# Stability of Stochastically Switched and Stochastically Time-Delayed Systems

Camille Carter  Jacob Murri  David Reber  Benjamin Webb  Department of
Mathematics, Brigham Young University, Provo, UT 84602, USA  Department of
Mathematics, Brigham Young University, Provo, UT 84602, USA  Independent
Researcher<EMAIL_ADDRESS>Department of Mathematics,
Brigham Young University, Provo, UT 84602, USA<EMAIL_ADDRESS>
###### Abstract

In this paper we introduce the notion of a patient first-mean stable system.
Such systems are switched systems that are first-mean stable meaning that they
converge to a globally attracting fixed point on average. They are also
patient so that they do not lose their first-mean stability when time-delays
are introduced into the system. As time-delays are, in general, a source of
instability and poor performance patient first-mean stability is a much
stronger condition than first-mean stability. This notion of patient stability
allows one to design systems that cannot be destabilized via time-delays. It
also significantly reduces the difficulty of modeling such systems since in
patient systems time-delays can, to a large extent, be safely ignored. The
paper’s main focus is on giving a sufficient criteria under which a system is
patient first-mean stable and we give a number of examples that demonstrate
the simplicity of this criteria.

###### keywords:

switched systems, time-delays, stability, dynamical networks

##  1 Introduction

Systems whose governing rules change over time are widely used to model and
control real-world systems. For example, daily/annual cycles can be
incorporated to refine population models [13]. However, switching can be
destabilizing, and traditional stability analysis can become intractable in a
switching regime (see [17] for a detailed overview of the stability theory of
switched systems). In addition, real-world systems are often stochastically
time-delayed due to traffic [14], finite transmission speeds and spatial
separation [4]. These delays can have a destabilizing effect on the system [5,
6]. In addition, many social, biological, information, and technological
systems have network-like structure [3], where the state space is in fact a
product space. In this paper we propose new tools to analyze discrete-time,
delayed, switched systems on a product space.

When considering the stability of a stochastically switched system, one
question which arises is whether to focus on bounding all possible
trajectories, or only those most frequently encountered. If we want to
consider all trajectories then we can, for example, consider the systems the
joint spectral radius [25], which can be used to determine the stability of
linear switched systems [26], by bounding the norms of all trajectories
(although it is in general very difficult to compute or approximate). On the
other hand, $p^{th}$ mean stability [18] is less demanding, requiring only
that, for linear systems, the $p^{th}$ power of the norm of the trajectory
converge to 0 on average.

In [2] the $p$-radius is shown to characterize $p^{th}$ mean-stability in the
case that switching is stochastically independent at each time step. The
benefit is that using this criteria stability is relatively easy to compute.
In particular, the stability of positive switched systems [19, 23, 24] is
easily analyzed using the $p$-radius.

For delayed systems, the most common approaches for characterizing stability
are Lyapunov methods, Linear Matrix Inequalities, and Semi-Definite
Programming (see, for example, [15, 16]. In [10, 11] the notion of intrinsic
stability was developed as an alternative to these approaches for studying
delayed systems. The approach used there consists of bounding the trajectories
of a possibly nonlinear system by a positive linear system, which we refer to
here as the associated _Lipschitz_ system. If this corresponding Lipschitz
system is stable, then [10] demonstrated that any delayed version of the
nonlinear system will also be stable (including the original undelayed
system), so long as the delays involved are constant in time. Thus, one can
show intrinsic stability is relatively simple to verify using, for instance,
the power method [12].

The approach of considering the intrinsic stability of a system has recently
been extended to switched systems. [1] demonstrated that intrinsically stable
unswitched systems are invariant to switching delays. [1] also considered
switched systems that are delayed, but since the primary proof technique
relied on the joint spectral radius, additional constraints were required
which increase the computational complexity of this approach.

In this paper we demonstrate that these issues can be avoided when we use the
$1$-radius, which is the $p$-radius for $p=1$. To this end we define a general
notion of _patient first-mean stability_ as the case when a stochastic
switched (and possibly nonlinear) system retains first-mean stability even
when time-delayed. We show that the computational efficiency of the $p$-radius
can be extended to nonlinear switched systems (Theorem 2), and that in the
case of the $1$-radius this condition is strong enough to imply the patient
first-mean stability of the nonlinear switched system (Theorem 3). Thus, the
difficulty of modeling time delays in stochastic switched systems can be
significantly reduced if the systems is known to be patient first mean-stable.
That is, if time-delays exist in the systems they need not be modeled in that
delays will not qualitatively effect the dynamics of the system and this is
true whether the system is linear or nonlinear, positive or mixed-sign.

This paper is organized as follows. Section 2 formally introduces switched
systems. Section 3 formalizes time-delayed switched systems, and illustrates
some of the dynamics which consequently arise. Section 4 introduces the tools
used to check patient first-mean stability, and presents our main results
(Theorems 2 and Theorem 3).

##  2 Switched Systems

######  Definition 1.

_(Dynamical System on a Product Space)_ Let $(X_{i},\|\cdot\|_{X_{i}})$ be a
Banach space for $1\leq i\leq n$, and let $d_{i}$ be the metric induced by
$\|\cdot\|_{X_{i}}$. Let $(X,\|\cdot\|)$ be the Banach space formed by giving
the product space $X=\prod_{i=1}^{n}X_{i}$ the norm

$\displaystyle\|\mathbf{x}\|=\max_{1\leq i\leq n}\|x_{i}\|\text{ where
}\mathbf{x}=[x_{1},x_{2},\dots,x_{n}]^{T}\in X\text{ and }x_{i}\in X_{i},$

and let $d$ be the metric induced by $\|\cdot\|$ on $X$. Let $F:X\to X$ be a
Lipschitz continuous map with $i^{th}$ component $F_{i}:X\rightarrow X_{i}$.
We refer to the dynamical system $(F,X)$ generated by iterating the function
$F$ on $X$ as a dynamical system on a product space. For the initial condition
$\mathbf{x}^{0}\in X$ we define the $k^{th}$ iterate of $\mathbf{x}^{0}$ to be
$\mathbf{x}^{k}=F^{k}(\mathbf{x}^{0})$, with orbit
$\\{F^{k}(\mathbf{x}^{0})\\}^{\infty}_{k=0}=\\{\mathbf{x}^{0},\mathbf{x}^{1},\ldots\\}$
in which $\mathbf{x}^{k}$ is the state of the system at time $k\geq 0$.

The dynamical systems we consider in this paper are all of the form given in
Definition 1. Specifically, each is a system consisting of a Lipschitz
continuous map $F$ on a product space $X$. In this section we also define the
notion of a switched system in which, at each time step, we switch between
systems of this type. In the following section we investigate how time delays
affect a system’s dynamics as a special case of these switched systems.

######  Definition 2.

_(Stochastic Switched System)_ Let $M$ be a set of Lipschitz continuous
mappings on the product space $X$ such that for every $F\in M$, $(F,X)$ is a
dynamical system. For a probability space $(M,\mathscr{A},\mu)$ we call
$(M,X,\mu)$ a _stochastic switched system_ on $X$. Given some sequence
$\\{F^{(k)}\\}^{\infty}_{k=1}$, where each $F^{(k)}\in M$ is drawn
independently with respect to $\mu$, we say that
$(\\{F^{(k)}\\}^{\infty}_{k=1},X)$ is an instance of $(M,X,\mu)$ with orbit
determined at time $k$ by
$\mathbf{x}^{k}=\mathscr{F}^{k}(\mathbf{x}^{0})=F^{(k)}\circ\ldots
F^{(2)}\circ F^{(1)}(\mathbf{x}^{0})$ for the initial condition
$\mathbf{x}^{0}\in X$.

######  Example 2.1.

Consider the dynamical systems $(F,\mathbb{R}^{2})$ and $(G,\mathbb{R}^{2})$
given by

$F\left(\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}\right)=\begin{bmatrix}\epsilon&1\\\
0&\epsilon\end{bmatrix}\begin{bmatrix}\tanh(x_{1})\\\
\tanh(x_{2})\end{bmatrix}\hskip 14.22636pt\text{and}\hskip
14.22636ptG\left(\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}\right)=\begin{bmatrix}\epsilon&0\\\
1&\epsilon\end{bmatrix}\begin{bmatrix}\tanh(x_{1})\\\
\tanh(x_{2})\end{bmatrix}$

where $\epsilon\in\mathbb{R}$. Note that both $F$ and $G$ are Lipschitz
continuous functions on $\mathbb{R}^{2}$. For $M=\\{F,G\\}$ and $\epsilon=1/2$
consider the three different switching distributions defined by
$\mu_{1}(F)=1$, $\mu_{1}(G)=0$; $\mu_{2}(F)=9/10$, $mu_{2}(G)=1/10$; and
$\mu_{3}(F)=1/2$, $\mu_{3}(G)=1/2$. This gives us three different switched
systems in which we always select $F$, mostly select $F$, and evenly choose
between $F$ and $G$, respectively.

