# On the spectral radius properties of a key matrix in periodic impulse
control

Swati Patel

Department of Mathematics

Oregon State University

Corvallis, OR 97331

Patrick De Leenheer

Department of Mathematics

Oregon State University

Corvallis, OR 97331

## Abstract

In this work, we consider the periodic impulse control of a system modeled as
a set of linear differential equations. We define a matrix that governs the
qualitative behavior of the controlled system. This matrix depends on the
period and effects of the control interventions. We investigate properties of
the spectral radius of this matrix and in particular, how it depends on the
period of the interventions. Our main result is on the convexity of the
spectral radius with respect to this period. We discuss implications of this
convexity on establishing an optimal and maximum period for effective control.
Finally, we provide an example motivated from a real-life scenario.

## Keywords

Impulse differential equations, spectral radius, matrix perturbations,
biological control, convexity

##  1 Introduction

The main result of this paper is on the convexity of the spectral radius of
the matrix $D\operatorname{e}^{\tau A}$, with respect to the scalar $\tau$
under certain conditions on the matrices $D$ and $A$. This matrix arises
naturally from considering a linear system with $\tau$-periodic impulse
control and its spectral radius allows us to distinguish between distinct
qualitative behaviors. Hence, understanding the behavior of the spectral
radius with respect to $\tau$ is important for identifying effective and
efficient control strategies.

The convexity of the spectral radius (or eigenvalues) of a given matrix with
respect to specific elements or matrix parameters has been a topic of
mathematical interest and examined in other contexts. [4] showed that the
spectral radius of a non-negative matrix is convex with respect any single
diagonal element and provides applications in demography. These results were
extended to show convexity of the spectral radius of non-negative matrices
with respect to all diagonal elements by [5], [6], and [8], each using
distinct approaches which rely on the Feynman-Kac formula, Trotter product
forumla [12], and the Donsker-Varadhan variational formula [7], respectively.
Additionally, [8] showed the log convexity of the spectral radius of
$\operatorname{e}^{D}A$ for non-negative matrices $A$ with respect to diagonal
matrices $D$. While these previous convexity results focused on non-negative
matrices, our results consider a different subset of matrices, namely
diagonally symmetrizable matrices, and we focus on convexity with respect to a
different parameter.

Our problem was motivated from ongoing control interventions of a real
biological scenario: massive drug administration (MDA) practices to eliminate
parasitic helminth diseases. For this scenario, we have a linear population
model, structured into discrete spatial/life classes, in which the control
interventions impact the classes differently. Additionally, we want to know
how often to implement control interventions to drive both harmful classes to
zero.

Parasitic helminths are a class of worms that infect and inflict harm on their
hosts, including humans [16] and several animal species of agricultural
importance [9]. These parasites typically spend part of their life cycle in an
environment external to the hosts developing before finding their way into a
host, where they reproduce and release offspring back into the external
environment. Hence, there is exchange between these two spatial locations. In
the 1980s, public health organizations began implementing MDA, the preemptive
distribution of anthelmintic drugs to populations where the disease was
thought to be prevalent. The World Health Organization has developed
guidelines that recommend MDA periodically 1-2 times a year depending on the
prevalence of the disease [15]. As drugs are given to hosts, this control
intervention kills a subset of the parasitic population within hosts, while
the population within the external environment is untouched.

This example motivated us to examine the spectral radius of the matrix
$D\operatorname{e}^{\tau A}$, where $D$ is a diagonal matrix that captures the
impact of the impulse control intervention on the distinct classes, $A$ is a
matrix that represents the continual transitions between classes coming from
the linear model, and $\tau$ is the period of the control interventions. In
the following section, we provide more details on the general setting and the
extraction of this matrix as well as some assumptions. In section 3, we give
the main convexity result. In section 4, we give implications of the main
result, in particular as they relate to the practical applications. In section
5, we return to our motivating example and provide insight on their control by
directly applying our result. Finally, in section 6, we frame these results in
a broader context and provide some open questions.

##  2 Setting

###  2.1 Model

Let $x(t)$ be the vector of $N$ state variables at time $t$, which are
governed naturally by a linear differential equation, captured in the $N\times
N$ matrix $A$. It is well-known that if $A$ has an eigenvalue with positive
real part, then the system is unstable and most solutions will grow unbounded
as $t\to+\infty$.

