {e^{i\chi(x)}+(-1)^pe^{-i\chi(x)}}{2}\frac {e^{i\chi(y)}+(-1)^pe^{-i\chi(y)}}{2} \prod_{m=n_1,\dots,n_p,n} \cos q_m x \cos  q_m y \,.
\end{equation}
Within the Rayleigh-Schr\"odinger perturbation theory, $G^>_{l}(t)\approx -i e^{-i\omega_l t}e^{-\Gamma_l |t|/2}$ with $\omega_l$ evaluated in the absence of phase slips, $\omega_l \to \varepsilon_l=(l+1/2)\Delta$, and the decay rate coming solely from radiation of the mode $i$  into the waveguide, $\Gamma_l\to \Gamma$. Assuming that the internal dissipation does occur, we modify $G^>_{l}(t)$ by allowing for $\Gamma_l\neq\Gamma$  and for $\omega_l$ corrected by the phase slips in the presence of a specific configuration of disorder. This way, Eq.~(\ref{Gamman1}) becomes a self-consistent equation for finding $\Gamma_n$:
\begin{equation}
\Gamma_n=
\frac {2 d K^2\lambda^2}a
\left(\frac{R_\star}{\ell_{\rm sc}}\right)^{-2K}
\sum_{p\geq 2} \frac{(2K)^{p-1}}{p!}
\sum_{n_1\dots n_p}
\frac 1{n_1\dots n_p n}
\frac{\Gamma_{n_1}+\dots+\Gamma_{n_p}+\Gamma_n}{(\omega_{n_1}+\dots+\omega_{n_p}-\omega_n)^2+\frac 14 (\Gamma_{n_1}+\dots+\Gamma_{n_p}+\Gamma_n)^2}
\,.
\label{Gamman2}
\end{equation}
The disorder-induced frequency shifts $\omega_l-\varepsilon_l$ are important, as they destroy multiple resonances which would occur for the equidistant unperturbed spectrum $\varepsilon_l$ appearing in the denominators of the summand of Eq.~(\ref{Gamman2}). It is clear from Eq.~(15) of the main text that the typical shifts $|\omega_l-\varepsilon_l|\sim\omega_{\rm cr}/l$ depend on $l$, obstructing the resonances unless the sum of $\Gamma_l$ in the denominators of  Eq.~(\ref{Gamman2}) exceeds ${\rm min}\{\Delta,\omega_{\rm cr}/l\}$.

We are interested in the estimate of $\Gamma_n$ for modes with relatively high frequencies, $\varepsilon_n\gtrsim\omega_{\rm cr}$, at which $\omega_{\rm cr}/n\lesssim\Delta$. Our analysis of Eq.~(\ref{Gamman2}) at $K\ll 1$ points to $\Gamma_n\ll\omega_{\rm cr}/n$ in that frequency range, while $\Gamma_l\sim\Delta$ at $\omega_l\sim\sqrt{ K}\omega_{\rm cr}$ (here we assume that albeit being small, $K$ is large enough to allow for $\sqrt{K}\omega_{\rm cr}\gtrsim\omega_\star$, {\it i.e.}, $R_\star/d\lesssim K\ll 1$). At small $K$, the main contribution to the sum over $p$ comes from the first term ($p=2$) corresponding to a decay of the plasmon mode $n$ into another high-frequency mode $n_1\approx n$ accompanied by emission of one broadened plasmon ($\Gamma_l\gtrsim\Delta$) with frequency below $\sqrt{K}\omega_{\rm cr}$. That allows us to replace summation over $\{n\}$ in Eq.~(\ref{Gamman2}) by integration. Dispensing with a factor $\sim\ln(\sqrt{K}\omega_{\rm cr}/\omega_\star)$ and with a numerical factor which do not alter our conclusions, we obtain
\begin{equation}
\label{eq:inel}
\Gamma(\omega)\sim
K\frac{\omega^3_\star}{\omega^2}
\sim \Delta \frac{K\omega^2_\text{cr}}{\omega^2}\quad\text{at}\quad \omega \gtrsim \sqrt{K}\omega_\text{cr}\,.
\end{equation}
Note that the frequency dependence here agrees with the one found in \cite{Rosenow2007}, as well as \cite{Bard2018} [cf.~the second line in their Eq.~(43)], at $K\ll 1$. Using Eq.~(\ref{eq:inel}) and Eq.~(15) of the main text, the ratio $\Gamma(\omega)/\delta(\omega)$ can be presented in the form
\begin{equation}
\frac{\Gamma(\omega)}{\delta(\omega)}\sim K\,\frac{\omega_{\rm cr}}{\omega}\,.
\label{Gamman3}
\end{equation}
This confirms that:\\ (i) at $\omega\gtrsim\omega_{\rm cr}$ the main contribution to the widths of the disorder-averaged plasmon resonances comes from the inhomogeneous broadening $\delta(\omega)$ considered in the main text, rather than from the internal dissipation;\\(ii) due to the internal dissipation, $\Gamma(\omega)$ may reach a value $\sim\Delta$ at low frequencies, $\omega\sim \sqrt{K}\omega_{\rm cr}$.