In the stochastic switched system $(M,\mathbb{R}^{2},\mu_{1})$ the function
$F$ is selected at every time step so there is, in fact, no switching.
Therefore there is only one instance of this system given by
$(\\{F,F,F,\dots\\},\mathbb{R}^{2})$. It is possible to show that for any
initial condition, the state of this system is asymptotic to the origin (see
Figure 1, (left)). Likewise, the switched system where the function $G$ is
selected at every time step shares this property.

In the switched system $(M,\mathbb{R}^{2},\mu_{2})$ the function $F$ is chosen
with probability $9/10$ and $G$ is chosen with probability $1/10$ at each time
step. This system also has the property that its state is asymptotic to the
origin irrespective of its initial condition (see Example 4.6), although the
convergence to the origin is typically slower than in
$(M,\mathbb{R}^{2},\mu_{1})$ (see Figure 1, center).

In $(M,\mathbb{R}^{2},\mu_{3})$ the situation is quite different. Here at each
time step we randomly select either the function $F$ or the function $G$ with
equal probability. The result is that the system’s state no longer converges
to any point (see Figure 1 (right)). That is, if we switch frequently enough
between $(F,\mathbb{R}^{2})$ and $(G,\mathbb{R}^{2})$ the state of the system
no longer tends to a single point and the system’s dynamics are no longer
stable.

\begin{overpic}[scale={0.33}]{IntrinsicStabExp11.eps}
\put(10.0,-2.0){$(M,\mathbb{R}^{2},\mu_{1})$}
\put(45.0,-2.0){$(M,\mathbb{R}^{2},\mu_{2})$}
\put(80.0,-2.0){$(M,\mathbb{R}^{2},\mu_{3})$} \end{overpic} Figure 1: The
dynamics of the switched system $(M,\mathbb{R}^{2},\mu_{i})$ from Example 2.1
is shown left, center, and right for $i=1$, $i=2$, and $i=3$, respectively,
for the case in which $\epsilon=1/2$.

Here we say that a dynamical system is stable if it has a globally attracting
fixed point. Specifically, $(F,X)$ is stable if there is a point
$\mathbf{x}^{*}\in X$ such that

$\lim_{k\rightarrow\infty}F^{k}(\mathbf{x})=\mathbf{x}^{*}\quad\text{for
any}\quad\mathbf{x}^{0}\in X.$

Such a point $\mathbf{x}^{*}$ is referred to as a globally attracting fixed
point.

In a switched system the notion of convergence to such a point is more
complicated since for some instances the state of the system may evolve toward
some $\mathbf{x}^{*}\in X$ while for others it may not. To illustrate, in
Example 2.1 $(\\{F,F,F,\dots\\},\mathbb{R}^{2})$ is a possible instance of the
system $(M,\mathbb{R}^{2},\mu_{3})$. This particular instance has the globally
attracting fixed point $\mathbf{x}^{*}=(0,0)$ but not every instance of this
switched system is asymptotic to this point (see Figure 1 (right)). Later, in
Section 4, we define the notion of $p^{th}$-mean stability (see Definition 7)
which extends the notion of stability to switched systems.

##  3 Time-Delayed Systems

For any dynamical system $(F,X)$ one can consider time-delayed versions of the
system. In many real-world systems, time delays are an inherent part of the
system’s dynamics. For example, one can consider the dynamical system $(F,X)$
given in Definition 1 to be a model for the dynamics of real-world networks.
In this setting the network is comprised of $n$ elements where the dynamics of
the $i^{th}$ element is given by the function $F_{i}:X\rightarrow X_{i}$.
Although the function $F_{i}=F_{i}(x_{1},\dots,x_{n})$ formally depends on
each $x_{j}$ for $j=1,\dots,n$ it may, in fact, only depend on a subset of
these variables. The network elements corresponding to this subset are the
elements that directly interact, or influence the dynamics of the $i^{th}$
network element.

In many real-world networks these interactions do not happen instantaneously
but are time-delayed. For example, network elements are often separated by
distance, or there may be other processes that must occur before a network
element can influence another.

We define a time-delayed version of a dynamical system $(F,X)$ as follows.

######  Definition 3.

_(Time-Delayed Dynamical System)_ Let $(F,X)$ be a dynamical system given in
Definition 1 on the product space $X=\prod_{i=1}^{n}X_{i}$. Let
$D=[d_{ij}]\in\mathbb{N}^{n\times n}_{L}$ be a delay matrix with
$\max_{i,j}d_{ij}\leq L$, a bound on the delay length. Define $X_{L}$, the
_extension of $X$ to delay-space_, to be

$X_{L}=\prod_{\ell=0}^{L}\prod_{i=1}^{n}X_{i,\ell}\quad\text{where}\quad
X_{i,\ell}=X_{i}\quad\text{for}\quad 1\leq i\leq n,\,0\leq\ell\leq L.$

Componentwise, define $F_{D}:X_{L}\to X_{L}$ by

$(F_{D})_{i,\ell+1}(x_{i,\ell})=x_{i,\ell}\quad\text{ for }\quad 0\leq\ell\leq
L-1,\,1\leq i\leq n,$ (1)

that is the identity map, and

$(F_{D})_{i,0}(x_{1,d_{i1}},x_{2,d_{i2}},\ldots,x_{n,d_{in}})=F_{i}(x_{1,d_{i1}},x_{2,d_{i2}},\ldots,x_{n,d_{in}})$
(2)

where $F_{i}:X\to X_{i}$ is the $i^{th}$ component function of $F$ for
$i=1,2,\ldots,n$. Then $(F_{D},X_{L})$ is the _delayed version_ of $(F,X)$
corresponding to the fixed delay matrix $D$ with _delay bound_ $L$.

This definition of a time-delayed dynamical system uses the product structure
of $X$ to create the new larger delay space $X_{L}$. For every dimension of
original space $X$ we add $L$ extra dimensions to our product space to allow
for the “storage” of the time-delayed interactions. If the space $X$ has
dimension $n$, the space $X_{L}$ has dimension $n(L+1).$ Viewing the dynamical
system as a network sheds some light on this construction. When an interaction
between element $i$ and element $j$ is delayed by $\tau$ time-steps (i.e.
$d_{ij}=\tau$), the state of the element $i$ does not immediately affect the
state of element $j$. Rather the “effect” $i$ has on $j$ is passed through
$\tau$ “filler” elements before reaching element $j$. In other words, the
state of the $i$th element $\ell$ time-steps in the past is stored as
$x_{i,\ell}$. The information about which interactions are delayed and by how
much is stored in the delay matrix $D=[d_{ij}]\in\mathbb{N}_{L}^{n\times n}$.

\begin{overpic}[scale={0.3}]{StochTimeDelayFig1.eps}
\put(12.0,-2.0){$(H,\mathbb{R}^{3})$}
\put(47.0,-2.0){$(H_{D},\mathbb{R}^{6})$}
\put(80.0,-2.0){$(H_{L},\mathbb{R}^{6},\pi)$} \end{overpic} Figure 2: The
stable dynamics of the system $(H,\mathbb{R}^{3})$ from Example 3.2 is shown
left. Adding the constant time delays $D$ given by Equation 3 results in the
system $(H_{D},\mathbb{R}^{6})$ with periodic dynamics, shown center. Adding
stochastically time-varying time-delays of size no larger than $L=1$ similarly
destabilizes the system’s dynamics. This is shown right for the system
$(H_{L},\mathbb{R}^{6},\pi)$ (see Example 4.6) for details.

######  Example 3.2.

(Constant Time-Delays) Consider the dynamical system $(H,\mathbb{R}^{3})$
given by

$H\left(\begin{bmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{bmatrix}\right)=\begin{bmatrix}\frac{1}{4}x_{1}-\tanh(x_{1})+\tanh(x_{2})-\tanh(x_{3})\\\
\frac{1}{4}x_{2}-\tanh(x_{1})+\tanh(x_{2})-\tanh(x_{3})-1/2\\\
\frac{1}{4}x_{3}+\tanh(x_{1})-\tanh(x_{3})\end{bmatrix}.$

For the delay matrix

$D=\begin{bmatrix}0&1&1\\\ 0&0&1\\\ 0&0&0\end{bmatrix},$ (3)

which has delay bound $L=1$, the delayed version is given by the function
$H_{D}:X_{L}\rightarrow X_{L}$ where $X_{L}=\mathbb{R}^{6}$, which has the
form

$H_{D}\left(\begin{bmatrix}x_{1,0}\\\ x_{2,0}\\\ x_{3,0}\\\ x_{1,1}\\\
x_{2,1}\\\
x_{3,1}\end{bmatrix}\right)=\begin{bmatrix}\frac{1}{4}x_{1,0}-\tanh(x_{1,0})+\tanh(x_{2,1})-\tanh(x_{3,1})\\\
\frac{1}{4}x_{2,0}-\tanh(x_{1,0})+\tanh(x_{2,0})-\tanh(x_{3,1})-1/2\\\
\frac{1}{4}x_{3,0}+\tanh(x_{1,0})-\tanh(x_{3,0})\\\ x_{1,0}\\\ x_{2,0}\\\
x_{3,0}\\\ \end{bmatrix}.$

Here the variables $x_{1,1},x_{2,1},x_{3,1}$ represent the “filler” elements
which store delay information.