Periodic impulsive control is a specific control protocol in which control
interventions occur at periodically spaced times $n\tau$, where $n$ is a non-
negative integer, and where $\tau>0$ is the period of the control. Here we
shall consider the scenario where, at the control instants, the current state
of the system is modified multiplicatively and instantaneously. Altogether, we
express the dynamics of $x$ as the following impulse differential equation:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=Ax\hskip 56.9055ptt\not\in n\tau$
(1) $\displaystyle x(n\tau^{+})$ $\displaystyle=Dx(n\tau^{-})$

where $x(n\tau^{+})$ and $x(n\tau^{-})$ denote right and left limits of $x(t)$
as $t$ approaches the control time $n\tau$ from the right and left
respectively. Here, $D$ is a fixed $N\times N$ diagonal matrix with positive
entries, which often take values in the interval $(0,1]$. In this case, the
control amounts to re-initializing the state components $x_{i}$ to a certain
fraction $D_{ii}$ of their current value. A state component $x_{i}$ is
uncontrolled if $D_{ii}=1$. Some of the $D_{ii}$ could coincide, but they
could also all be distinct.

The main control objective is to stabilize the zero solution of the controlled
system. We will soon show that this happens if the spectral radius of the
matrix111The spectral radius of a matrix $A$ is defined as
$\max\\{|\lambda|\,|\,\lambda\textrm{ is an eigenvalue of }A\\}$.
$D\operatorname{e}^{\tau A}$, which we denote $r(D\operatorname{e}^{\tau A})$,
is less than one. Note that there are two control parameters to achieve this
goal: the period $\tau$ between consecutive control interventions, and the
diagonal matrix $D$ which encodes how much the state is affected by the
control. Thus, we are interested in understanding the behavior of the map
$(\tau,D)\to r(D\operatorname{e}^{\tau A})$, and in particular whether the
range of this map includes values below $1$. A secondary control objective
might be to minimize $r(D\operatorname{e}^{\tau A})$, especially when the aim
is to stabilize the system as efficiently as possible, i.e., optimize the
control.

In this paper, we focus on the restricted scenario in which the matrix $D$ is
fixed, and only the period $\tau$ is variable. In other words, we shall
consider the map $\tau\to r(D\operatorname{e}^{\tau A})$, assuming that $D$
(and of course $A$) is fixed. In many practical applications, $D$ is indeed
fixed. For example, some of the diagonal entries of $D$ might reflect
efficiencies of existing drugs used to treat a disease, or the potencies of
insecticides used to protect an agricultural crop. In such cases, the only
control parameter available for tuning is the period $\tau$. For scenarios
where $\tau$ is fixed, but $D$ is tunable, we refer to the theory developed in
[8, 6] which shows how $r(D\operatorname{e}^{\tau A})$ varies qualitatively
with respect to $D$.

To conclude this section, we discuss how the periodic impulsive control
formalism can be analyzed by means of Floquet’s theory [2] and lead us to
consider $r(D\operatorname{e}^{\tau A})$. We will show that the behavior of
the impulsively controlled model (1) can also be described by that of the
periodically time-varying system

${\dot{y}}=B(t)y,$

where

$B(t)=\begin{cases}\ln(D),\textrm{ if }t\in[n\tau,n\tau+1)\\\ A,\textrm{ if
}t\in[n\tau+1,(n+1)\tau+1)\end{cases}$

for all non-negative integers $n$, and $\ln(D)$ is a diagonal matrix defined
by $[\ln D]_{ii}=\ln(D_{ii})$ for all $i=1,\dots,N$. Note that $B(t)$ is
periodic with period $\tau+1$, and piece-wise constant with time $t$.

We have replaced the instantaneous control map which maps the state
$x(n\tau-)$ of the impulsively controlled system to the state
$x(n\tau+)=Dx(n\tau-)$, by a continuous-time system ${\dot{y}}=\ln(D)y$ which
acts on the time intervals $[n\tau,n\tau+1)$. During these intervals, the
state $y(n\tau)$ evolves to the state
$y(n\tau+1)=\operatorname{e}^{\ln(D)}y(n\tau)=Dy(n\tau)$. Therefore, the image
of the state $y(n\tau)$ to $y(n\tau+1)$ for the above periodically time-
varying system is the same as the map of $x(n\tau-)$ to $x(n\tau+)=Dx(n\tau-)$
for the impulsively controlled system. Also, note that if $x(t)$ is a solution
of the impulsively controlled system, defined on some interval
$[n\tau,(n+1)\tau]$, then $x(t)$ coincides with a solution $y(t)$ of the
system ${\dot{y}}=B(t)y$, but defined on the interval $[n\tau+1,(n+1)\tau+1]$,
whenever the initial condition of both functions coincides. Indeed, on these
intervals, both functions are solutions of the same linear system
${\dot{z}}=Az$, which is known to have unique solutions for every initial
condition.