Note that the original undelayed system $(H,\mathbb{R}^{3})$ is stable with
the globally attracting fixed point $\mathbf{x}\approx[-1.05,0.62,0.14]^{T}$
but the delayed version $(H_{D},\mathbb{R}^{6})$ is not (see Figure 2 left and
center, respectively). That is, a system’s dynamics can be destabilized by the
introduction of time-delays.

Not only can time delays be introduced into dynamical systems they can also be
introduced into switched systems.

######  Definition 4.

_(Delayed Stochastic Switched System)_ Let $(M,X,\mu)$ be a stochastic
switched system with corresponding probability space $(M,\mathscr{A},\mu)$.
For the delay bound $L$ let $M_{L}$ be the set of all delayed versions of
functions in $M$ with maximum delay length of $L$, i.e.

$M_{L}=\\{F_{D}:F\in M\text{ and }\ D\in\mathbb{N}^{n\times n}_{L}\\}.$

Let $(M_{L},\mathscr{A}_{D},\pi)$ be a probability space such that for a
measurable set $E\subset M$, the set of all delayed versions of each $F\in E$,
given by

$E_{L}=\\{F_{D}:F\in E,D\in\mathbb{N}^{n\times n}_{L}\\},$

we have $\pi(E_{L})=\mu(E)$. If this is the case we call the corresponding
switched system $(M_{L},X_{L},\pi)$ a delayed version of $(M,X,\mu)$.

In the delayed version $(M_{L},X_{L},\pi)$ of the switched system $(M,X,\mu)$
the idea is that we choose at each time step $k\geq 1$ some $F^{(k)}\in M$ and
some delay matrix $D^{(k)}\in\mathbb{N}^{n\times n}_{L}$. The function
$F^{(k)}_{D^{(k)}}\in M_{L}$ is chosen in this system at time $k$ with
probability $\pi(F^{(k)}_{D^{(k)}})$. The one stipulation is the probability
distribution $\pi$ has the property

$\sum_{D^{(k)}\in\mathbb{N}^{n\times
n}_{L}}\pi(F^{(k)}_{D^{(k)}})=\mu(F^{(k)}).$ (4)

That is, delayed versions of a function $F\in M$ are as likely in
$(M_{L},X_{L},\pi)$ as $F$ is in the undelayed system $(M,X,\mu)$. This gives
us an instance $(\\{F^{(k)}_{D^{(k)}}\\}_{k=1}^{\infty},X_{L})$ of the delayed
stochastic switched system $(M_{L},X_{L},\pi)$.

It is also worth noting that if we have a delayed system $(M_{L},X_{L},\pi)$
then we can uniquely recover the original undelayed system $(M,X,\mu)$. The
reason for this is that the probability distribution $\mu$ that determines the
frequency at which we choose a function $F\in M$ is given to us in terms of
$\pi$ by Equation (4).

\begin{overpic}[scale={0.33}]{IntrinsicStabExp12.eps}
\put(20.0,-2.0){$(M,\mathbb{R}^{2},\mu_{2})$}
\put(70.0,-2.0){$(M_{L},\mathbb{R}^{2},\pi_{2})$} \end{overpic} Figure 3: The
undelayed dynamics of the system $(M,\mathbb{R}^{2},\mu_{2})$ from Example 2.1
is shown left. The delayed version of this system
$(M_{L},\mathbb{R}^{2},\pi_{2})$ for $L=1$ is shown right. Both are stable.

######  Example 3.3.

(Time-Varying Time-Delays) Let $M$ be the single mapping $F:X\rightarrow X$,
i.e. $M=\\{F\\}$. Then $(M,X,\mu)$ with $\mu(F)=1$ is a switched system,
although there is no system to switch to apart from $(F,X)$. For a delay bound
of $L$ the corresponding delayed switched system is $(M_{L},X_{L},\pi)$ in
which an instance $(\\{F_{D^{(k)}}\\}_{k=1}^{\infty},X_{L})$ is composed of
functions $F_{D^{(k)}}:X_{L}\rightarrow X_{L}$ each of which are time-delayed
versions of the original function $F$.

Since we switch between different time-delayed versions of the same function
$F$ the result is what is referred to as a dynamical system with time-varying
time-delays where each time-delayed version of $F$ is chosen at each time step
with probability determined by $\pi$. We write this system as
$(F_{D},X_{L},\pi)$.

Taking the system $(H,\mathbb{R}^{3})$ from Example 3.2 we can add time-
varying time-delays to this system as follows. Let $L=1$. Then there are
$2^{9}$ different delay matrices $D=[d_{ij}]\in\mathbb{N}^{3\times 3}_{L}$,
which can be used to create delayed versions $(H_{D},\mathbb{R}^{6})$ of
$(H,\mathbb{R}^{3})$. Suppose $\pi$ is the probability distribution that
chooses each of these $2^{9}$ delayed systems with equal probability. The
dynamics of the resulting system $(H_{L},\mathbb{R}^{6},\pi)$ is shown in
Figure 2 (right). Note that similar to adding constant time delays adding
these time-varying time-delays to the system similarly has a destabilizing
effect to the system’s dynamics.

Adding time delays can also destabilize switched systems but this is not
always the case. For some switched systems it can happen that the addition of
time-delays cannot destabilize the system so long as the delay bound
$L<\infty$. These “patient” systems will be introduced later in Section 4 (see
Definition 9).

######  Example 3.4.

Consider the switched system $(M,\mathbb{R}^{2},\mu_{2})$ given in Example
2.1. For $L=1$ let $(M_{L},\mathbb{R}^{4},\pi_{2})$ be the delayed version of
this system in which the probability distribution $\pi_{2}$ is given by

$\pi(F_{D})=\frac{1}{16}\mu_{2}(F)=\frac{9}{160}\quad\text{and}\quad\pi(G_{D})=\frac{1}{16}\mu_{2}(G)=\frac{1}{160}\quad\text{
for }\text{ any }\quad D\in\mathbb{N}^{2\times 2}_{L}.$

Note that there are $2^{4}$ delayed versions of each of $(F,\mathbb{R}^{2})$
and $(G,\mathbb{R}^{2})$, making for 32 total systems in $M_{L}$. Here each
delayed version of $F$ is equally likely, and each delayed version of $G$ is
equally likely, but delayed versions of $F$ are more likely than delayed
versions of $G$.

As is shown in Figure 3 the delayed switched system
$(M_{L},\mathbb{R}^{4},\pi_{2})$ is stable. In fact, any time-delayed version
of $(M,\mathbb{R}^{2},\mu_{2})$ will be stable (see Example 4.6). That is,
irrespective of the maximal delay length $L<\infty$ and of the way in which we
create the probability distribution $\pi$, this switched system cannot be
destabilized by time delays.

In contrast, consider the dynamical systems $(\bar{F},\mathbb{R}^{2})$ and
$(\bar{G},\mathbb{R}^{2})$ given by

$\bar{F}\left(\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}\right)=\begin{bmatrix}-\epsilon&1\\\
0&-\epsilon\end{bmatrix}\begin{bmatrix}\tanh(x_{1})\\\
\tanh(x_{2})\end{bmatrix}\hskip 14.22636pt\text{and}\hskip
14.22636pt\bar{G}\left(\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}\right)=\begin{bmatrix}-\epsilon&0\\\
-1&-\epsilon\end{bmatrix}\begin{bmatrix}\tanh(x_{1})\\\
\tanh(x_{2})\end{bmatrix}$

where $\epsilon=\frac{2}{3}$. Let $\bar{\mu}$ be the distribution given by
$\bar{\mu}(F)=\bar{\mu}(G)=\frac{1}{2}$. For $\bar{M}=\\{\bar{F},\bar{G}\\}$,
the stochastic switched system $(\bar{M},\mathbb{R}^{2},\bar{\mu})$ is stable
(see Figure 4, left).

\begin{overpic}[scale={0.33}]{IntrinsicStabExp13.eps}
\put(20.0,-2.0){$(\bar{M},\mathbb{R}^{2},\bar{\mu})$}
\put(70.0,-2.0){$(\bar{M}_{L},\mathbb{R}^{2},\bar{\pi})$} \end{overpic} Figure
4: The undelayed dynamics of the system $(\bar{M},\mathbb{R}^{2},\bar{\mu})$
from Example 3.4 is shown left. A delayed version of this system
$(\bar{M}_{L},\mathbb{R}^{2},\bar{\pi})$ for $L=1$ is shown right. In this
case time-delays destabilize the switched system.