The advantage is that we now can appeal to Floquet’s theory to analyze the
periodic linear system ${\dot{y}}=B(t)y$. It has the following principal
fundamental matrix solution

$\phi(t)=\begin{cases}\operatorname{e}^{t\ln(D)},\textrm{ if }t\in[0,1)\\\
\operatorname{e}^{(t-1)A}D,\textrm{ if }t\in[1,1+\tau]\end{cases},$

and the monodromy matrix associated to $\phi(t)$ is

$\phi^{-1}(0)\phi(\tau+1)=\operatorname{e}^{\tau A}D.$

The eigenvalues of this monodromy matrix are the characteristic multipliers.
If all characteristic multipliers have modulus less than $1$, or
equivalently222For any pair of $N\times N$ matrices $A$ and $B$, the
eigenvalues, and hence the spectral radius, of $AB$ and $BA$ coincide., if
$r(\operatorname{e}^{\tau A}D)=r(D\operatorname{e}^{\tau A})<1$, then the zero
solution of ${\dot{y}}=B(t)y$ is asymptotically stable. In this case, the zero
solution of the original impulsively controlled system (1) is asymptotically
stable as well. Hence, we are motivated to understand the properties of
$r(D\operatorname{e}^{\tau A})$.

###  2.2 Definitions and Preliminaries

To set up our main result, we provide some definitions and useful
preliminaries. We begin with a brief discussion of diagonally symmetrizable
matrices and then give three different notions of convexity, which we use in
our main results.

######  Definition 1.

A real $N\times N$ matrix $A$ is diagonally symmetrizable 333Our definition
and terminology differ slightly from those in [10], but it is not hard to show
that both definitions are equivalent. if there exists a real, invertible
diagonal matrix $T$ such that $T^{-1}AT$ is a symmetric matrix.

As a diagonally symmetrizable matrix is similar to a symmetric matrix, all its
eigenvalues must be real. This is a strong restriction on such matrices. On
the other hand, we will see that they can be characterized in an elegant way,
enabling us to recognize them fairly easily.

For any real number $x$ we define the sign function

$\textrm{sgn}(x)=\begin{cases}+1,\textrm{ if }x>0\\\ 0,\textrm{ if }x=0\\\
-1,\textrm{ if }x<0\end{cases}$

######  Definition 2.

A real $N\times N$ matrix $A$ is sign-symmetric if
$\textrm{sgn}(A_{ij})=\textrm{sgn}(A_{ji})$, for all $i\neq j$.

It follows from the definition that if $A$ is diagonally symmetrizable, then
there is an invertible diagonal matrix $T$ such that

$A_{ij}=A_{ji}\left(\frac{T_{ii}}{T_{jj}}\right)^{2},\textrm{ for all }i\neq
j.$ (2)

This implies that $\textrm{sgn}(A_{ij})=\textrm{sgn}(A_{ji})$, for all $i\neq
j$, and therefore $A$ is sign-symmetric. Furthermore, for any distinct
elements $i_{1},\dots,i_{k}$ in $\\{1,\dots,N\\}$, it follows from
$(\ref{balance})$ that

$A_{i_{1}i_{2}}A_{i_{2}i_{3}}\dots
A_{i_{k-1}i_{k}}A_{i_{k}i_{1}}=A_{i_{1}i_{k}}A_{i_{k}i_{k-1}}\dots
A_{i_{3}i_{2}}A_{i_{2}i_{1}},$ (3)

known as the cycle condition [10]. To justify this terminology we offer a
simple graphical interpretation: Associating a directed and weighted graph to
$A$ by drawing an edge from vertex $i$ to vertex $j$ with weight $A_{ij}$, the
cycle condition $(\ref{cycle})$ expresses that in any directed cycle $i_{1}\to
i_{2}\to\dots\to i_{k-1}\to i_{k}\to i_{1}$ in this graph, the weight of the
cycle (defined as the product of the weights of all the directed edges in the
cycle) equals the weight of the reversed cycle $i_{1}\to i_{k}\to
i_{k-1}\to\dots i_{2}\to i_{1}$.