However, this system can be destabilized by time delays. Consider the delay
matrices

$D_{0}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}\quad\text{and}\quad
D_{1}=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix},$

which correspond to no delays and to a single delay, respectively. Let
$\bar{\pi}$ be the probability distribution on $\bar{M}_{L}$ such that for
$H_{D}\in\bar{M}_{L}$, we have

$\displaystyle\bar{\pi}(H_{D})=\begin{cases}\frac{1}{2}\text{ if
}H=\bar{F},D=D_{0}\\\ \frac{1}{20}\text{ if }H=\bar{G},D=D_{0}\\\
\frac{9}{20}\text{ if }H=\bar{G},D=D_{1}\\\ 0\text{ otherwise}.\end{cases}$

The stochastically delayed version $(\bar{M}_{L},\mathbb{R}^{4},\bar{\pi})$ of
the system $(\bar{M},\mathbb{R}^{2},\bar{\mu})$ switches between three
systems: $\bar{F},\bar{G}$, and $\bar{G}$ with a single delay. Including this
version of $\bar{G}$ with a single delay destabilizes the system (see Figure
4, right). Technically, the system is no longer first mean stable (see
Definition 7).

The question we answer in the following sections is why do certain switched
systems remain stable when they are delayed while others do not. Specifically,
we give a sufficient condition under which a switched system cannot be
destablized under the addition of time-delays.

##  4 Stability of Stochastically Switched and Delayed Systems

In this section we examine the $p^{th}$-mean stability of nonlinear stochastic
systems. To do this, we create a linear system that bounds the nonlinear
system and use tools developed to find stability of linear systems to
determine the stability of the nonlinear system.

######  Definition 5.

_(Lipschitz Matrix)_ Let $F:X\to X$ be a Lipschitz continuous mapping on $X$.
Then there exist real constants $a_{ij}\geq 0$ such that

$\displaystyle\|F_{i}(x)-F_{i}(y)\|\leq\sum_{j=1}^{n}a_{ij}\,\|x_{j}-y_{j}\|$

for all $x,y\in X$. We call $A=[a_{ij}]\in\mathbb{R}^{n\times n}$ a _Lipschitz
matrix_ of the mapping $F$.

Note that if $F$ is Lipschitz continuous with Lipschitz matrix $A=[a_{ij}]$,
and $A\preceq B\in\mathbb{R}^{n\times n}$ entrywise, then $B$ is also a
Lipschitz matrix of $F$. That is, $F$ has infinitely many Lipschitz matrices.
If $F$ is differentiable, the entrywise smallest Lipschitz matrix $A$ of $F$
is efficiently obtained as

$a_{ij}=\sup_{x\in X}\left\lvert\frac{\partial F_{i}}{\partial
x_{j}}(\mathbf{x})\right\rvert.$ (5)

While using entrywise smaller Lipschitz matrices provides tighter bounds on
the nonlinear system $F$, the results of this paper apply to any choice of
Lipchitz matrix $A$ of $F$.

######  Definition 6.

_(Stochastic Lipschitz System)_ Let $(M,X,\mu)$ be a stochastic switched
system where each $F\in M$ is Lipschitz continuous. We construct a stochastic
Lipschitz set $S\subseteq\mathbb{R}^{n\times n}$ as follows: For each $F\in
M$, contribute exactly one Lipschitz matrix $A\in\mathbb{R}^{n\times n}$ of
$F$ to the set $S$. Define a probability distribution $\mu^{\prime}$ on $S$ by
setting $\mu^{\prime}(A)=\mu(F)$ for each $A\in S$. The system
$(S,\mathbb{R}^{n},\mu^{\prime})$ is a linear stochastic switched system.

A fact that will be useful to us is that if $F:X\rightarrow X$ is Lipschitz
continuous then any delayed version $F_{D}:X_{L}\rightarrow X_{L}$ of this
function is also Lipschitz continuous. The following lemma, proved in [1],
describes how to construct a Lipschitz matrix of a delayed version $F_{D}$ of
$F$ from a Lipschitz matrix of $F$.

######  Lemma 1.

_(Lipschitz Set for Stochastic Switched Systems)_ Let $(M,X,\mu)$ be a
stochastic switched system with Lipschitz set $S$. For every $A\in S$ and
$D\in\mathbb{N}^{n\times n}_{L}$ define $A_{D}$ as

$\displaystyle A_{D}=\begin{bmatrix}A_{0}&A_{1}&\ldots&A_{L-1}&A_{L}\\\
I_{n}&0&\ldots&0&0\\\ 0&I_{n}&\ldots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\ldots&I_{n}&0\end{bmatrix}\in\mathbb{R}^{n(L+1)\times n(L+1)}$

where $I_{n}\in\\{0,1\\}^{n\times n}$ is the identity matrix and each
$A_{\ell}\in\mathbb{R}^{n\times n}$ is defined element-wise by
$A_{\ell}=[a_{ij}\mathbbm{1}_{d_{ij}=\ell}]$, with the indicator function
$\mathbbm{1}_{d_{ij}=\ell}$ defined as

$\displaystyle\mathbbm{1}_{d_{ij}=\ell}=\begin{cases}1\text{ if
$d_{ij}=\ell$}\\\ 0\text{ otherwise}\end{cases}\text{ for }\ 0\leq\ell\leq L.$

Furthermore, let $S_{L}=\\{A_{D}:A\in S,D\in\mathbb{N}^{n\times n}_{L}\\}$.
Then each $A_{D}$ is a Lipschitz matrix of $F_{D}$ and $S_{L}$ is a stochastic
Lipschitz set of the corresponding delayed system $(M_{L},X_{L},\pi)$. This
forms a delayed linear stochastic switched system
$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$, where we define
$\pi^{\prime}(A_{D})=\pi(F_{D})$ for every $F_{D}\in M_{L}$ where $A_{D}$ is
the Lipschitz matrix of $F_{D}$, so that the condition
$\sum_{D\in\mathbb{N}^{n\times n}_{L}}\pi^{\prime}(A_{D})=\pi^{\prime}(A)$
holds.

\begin{overpic}[scale={0.33}]{DelayComm.eps}
\put(-10.0,29.5){\large{$(M,X,\mu)$}}
\put(86.0,29.5){\large{$(M_{L},X_{L},\pi)$}}
\put(-10.0,0.5){\large{$(S,\mathbb{R}^{n},\mu^{\prime})$}}
\put(82.0,0.5){\large{$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$}}
\par\put(46.0,5.5){\large{$\tau_{L}$}} \put(46.0,34.5){\large{$\tau_{L}$}}
\put(-7.0,16.0){\large{$\Lambda$}} \put(102.0,16.0){\large{$\Lambda$}}
\end{overpic} Figure 5: For the delay operator $\tau_{L}$ and the
linearization operator $\Lambda$ defined by Equations ?? the diagram shown
above commutes. Therefore, the order in which we linearize and delay the
switched system $(M,X,\mu)$ does not matter.

In the following example we show how to construct the matrix $A_{D}$ of a
system delayed by the matrix $D$.

######  Example 4.5.

Consider the system $(H,\mathbb{R}^{3})$ in Example 3.2, which can be written
in vector form as

$H\left(\begin{bmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{bmatrix}\right)=\frac{1}{4}\begin{bmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{bmatrix}+\begin{bmatrix}-1&1&-1\\\ -1&1&-1\\\
1&0&-1\end{bmatrix}\begin{bmatrix}\tanh(x_{1})\\\ \tanh(x_{2})\\\
\tanh(x_{3})\end{bmatrix}-\begin{bmatrix}0\\\ 1/2\\\ 0\end{bmatrix}.$

A Lipschitz matrix $A=[a_{ij}]\in\mathbb{R}^{3\times 3}$ for the function
$H:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ using
$\sup_{\mathbf{x}\in\mathbb{R}^{3}}|\partial H_{i}/\partial
x_{j}(\mathbf{x})|=a_{ij}$ is

$A=\begin{bmatrix}3/4&1&1\\\ 1&5/4&1\\\ 1&0&3/4\end{bmatrix}.$

Using the delay matrix $D\in\mathbb{N}^{3\times 3}_{1}$ from Equation (3) the
delayed version $F_{D}:\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ can be written
in vector form as

$H_{D}\left(\begin{bmatrix}x_{1,0}\\\ x_{2,0}\\\ x_{3,0}\\\ x_{1,1}\\\
x_{2,1}\\\
x_{3,1}\end{bmatrix}\right)=\frac{1}{4}\begin{bmatrix}1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
\end{bmatrix}\begin{bmatrix}x_{1,0}\\\ x_{2,0}\\\ x_{3,0}\\\ x_{1,1}\\\
x_{2,1}\\\ x_{3,1}\end{bmatrix}+\begin{bmatrix}-1&0&0&0&1&-1\\\
-1&1&0&0&0&-1\\\ 1&0&-1&0&0&0\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\
\end{bmatrix}\begin{bmatrix}\tanh(x_{1,0})\\\ \tanh(x_{2,0})\\\
\tanh(x_{3,0})\\\ x_{1,1}\\\ x_{2,1}\\\
x_{3,1}\end{bmatrix}-\begin{bmatrix}0\\\ 1/2\\\ 0\\\ 0\\\ 0\\\
0\end{bmatrix}.$

From Lemma 1 we can create the Lipschitz matrix $A_{D}$ of $H_{D}$ where

$A_{D}=\left[\begin{array}[]{ccc|ccc}3/4&0&0&0&1&1\\\ 1&5/4&0&0&0&1\\\
1&0&3/4&0&0&0\\\ \hline\cr 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\
\end{array}\right]\text{ in which }A_{0}=\begin{bmatrix}3/4&0&0\\\ 1&5/4&0\\\
1&0&3/4\end{bmatrix}\text{ and }A_{1}=\left[\begin{array}[]{ccc}0&1&1\\\
0&0&1\\\ 0&0&0\end{array}\right].$

In Section 2 we said that a system $(F,X)$ is stable if it has a globally
attracting fixed point. Stability is more complicated for a switched system
since there may be instances for which the system’s state tends towards a
single point, irrespective of its initial condition, but not for other
instances. To define stability for a stochastically switched system we use the
notion of $p^{th}$-mean stability.