Remarkably, it turns out that if a sign-symmetric matrix satisfies
$(\ref{cycle})$, then it is diagonally symmetrizable [10]. In summary, the
following elegant characterization of diagonally symmetrizable matrices is
available.

######  Proposition 1.

(Proposition 5.15 in [10]) A real $N\times N$ matrix $A$ is diagonally
symmetrizable if and only if $A$ is sign-symmetric and satisfies the cycle
condition $(\ref{cycle})$.

######  Definition 3.

An $N\times N$ matrix $A$ is tri-diagonal if $A_{ij}=0$ for all $|i-j|>1$.

Proposition 1 immediately implies

######  Corollary 1.

Every sign-symmetric, tri-diagonal matrix is diagonally symmetrizable.

The following provides a sense of how restrictive it is for a matrix to be
diagonally symmetrizable. It turns out that it is not very restrictive for
matrices of size two, but is restrictive for matrices of size larger than two,
in a sense made precise below.

  * •

We say that a real $N\times N$ matrix is strictly cooperative444Cooperative
matrices are also known as Metzler matrices, especially in control theory.
(respectively strictly competitive) if $A_{ij}>0$ (respectively $A_{ij}<0$)
for all $i\neq j$. Note that being a strictly cooperative (or competitive)
matrix is an open condition: Any strictly cooperative (or competitive) matrix
has a neighborhood of strictly cooperative (competitive) matrices, under a
standard topology.

Strictly cooperative and competitive matrices frequently occur in the analysis
of biological and chemical systems.

Corollary 1 implies that all strictly cooperative and all strictly competitive
$2\times 2$ matrices are diagonally symmetrizable.

  * •

Assume that $A$ is a $3\times 3$ matrix which is sign-symmetric. For instance,
$A$ may be strictly cooperative or competitive, although it doesn’t have to be
of either type; for example, $A_{12}$ and $A_{21}$, and $A_{13}$ and $A_{31}$
could all be positive, while $A_{23}$ and $A_{32}$ could be negative.

Proposition 1 then implies that $A$ is diagonally symmetrizable if and only if

$A_{12}A_{23}A_{31}=A_{13}A_{32}A_{21}.$

It follows that being a diagonally symmetrizable $3\times 3$ matrix is non-
generic. Indeed, every neighborhood of such a matrix contains matrices which
violate the above cycle condition.

For more on properties and the structure of diagonally symmetrizable matrices,
see [10].

To state our main convexity results, we define three commonly used notions of
convexity; further relevant properties of these notions are reviewed in the
Appendix.

######  Definition 4.

Let $C$ be a convex set in $\mathbb{R}^{N}$. A function
$f:C\rightarrow\mathbb{R}$ is

  * •

convex if for all $x_{1},x_{2}\in C$ and $\lambda\in[0,1]$

$f(\lambda x_{1}+(1-\lambda)x_{2})\leq\lambda f(x_{1})+(1-\lambda)f(x_{2})$

  * •

strictly convex if for all $x_{1}\neq x_{2}\in C$ and $\lambda\in(0,1)$

$f(\lambda x_{1}+(1-\lambda)x_{2})<\lambda f(x_{1})+(1-\lambda)f(x_{2})$

  * •

strongly convex with parameter $m>0$ if $f(x)-\frac{m}{2}||x||^{2}$ is convex.

Here, $||x||$ denotes the Euclidean norm of the vector $x$. These
characterizations are ordered from weakest to strongest. It is easy to see
that strict convexity implies convexity and that strong convexity implies
strict convexity. None of the converse implications hold. For example, if
$f:\mathbb{R}\to\mathbb{R}$ is defined as $f(x)=x$, then $f$ is convex, but
not strictly convex. If $f:(-1,1)\to\mathbb{R}$ is defined as $f(x)=x^{4}$,
then $f$ is strictly convex, but we show in the Appendix that $f$ is not
strongly convex for any positive parameter $m$.

##  3 Main Results

We are interested in the properties, including the convexity, of the map
$\tau\rightarrow r(De^{\tau A})$, because of its implications for identifying
how frequently one needs to intervene to effectively and efficiently control
the state variables.

The main technical result of this paper is as follows.

######  Theorem 1.

Assume that

  1. 1.