######  Definition 7.

_( $p^{th}$-mean stability)_ We say that a stochastic switched system
$(M,X,\mu)$ is exponentially stable in $p^{th}$ mean to a fixed point
$\mathbf{\tilde{x}}$ if there exist real constants $C>0$ and $\beta>0$ such
that for any initial condition $\mathbf{x}^{0}$ we have

$\displaystyle\mathbb{E}\left[||\mathbf{x}^{k}-\mathbf{\tilde{x}}||^{p}\right]\leq
Ce^{-\beta k}||\mathbf{x}^{0}-\mathbf{\tilde{x}}||^{p}.$

For simplicity, we say that such a system is $p^{th}$-mean stable. Though it
may be hard to verify that a nonlinear system is $p^{th}$-mean stable we can
use a generalization of the spectral radius called the $p$-radius to determine
the stability of an associated linear stochastic switched system.

######  Definition 8.

_( $p$-radius)_ For a linear stochastic switched system
$(S,\mathbb{R}^{n},\mu)$ (where $\mathbf{x}^{k+1}=A_{k}\mathbf{x}^{k}$ and
each $A_{k}$ is drawn from $S$ with distribution $\mu$) and constant $p>0$,
the generalized joint spectral radius of $(S,\mathbb{R}^{n},\mu)$ with
exponent $p$ ($p$-radius for short) is defined by

$\displaystyle\rho_{p,\mu}=\lim_{k\to\infty}(\mathbb{E}[||A_{k}\cdots
A_{1}||]^{p})^{1/pk}.$

We cite a theorem from [2] that says that under certain conditions on $S$ and
$\mu$, the $p$-radius can be computed as the spectral radius of a single
matrix. Sets of nonnegative matrices, such as the Lipschitz sets we are
working with, satisfy these conditions.

Given a probability distribution $\mu$, we define the support of $\mu$ as the
closed set supp $\mu$ such that

$\mu((\text{supp }\mu)^{c})=0$

and for any open set $G$ with $G\cap\text{supp }\mu\neq\emptyset$, we have
$\mu(G\cap\text{supp }\mu)>0$.

Next we given some definitions related to Kronecker products. The _Kronecker
product_ $A\otimes B$ of two matrices $A\in\mathbb{R}^{m\times n}$ and
$B\in\mathbb{R}^{q\times r}$ is defined as the $mq\times nr$ matrix

$\displaystyle A\otimes B=\begin{bmatrix}a_{11}B&\cdots&a_{1n}B\\\
\vdots&\ddots&\vdots\\\ a_{m1}B&\cdots&a_{mn}B\end{bmatrix}.$

The _Kronecker power_ $A^{\otimes p}$ is defined inductively by $A^{\otimes
1}=A$ and $A^{\otimes p}=A^{\otimes(p-1)}\otimes A$ for any integer $p>1$.

######  Theorem 1.

_(Computing $p$-radius)_ If the support of $\mu$ is bounded, and the pair
$(\mu,p)$ satisfies either

  1. (1)

$p$ is an integer and $\text{supp }\mu$ leaves a proper cone invariant, or

  2. (2)

$p$ is an even integer

then the $p$-radius of $(S,\mathbb{R}^{n},\mu)$ is given by

$\displaystyle\rho_{p,\mu}=\rho(\mathbb{E}_{\mu}[S^{\otimes p}])^{1/p},$

where $S^{\otimes p}=\\{A^{\otimes p}:A\in S\\}$ and the expectation is taken
entrywise according to the distribution $\mu$ on $S$.

Now, we relate the stability of a nonlinear system to that of its
corresponding Lipschitz system.

######  Theorem 2.

_(Lipschitz Linearization)_ Let $(M,X,\mu)$ be a stochastic switched system.
If each $F\in M$ shares a fixed point $\mathbf{\tilde{x}}$, and there exists a
corresponding Lipschitz system $(S,\mathbb{R}^{n},\mu^{\prime})$ with
$\rho_{\mu,p}(S)<1$, then $(M,X,\mu)$ is exponentially stable in $p^{th}$ mean
to the fixed point $\mathbf{\tilde{x}}$.

###### Proof.

Assume the hypothesis. Definition 2 implies $\mathbf{\tilde{x}}$ is a fixed
point of $(M,X,\mu)$. Since $\rho_{\mu,p}(S)<1$, there exist real constants
$C>0$ and $\beta>0$ such that for any initial condition
$\mathbf{y}^{0}\in\mathbb{R}^{n}$ we have
$\mathbb{E}\left[||\mathbf{y}^{k}||^{p}\right]\leq Ce^{-\beta
k}||\mathbf{y}^{0}||^{p}$. Let $\mathbf{x}^{0}\in X$ be given. Set

$\mathbf{y}^{0}=\begin{bmatrix}||\mathbf{x}^{0}_{1}-\mathbf{\tilde{x}}_{1}||\\\
||\mathbf{x}^{0}_{2}-\mathbf{\tilde{x}}_{2}||\\\ \vdots\\\
||\mathbf{x}^{0}_{n}-\mathbf{\tilde{x}}_{n}||\end{bmatrix}.$

Then $||\mathbf{x}^{0}-\mathbf{\tilde{x}}||=||\mathbf{y}^{0}||$ by Definition
1 and $||\mathbf{x}^{1}-\mathbf{\tilde{x}}||\leq||\mathbf{y}^{1}||$ by
Definitions 1 and 5. Thus
$\mathbb{E}_{\mu}\left[||\mathbf{x}^{1}-\mathbf{\tilde{x}}||^{p}\right]\leq\mathbb{E}_{\mu^{\prime}}\left[||\mathbf{y}^{1}||^{p}\right]$
by Definition 6. Inductively we have

$\displaystyle\mathbb{E}_{\mu}\left[||\mathbf{x}^{k}-\mathbf{\tilde{x}}||^{p}\right]$
$\displaystyle\leq\mathbb{E}_{\mu^{\prime}}\left[||\mathbf{y}^{k}||^{p}\right]$
$\displaystyle\leq Ce^{-\beta k}||\mathbf{y}^{0}||^{p}$
$\displaystyle=Ce^{-\beta k}||\mathbf{x}^{0}-\mathbf{\tilde{x}}||^{p}$

so $(M,X,\mu)$ is exponentially stable in $p^{th}$ mean to
$\mathbf{\tilde{x}}$ as claimed. ∎

Now that we have established tools for determining first-mean stability of
nonlinear switched systems, we focus on determining which of these systems
will remain stable even when time delays are introduced. We refer to such
systems as “patiently stable”, as they cannot be destabilized by bounded time-
delays of any kind.

######  Definition 9.

(Patient First Mean Stability) A stochastic switched system $(M,X,\mu)$ is
_patiently first-mean stable_ if $(M,X,\mu)$ as well as all of its delayed
versions $(M_{L},X_{L},\pi)$ are first-mean stable.

Our main result (Theorem 3) gives a sufficient condition for patient first
mean stability. In order to prove our main result (see Theorem 3) we will need
to relate the spectral radius of the matrix $A_{D}$ in Lemma 1 with the
spectral radius of the matrix $B=\sum_{\ell=0}^{L}A_{\ell}$. To do this we use
a matrix transform referred to as an isoradial matrix reduction, or the Perron
compliment [22]. An isoradial reduction is a special type of isospectral
reduction that preserves the matrix’ spectral radius while reducing its size.
(See [21, 11] for details.)

To define an isoradial matrix reduction we require the following. For a matrix
$M\in\mathbb{R}^{n\times n}$ let $N=\\{1,\ldots,n\\}$. If the sets $R,C\subset
N$ are proper subsets of $N$, we denote by $M_{RC}$ the $|R|\times|C|$
_submatrix_ of $M$ with rows indexed by $R$ and columns indexed by $C$. We
denote the subset of $N$ not contained in $S$ by $\bar{S}$, that is $\bar{S}$
is the _complement_ of $S$ in $N$. Using this notation, an isoradial reduction
of a square real-valued matrix is defined as follows.

######  Definition 10.

(Isoradial Reductions) The _isoradial reduction_ of a matrix
$M\in\mathbb{R}^{n\times n}$ over a nonempty subset $S\subset N$ is the matrix

$\mathcal{I}_{S}(M)=M_{SS}-M_{S\bar{S}}\left(M_{\bar{S}\bar{S}}-\rho(M)I\right)^{-1}M_{\bar{S}S}\in\mathbb{R}^{|S|\times|S|},$
(6)

where $\rho(M)=\max\\{|\lambda|:\lambda\textrm{ is an eigenvalue of }M\\}$ is
the _spectral radius_ of $M$.

The isoradial reduction $\mathcal{I}_{S}(M)$ does not exist for every square
real-valued matrix $M$ and every subset $S\subseteq N$, due to the fact that
the inverse taken in Equation (6) may not exist. If the isoradial reduction
$\mathcal{I}_{S}(M)$ exists then

$\rho(\mathcal{I}_{S}(M))=\rho(M).$ (7)

Using equation (7) we can prove the following lemma.

######  Lemma 2.

Assume $A\in\mathbb{R}^{n(L+1)\times n(L+1)}$ is a nonnegative matrix given in
Lemma 1 of the form

$\displaystyle A=\begin{bmatrix}A_{0}&A_{1}&\ldots&A_{L-1}&A_{L}\\\
I_{n}&0&\ldots&0&0\\\ 0&I_{n}&\ldots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\ldots&I_{n}&0\end{bmatrix}$

where each $A_{ij}\in\mathbb{R}^{n\times n}$ and $I_{n}\in\mathbb{R}^{n\times
n}$ is the identity matrix. Then $\rho(A)<1$ if and only if
$\rho(\sum_{\ell=0}^{L}A_{\ell})<1$.

###### Proof.

Let $\rho(A)=\alpha$ and suppose $0<\alpha<1$. We choose $S=\\{1,2,\dots,n\\}$
in which case

$A_{SS}=A_{0},\quad
A_{S\bar{S}}=\begin{bmatrix}A_{1}&\ldots&A_{L-1}&A_{L}\end{bmatrix},\quad
A_{\bar{S}\bar{S}}=\begin{bmatrix}0&\ldots&0&0\\\ I_{n}&\ldots&0&0\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\ldots&I_{n}&0\end{bmatrix},\text{ and
}A_{\bar{S}S}=\begin{bmatrix}I_{n}\\\ 0\\\ \vdots\\\ 0\end{bmatrix}.$

The isoradial reduction of $A$ over the set $S$ is then

$\displaystyle\mathcal{I}_{S}(A)=$ $\displaystyle
A_{SS}-A_{S\bar{S}}\left(A_{\bar{S}\bar{S}}-\rho(A)I\right)^{-1}M_{\bar{S}S}$
$\displaystyle=$ $\displaystyle
A_{0}-\begin{bmatrix}A_{1}&\ldots&A_{L-1}&A_{L}\end{bmatrix}\begin{bmatrix}-\alpha
I_{n}&\ldots&0&0\\\ I_{n}&-\alpha I_{n}&0&0\\\ \vdots&\ddots&\ddots&\vdots\\\
0&\ldots&I_{n}&-\alpha I_{n}\end{bmatrix}^{-1}\begin{bmatrix}I_{n}\\\ 0\\\
\vdots\\\ 0\end{bmatrix}.$

Since $\alpha>0$ the matrix $A_{\bar{S}\bar{S}}-\alpha I$ is invertible with
inverse

$\begin{bmatrix}-\alpha I_{n}&\ldots&0&0\\\ I_{n}&-\alpha I_{n}&0&0\\\
\vdots&\ddots&\ddots&\vdots\\\ 0&\ldots&I_{n}&-\alpha
I_{n}\end{bmatrix}^{-1}=\begin{bmatrix}-\alpha^{-1}I_{n}&0&0&\ldots&0\\\
-\alpha^{-2}I_{n}&-\alpha^{-1}I_{n}&0&\ldots&0\\\
-\alpha^{-3}I_{n}&-\alpha^{-2}I_{n}&-\alpha^{-1}I_{n}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\alpha^{-L}&-\alpha^{-L+1}I_{n}&-\alpha^{-L+2}I_{n}&\ldots&-\alpha^{-1}I_{n}\end{bmatrix}.$

Combining this with the previous equation we have

$\displaystyle\mathcal{I}_{S}(A)=$ $\displaystyle
A_{0}-\begin{bmatrix}A_{1}&\ldots&A_{L-1}&A_{L}\end{bmatrix}\begin{bmatrix}-\alpha^{-1}I_{n}&0&0&\ldots&0\\\
-\alpha^{-2}I_{n}&-\alpha^{-1}I_{n}&0&\ldots&0\\\
-\alpha^{-3}I_{n}&-\alpha^{-2}I_{n}&-\alpha^{-1}I_{n}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\alpha^{-L}&-\alpha^{-L+1}I_{n}&-\alpha^{-L+2}I_{n}&\ldots&-\alpha^{-1}I_{n}\end{bmatrix}\begin{bmatrix}I_{n}\\\
0\\\ 0\\\ \vdots\\\ 0\end{bmatrix}$ $\displaystyle=$ $\displaystyle
A_{0}+\sum_{\ell=1}^{L}\alpha^{-\ell}A_{\ell}=\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell}.$

By Equation (7) we have
$\rho(\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell})=\rho(A)=\alpha$. Note that
entrywise the matrix
$\sum_{\ell=0}^{L}A_{\ell}\preceq\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell}$
since each $A_{i}$ is nonnegative and $\alpha<1$. By Proposition 3.3 in [11]
it follows that

$\rho\left(\sum_{\ell=0}^{L}A_{\ell}\right))\leq\rho\left(\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell}\right)=\rho(A)<1.$
(8)

Therefore, if $0<\rho(A)<1$ then $\rho(\sum_{\ell=0}^{L}A_{\ell})<1$.

If it is the case that $\rho(A)=0$ then the claim is that
$\rho(\sum_{\ell=0}^{L}A_{\ell})=0$. To see this suppose $\rho(A)=0$. For
$\epsilon>0$ let $A_{\epsilon}$ be the matrix $A$ where we replace
$A_{0}\in\mathbb{R}^{n\times n}$ with the matrix $A_{0}+\epsilon
I\in\mathbb{R}^{n\times n}$. Since $0\preceq A_{\epsilon}\preceq A+\epsilon I$
and $\rho(A+\epsilon I)=\rho(A)+\epsilon=\epsilon$ then Proposition 3.3 in
[11] implies $\rho(A_{\epsilon})\leq\epsilon$. Choosing $\epsilon<1$ then

$\rho(\sum_{\ell=0}^{L}A_{\ell})\leq\rho\left((A_{0}+\epsilon
I)+\sum_{\ell=1}^{L}A_{\ell}\right)\leq\rho(A_{\epsilon})\leq\epsilon$

where the first inequality follows from Proposition 3.3 and the second from
Equation (8) using $A_{\epsilon}$ instead of $A$. Letting $\epsilon\rightarrow
0^{+}$ it follows that $\rho(\sum_{\ell=0}^{L}A_{\ell})=0$. Thus, if
$\rho(A)=0$ then $\rho(\sum_{\ell=0}^{L}A_{\ell})=0$. Combining this with the
previous result, if $\rho(A)<1$ then $\rho(\sum_{\ell=0}^{L}A_{\ell})<1$.

To prove the converse, note that if $\alpha\geq 1$ then entrywise
$\rho(\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell})\preceq\sum_{\ell=0}^{L}A_{\ell}$
and Proposition 3.3 in [11] implies that

$\rho\left(\sum_{\ell=0}^{L}A_{\ell}\right)\geq\rho\left(\sum_{\ell=0}^{L}\alpha^{-\ell}A_{\ell}\right)=\alpha\geq
1$

so that $\rho(\sum_{\ell=0}^{L}A_{\ell})\geq 1$. Therefore, if
$\rho(\sum_{\ell=0}^{L}A_{\ell})<1$ then it is not possible for $\alpha\geq 1$
implying that if $\rho(\sum_{\ell=0}^{L}A_{\ell})<1$ then $\rho(A)<1$. This
completes the proof. ∎

For example, the matrix $A_{D}$ in Example 4.5 has the spectral radius
$\rho(A_{D})=2.0239$, which is greater than 1. Hence, Lemma 2 implies the same
is true of $\rho(\sum_{\ell=0}^{1}A_{\ell})=(6+\sqrt{17})/4$.

######  Theorem 3.

Let $(M,X,\mu)$ be a stochastic switched system where the support of $\mu$ is
bounded and each $F\in M$ shares a fixed point $\tilde{x}\in X$. If there is a
Lipschitz system $(S,\mathbb{R}^{n},\mu^{\prime})$ of $(M,X,\mu)$ that is
exponentially stable in first-mean, or if there is a delayed version
$(M_{L},X_{L},\mu)$ with corresponding Lipschitz system
$(S_{L},\mathbb{R}^{(L(n+1)},\pi^{\prime})$ that is exponentially stable in
first-mean, then $(M,X,\mu)$ is patiently first-mean stable.