$D$ is an $N\times N$ diagonal matrix with positive diagonal entries, and

  2. 2.

$A$ is a diagonally symmetrizable $N\times N$ matrix.

Then the map $\tau\to r(D\operatorname{e}^{\tau A})$, is convex for
$\tau\in[0,+\infty)$.

If in addition, $A$ is non-singular, then for every $p>0$, there exists some
$m_{p}>0$ such that the map $\tau\to r(D\operatorname{e}^{\tau A})$ is
strongly convex with parameter $m_{p}$ for $\tau\in[0,p]$.

This result is easily shown when $D_{ii}=D_{jj}$ for all $i=1,...,N$ and
$j=1,...,N$. In this case,

$r(D\operatorname{e}^{\tau A})=D_{11}e^{\tau\lambda_{\max}(A)}$

where for any diagonally symmetrizable matrix $S,\lambda_{\max}(S)$ denotes
the largest eigenvalue of $S$. (Recall that since $S$ is diagonally
symmetrizable, all eigenvalues of $S$ are real.) Moreover, if $A$ is non-
singular, then $\lambda_{\max}(A)\neq 0$ and this map is either a strictly
increasing or strictly decreasing exponential, which is strongly convex on
every compact interval $[0,p]$. The more interesting case is when the diagonal
elements of $D$ are not all equal (from a practical stand point, when the
control intervention impacts different classes differently). The proof is as
follows.

###### Proof.

  * •

First we will show that $\tau\to r(D\operatorname{e}^{\tau A})$ is convex for
$\tau\in[0,+\infty)$.

Since $A$ is diagonally symmetrizable, there is a real diagonal and invertible
matrix $T$ such that the matrix ${\tilde{A}}:=T^{-1}AT$, is symmetric. Then
$D\operatorname{e}^{\tau A}$ is similar to

$T^{-1}D\operatorname{e}^{\tau A}T=D\operatorname{e}^{\tau{\tilde{A}}},$

where the equality holds because $T^{-1}$ and $D$ commute as both are
diagonal. Let $D^{1/2}$ be the unique, diagonal square root of $D$. Then
$D\operatorname{e}^{\tau{\tilde{A}}}$, and hence also $D\operatorname{e}^{\tau
A}$, is similar to

$D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2},$

which is a symmetric matrix. Furthermore, we claim that
$D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}$ is positive definite. To
see why, let $\lambda_{i}$, $i=\dots,N$, be the eigenvalues of ${\tilde{A}}$
(and also of $A$). Note that since ${\tilde{A}}$ is symmetric, all its
eigenvalues are real and additionally, $\operatorname{e}^{\tau{\tilde{A}}}$ is
also symmetric. By the Spectral Mapping Theorem,
$\operatorname{e}^{\tau\lambda_{i}}$, $i=1,\dots,N$, are the eigenvalues of
$\operatorname{e}^{\tau{\tilde{A}}}$ and they clearly are all positive for all
$\tau\in[0,\infty)$. Thus, $\operatorname{e}^{\tau{\tilde{A}}}$ is positive
definite for all $\tau\in[0,\infty)$. But then
$D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}$ is also positive definite,
as claimed. Consequently,

$r(D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2})=\lambda_{\max}(D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}),$

Since $D\operatorname{e}^{\tau A}$ is similar to
$D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}$, it follows that

$r(D\operatorname{e}^{\tau
A})=\lambda_{\max}(D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}).$

By the Raleigh quotient formula,

$r(D\operatorname{e}^{\tau A})=\max_{x:\langle x,x\rangle=1}\langle
x,D^{1/2}\operatorname{e}^{\tau{\tilde{A}}}D^{1/2}x\rangle,$

where $\langle x,y\rangle$ denotes the Euclidean inner product of any two
vectors $x$ and $y$ in $\mathbb{R}^{N}$.