Before we give a proof of Theorem 3, we recall the systems in Example 2.1 and
now answer the question of why some of these systems can be destabilized by
time delays and others remain stable when time delayed.

######  Example 4.6.

We return to the systems defined in Example 2.1 and examine their stability
under stochastic delays. For $M=\\{F,G\\}$, we have Lipschitz set
$S=\left\\{\begin{bmatrix}\epsilon&1\\\
0&\epsilon\end{bmatrix},\begin{bmatrix}\epsilon&0\\\
1&\epsilon\end{bmatrix}\right\\}$ with $\epsilon=1/2$. For the system
$(M,X,\mu_{1})$ with no switching we have

$\displaystyle E_{\mu_{1}}[S]=\begin{bmatrix}\epsilon&1\\\
0&\epsilon\end{bmatrix}$

and $\rho(E_{\mu_{1}}[S])=1/2<1$. Thus, $(M,X,\mu_{1})$ will be stable under
any stochastic switching between time-delayed versions of $F$ with bounded
time-delays.

For $(M,X,\mu_{2})$, we have

$\displaystyle E_{\mu_{2}}[S]=\begin{bmatrix}\epsilon&9/10\\\
1/10&\epsilon\end{bmatrix}$

for which $\rho(E_{\mu_{2}}[S])=4/5<1$, so any stochastically time-delayed
version of $(M,X,\mu_{2})$ is first-mean stable, although the convergence of
the system is slower than that of $(M,X,\mu_{1})$.

Lastly, for $(M,X,\mu_{3})$ we have

$\displaystyle E_{\mu_{3}}[S]=\begin{bmatrix}\epsilon&1/2\\\
1/2&\epsilon\end{bmatrix}$

which has spectral radius $\rho(E_{\mu_{3}}[S])=1$, so stochastically time-
delayed versions of $(M,X,\mu_{3})$ are not guaranteed to be first mean
stable.

We now give a proof of Theorem 3.

###### Proof.

Let $(M,X,\mu)$ be a stochastic switched system with corresponding Lipschitz
system $(S,\mathbb{R}^{n},\mu^{\prime})$ such that the assumptions of Theorem
3 are satisfied. Let $L>0$ and let $(M_{L},\mathbb{R}^{n(L+1)},\pi)$ be any
delayed version of $(M,X,\mu)$.

We determine the stability of $(M_{L},X_{L},\mathbb{R}^{n(L+1)})$ using the
system’s corresponding Lipschitz set $S_{L}$. We construct
$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$ from $(S,\mathbb{R},\mu^{\prime})$
using Lemma 1. We then use Theorem 1 to determine the stability of
$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$, which by Theorem 2 tells us the
stability of $(M_{L},X_{L},\pi)$.

First, we use Lemma 1 to construct the Lipschitz set
$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$ of $(M_{L},X_{L},\pi)$ from the
Lipschitz set of the undelayed system, $(S,R^{n},\mu)$. We have
$S_{L}=\\{A_{D}:A\in S,D\in\mathbb{N}^{n\times n}_{L}\\}$, where each $A_{D}$
is of the form defined in Lemma 1.

Since $S_{L}$ is a linear system, we determine the 1st mean stability of
$S_{L}$ using the 1-radius (Definition 8). Furthermore, $S_{L}$ is comprised
of nonnegative matrices. The set of nonnegative vectors in $\mathbb{R}^{n}$,
i.e. $\\{\mathbf{x}\in\mathbb{R}^{n}:\mathbf{x}\succeq\mathbf{0}\\}$ is a
proper cone because it is closed, convex, solid, and pointed. Since $S_{L}$ is
a set of nonnegative Lipschitz matrices, each matrix leaves this set
invariant, and therefore satisfies the assumptions of Theorem 1. We can
therefore apply Theorem 1 to compute the 1-radius as
$\rho_{1,\pi^{\prime}}(S_{L})=\rho(E_{\pi^{\prime}}[S_{L}])$. Thus, we proceed
by computing $E_{\pi^{\prime}}[S_{L}]$.

Note that each $A_{D}\in S_{L}$ has the same block subdiagonal form (as
defined in Lemma 1). The expectation is taken entrywise, so $E[S_{L}]$ will
have the same block subdiagonal form as well, as follows:

$\displaystyle
E_{\pi^{\prime}}[S_{L}]=\begin{bmatrix}B_{0}&B_{1}&\ldots&B_{L-1}&B_{L}\\\
I_{n}&0&\ldots&0&0\\\ 0&I_{n}&\ldots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\ldots&I_{n}&0\end{bmatrix},$

where each $B_{l}$ is a $n\times n$ matrix.

$E_{\pi^{\prime}}[S_{L}]$ is now in the form required to place an upper bound
on $\rho(E_{\pi^{\prime}}[S_{L}])$ using Lemma 2, so we compute
$\sum_{\ell=0}^{L}B_{\ell}$ to apply the lemma.

Note that given any $A_{D}\in S_{L}$, we can uniquely determine which $n\times
n$ matrix $A\in S$ that $A_{D}$ came from. Thus, we define the mapping
$f:S_{L}\to S$ by $f(A_{D})=A$, where $A$ is the unique matrix in $S$ of which
$A_{D}$ is a delayed version. Since $f$ is measurable, we can also use $f$ to
define the measure $f_{*}(\pi^{\prime})$ on $S$ given by

$\displaystyle f_{*}(\pi^{\prime})(E)=\pi^{\prime}(\left\\{A_{D}\in
S_{L}:f(A_{D})\in E)\right\\})\quad\text{ for }E\in\mathscr{A}.$

What’s more, we can show that $f_{*}\pi^{\prime}=\mu^{\prime}$. For any
$E\in\mathscr{A}$, we have

$f_{*}(\pi^{\prime})(E)=\pi^{\prime}(\left\\{A_{D}\in S_{L}:f(A_{D})\in
E)\right\\})=\pi^{\prime}(\left\\{A_{D}:A\in E,D\in\mathbb{N}_{L}^{n\times
n}\right\\})=\pi^{\prime}(E_{L})=\mu^{\prime}(E).$

.

Then, using a change of variables, we have

$\displaystyle\int_{S_{L}}f(A_{D})\>d\pi^{\prime}=\int_{S}A\>df_{*}(\pi^{\prime})=\int_{S}A\>d\mu^{\prime}=E_{\mu^{\prime}}[S].$

Recall that the $i,j+n\ell$ entry of $A_{D}\in S_{L}$ is
$f(A_{D})_{ij}=a_{ij}$ if $d_{i,j}=\ell$ and 0 otherwise, where $A=[a_{ij}]$
is the unique undelayed version of $A_{D}$, and $D=[d_{ij}]$ is a delay matrix
such that delaying $A$ with $D$ results in $A_{D}$. We have

$\displaystyle\left[\sum_{\ell=0}^{L}B_{\ell}\right]_{ij}=\sum_{\ell=0}^{L}[B_{\ell}]_{ij}$
$\displaystyle=\sum_{\ell=0}^{L}\left[E_{\pi^{\prime}}[S_{L}]\right]_{i,j+n\ell}$
$\displaystyle=\sum_{\ell=0}^{L}\int_{S_{L}}[A_{D}]_{i,j+n\ell}\>d\pi^{\prime}$
$\displaystyle=\sum_{\ell=0}^{L}\int_{S_{L}}\left[f(A_{D})\right]_{ij}1_{d_{ij}=l}\>d\pi^{\prime}$
$\displaystyle=\int_{S_{L}}\left[f(A_{D})\right]_{ij}\sum_{\ell=0}^{L}1_{d_{ij}=l}\>d\pi^{\prime}$
$\displaystyle=\int_{S_{L}}\left[f(A_{D})\right]_{ij}\>d\pi^{\prime}$
$\displaystyle=\int_{S}a_{ij}\>d\mu^{\prime}$
$\displaystyle=\left[E_{\mu^{\prime}}[S]\right]_{ij}.$

Therefore $\sum_{l=0}^{L}B_{l}=E_{\mu^{\prime}}[S]$. By Lemma 2,

$\displaystyle
E_{\pi^{\prime}}[S_{L}]=\begin{bmatrix}B_{0}&B_{1}&\ldots&B_{L-1}&B_{L}\\\
I_{n}&0&\ldots&0&0\\\ 0&I_{n}&\ldots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\ldots&I_{n}&0\end{bmatrix}.$

has spectral radius less than 1 if and only if
$\sum_{l=0}^{L}B_{l}=E_{\mu^{\prime}}[S]$ has spectral radius less than 1.
Since we assumed that $(S,\mathbb{R}^{n},\mu^{\prime})$ was exponentially
stable in first mean,
$\rho_{(1,\mu^{\prime})}(S)=\rho(E_{\mu^{\prime}}[S])<1.$ We conclude that
$\rho(E_{\pi^{\prime}}[S_{L}])=\rho_{1,\pi^{\prime}}(S_{L})<1$. Thus
$(S_{L},\mathbb{R}^{n(L+1)},\pi^{\prime})$ is exponentially stable in first
mean, and by Theorem 2, so is $(M_{L},X_{L},\pi)$. That is, if we have a
stochastic switched system with a Lipschitz system that is exponentially
stable in first mean, any stochastically delayed version of that system will
maintain that stability.

Now, suppose that instead of knowing of a first mean stable Lipschitz set for
$(M,X,\mu)$, we know that there exists a stochastically delayed version of the
system $(M_{L},X_{L},\pi)$ which has a first mean stable Lipschitz set
$(S_{L},\mathbb{R}^{n(L+1)}$. Define the set

$\displaystyle S=\\{f(A_{D}):A_{D}\in S_{L}\\}$

$(S,\mathbb{R}^{n(L+1)},\mu^{\prime})$ is a Lipschitz set for $(M,X,\mu)$.
Then, if we proceed with the same process outlined in the first case of the
proof, we can determine that $E_{\mu^{\prime}}[S]<1$ if and only if
$E_{\pi^{\prime}}[S_{L}]<1$, however, we know that
$E_{\pi^{\prime}}[S_{L}]<1$, so $(S,\mathbb{R}^{n(L+1)},\mu^{\prime})$ is
exponentially stable in first mean. We can then apply the first case of the
theorem to determine that $(M,X,\mu)$ is patiently stable.

∎

\begin{overpic}[scale={.4}]{IntrinsicStabExp14.eps}
\put(20.0,-2.0){$(M,\mathbb{R}^{2},\mu)$}
\put(70.0,-2.0){$(M_{L},\mathbb{R}^{12},\pi)$} \end{overpic} Figure 6: The the
stable dynamics of the switched system $(M,\mathbb{R}^{2},\mu)$ from Example
3.4 is shown (left). Since the system is patient first mean stable then any
delayed version of the system $(M_{L},\mathbb{R}^{2(L+1)},\pi)$ for any
$L<\infty$ is also stable. Shown right is the delayed version of this system
with $L=5$.

In all examples given thus far in this paper we have considered only discrete
probability spaces. It is worth noting that our results hold for any switched
system that satisfies the conditions of Theorem 3, including those systems
that switch over a continuous set of functions. An example of this is the
following.

######  Example 4.7.

Consider the linear switched system $(M,\mathbb{R}^{2},\mu)$ where the set
$M\subset\mathbb{R}^{2\times 2}$ is given by

$\displaystyle M=\begin{bmatrix}[-0.8,0]&[0.05,0.35]\\\
[0.05,0.35]&[0,0.08]\\\ \end{bmatrix}$

where a matrix $A\in M$ has entry $a_{ij}$ drawn independently and uniformly
from the interval in the $m_{ij}$ entry of $M$. We obtain a Lipschitz matrix
for each $A\in M$ using Equation 5, giving us a Lipschitz set
$(S,\mathbb{R^{2}},\mu^{\prime})$ for $M$ where

$\displaystyle S=\begin{bmatrix}[0,0.8]&[0.05,0.35]\\\ [0.05,0.35]&[0,0.08]\\\
\end{bmatrix}$

and where $\mu^{\prime}$ is similarly the distribution on $S$ where each
element is independently and uniformly distributed on the given interval.

To calculate the one-radius of $S$, we first take the entrywise expectation to
get

$\displaystyle E[S]=\begin{bmatrix}0.4&0.2\\\ 0.2&0.04\\\ \end{bmatrix}.$

As $\rho_{1}(S)=\rho(E[S])\approx 0.489<1$, the switched system $(M,X,\mu)$ is
first-mean stable. Furthermore, by Theorem 3, $M$ is also patiently first mean
stable meaning that any stochastically delayed version of $(M,X,\mu)$ will
also be exponentially stable in first mean (see Figure 6).

We now give a proof that the diagram in Figure 5 commutes.

###### Proof.

We prove that the diagram shown in Figure 5 commutes for a stochastic switched
system $(M,X,\mu)$ in the case where each $F\in M$ is differentiable. In other
words, we want to show that given a switched system, it doesn’t matter what
order in which you delay and ’linearize’ the system by creating the Lipschitz
set; the result is the same ($\Lambda\circ\tau=\tau\circ\Lambda$). Because we
are considering the case where each $F\in M$ is differentiable, we can use
Equation 5 to define the linearization operator $\Lambda$.

We define $\Lambda$ as

$\Lambda(F)=A=[a_{ij}]\quad\text{ where }\quad a_{ij}=\sup_{x\in
X}\left\lvert\frac{\partial F_{i}}{\partial x_{j}}(\mathbf{x})\right\rvert.$
(9)

The diagram still commutes even when some $F$ is not differentiable, however,
the operator $\Lambda$ will be a different bijective function.

We also define the delay operator $\tau_{L}$ as in Definition 3, i.e.,
$\tau_{L}(F)=F_{D}$ where $F_{D}$ is the equation defined in Definition 3.

Let $F\in M$. First, we calculate $\Lambda\circ\tau_{L}(F)$, which corresponds
to delaying the system and then computing the Lipschitz linearization.

Recall that for $1\leq l\leq L,1\leq i\leq n$, we have
$(F_{D})_{i,l}(\mathbf{x})=x_{i,l-1}.$ So for $1\leq l\leq L,1\leq i\leq
n,0\leq m\leq L,1\leq j\leq n$, we have

$\displaystyle[A_{D}]_{(i,l),(j,m)}=\sup_{x\in X}\left\lvert\frac{\partial
F_{D\>i,l}}{\partial x_{j,m}}(\mathbf{x})\right\rvert$
$\displaystyle=\begin{cases}1\text{ if $i=j,m=l-1$}\\\ 0\text{
otherwise}.\end{cases}$

Now, for $\ell=0$, we have
$(F_{D})_{i,0}(\mathbf{x_{\ell}})=F_{i}(x_{1,d_{i1}},x_{2,d_{i2}},\ldots,x_{n,d_{in}})$.

Then

$\displaystyle[A_{D}]_{(i,0),(j,m)}=\sup_{x_{\ell}}\in
X_{\ell}\left\lvert\frac{\partial F_{D\>i,0}}{\partial
x_{j,m}}(\mathbf{x_{\ell}})\right\rvert$ $\displaystyle=\sup_{x\in
X}\left\lvert\frac{\partial F_{i}}{\partial
x_{j,m}}(x_{1,d_{i1}},x_{2,d_{i2}},\ldots,x_{n,d_{in}})\right\rvert$
$\displaystyle=\begin{cases}a_{ij}\text{ if $m=d_{ij}$}\\\ 0\text{
otherwise}\end{cases}.$

Now, we calculate $\tau_{L}\circ\Lambda(F)$, which corresponds to linearizing
the system first and then delaying.

Let $A=\Lambda(F)$, and $A_{D}$ = $\tau_{L}(A)$. We have

$\displaystyle(A_{D})_{i,\ell}(\mathbf{x_{\ell}})=x_{i,\ell-1}\quad\text{ for
}\quad 1\leq\ell\leq L,\,1\leq i\leq n$

therefore

$\displaystyle[A_{D}]_{(i,l)(j,m)}=\begin{cases}1\text{ if $i=j,m=l-1$}\\\
0\text{ otherwise}\end{cases}.$

And for $l=0$, we have

$\displaystyle(A_{D})_{i,0}(\mathbf{x_{\ell}})=A_{i}(x_{1,d_{i1}},x_{2,d_{i2}},\ldots,x_{n,d_{in}})=a_{i1}x_{1,d_{i1}}+a_{i2}x_{2,d_{i2}}+\ldots+a_{in}x_{n,d_{in}}$

so

$\displaystyle[A_{D}]_{(i,0),(j,m)}=\begin{cases}a_{ij}\text{ if
$m=d_{ij}$}\\\ 0\text{ otherwise }\end{cases}.$

Therefore $\tau_{L}\circ\Lambda(F)=\Lambda\circ\tau_{L}(F)$. ∎

##  5 Conclusion

In this paper we defined the notion of patient first-mean stability for
nonlinear switched systems, and presented a computationally efficient means of
checking this form of stability. It is our hope that practitioners of control
and systems theory will apply this theory to novel applications. We expect
that these results will greatly simplify the stability analysis of
stochastically switched, time-delayed systems.

A significant research question which remains open is whether the requirement
that switching is iid. at each time step can be weakened. For example, it
seems reasonable that similar computationally efficient criteria might exist
for the patient $p^{th}$ mean stability of Markov systems, we look forward to
further developments in this area. We also encourage the incorporation of this
analysis into open-source software, automating the construction of the
Lipschtiz set $S$ and the expectation matrix $E_{\mu^{\prime}}[S]$ for a given
stochastic switched system $(M,X,\mu)$.

##  6 References

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