Since ${\tilde{A}}$ is symmetric, the Spectral Theorem implies that there
exists an orthogonal matrix $Q$ (i.e., $QQ^{T}=Q^{T}Q=I$) and a diagonal
matrix $\Lambda$ such that

$Q^{T}{\tilde{A}}Q=\Lambda,\textrm{ and
}\Lambda=\begin{pmatrix}\lambda_{1}&0&\dots&0\\\ 0&\lambda_{2}&\dots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\dots&\lambda_{n}\end{pmatrix},$

where the $\lambda_{i}$ are the (real) eigenvalues of ${\tilde{A}}$ (and also
of $A$). Inserting ${\tilde{A}}=Q\Lambda Q^{T}$ in the above Raleigh quotient
formula yields

$\displaystyle r(D\operatorname{e}^{\tau A})$ $\displaystyle=$
$\displaystyle\max_{x:\langle x,x\rangle=1}\langle
x,D^{1/2}Q\operatorname{e}^{\tau\Lambda}Q^{T}D^{1/2}x\rangle$ $\displaystyle=$
$\displaystyle\max_{x:\langle x,x\rangle=1}\langle
Q^{T}D^{1/2}x,\operatorname{e}^{\tau\Lambda}Q^{T}D^{1/2}x\rangle$
$\displaystyle=$ $\displaystyle\max_{y:\langle
D^{-1/2}Qy,D^{-1/2}Qy\rangle=1}\langle
y,\operatorname{e}^{\tau\Lambda}y\rangle$ $\displaystyle=$
$\displaystyle\max_{y:\langle y,Q^{T}D^{-1}Qy\rangle=1}\langle
y,\operatorname{e}^{\tau\Lambda}y\rangle.$

For each fixed $y$ in the constraint set $\\{y\,|\,\langle
y,Q^{T}D^{-1}Qy\rangle=1\\}$, the function

$\tau\to\langle
y,\operatorname{e}^{\tau\Lambda}y\rangle=\sum_{i=1}^{N}\operatorname{e}^{\lambda_{i}\tau}y_{i}^{2}$

is a linear combination of exponential functions with non-negative weights,
hence it is a convex function for $\tau\in[0,\infty)$. Since the function
$\tau\to r(D\operatorname{e}^{\tau A})$, where $\tau\in[0,\infty)$, is a
point-wise maximum of a collection of convex functions, it is also a convex
function by Theorem 7 in the Appendix.

  * •

Next we show that if in addition, $A$ is non-singular, then for any fixed
$p>0$, the map $\tau\to r(D\operatorname{e}^{\tau A})$ is strongly convex with
some parameter $m_{p}>0$, for $\tau\in[0,p]$.

We have shown that

$r(D\operatorname{e}^{\tau A})=\max_{y:\langle
y,Q^{T}D^{-1}Qy\rangle=1}\langle y,\operatorname{e}^{\tau\Lambda}y\rangle,$

but here, the diagonal matrix $\Lambda$ is non-singular because its diagonal
entries are the eigenvalues of $A$, which is non-singular by assumption.

Fix $p>0$. We will first show that there exists some $m_{p}>0$ such that for
every $y$ in the constraint set $\\{y\,|\,\langle y,Q^{T}D^{-1}Qy\rangle=1\\}$
(this set is an ellipsoid because $D$ is a diagonal, positive-definite
matrix), the map $f_{y}(\tau):=\langle
y,\operatorname{e}^{\tau\Lambda}y\rangle$, has the following property:

$f_{y}^{\prime\prime}(\tau)\geq m_{p},\textrm{ for all }\tau\textrm{ in
}[0,p].$

To see why, we calculate $f^{\prime\prime}_{y}$ for every $y$ in the
constraint set:

$f^{\prime\prime}_{y}(\tau)=\sum_{i=1}^{N}\lambda_{i}^{2}\operatorname{e}^{\lambda_{i}\tau}y_{i}^{2}.$

Since all the $\lambda_{i}$ are non-zero, we have that for all $\tau$ in
$[0,p]$

$\displaystyle f^{\prime\prime}_{y}(\tau)$ $\displaystyle\geq$
$\displaystyle\sum_{i:\,\lambda_{i}<0}\lambda_{i}^{2}\operatorname{e}^{\lambda_{i}p}y_{i}^{2}+\sum_{i:\,\lambda_{i}>0}\lambda_{i}^{2}y_{i}^{2},$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{N}b_{i}^{2}y_{i}^{2},$

where for all $i=1,\dots,N$, we defined the positive numbers

$b^{2}_{i}=\begin{cases}\lambda_{i}^{2}\operatorname{e}^{\lambda_{i}p},\textrm{
if }\lambda_{i}<0\\\ \lambda_{i}^{2},\textrm{ if }\lambda_{i}>0\end{cases}.$

The quadratic form $y\to\sum_{i=1}^{N}b_{i}^{2}y_{i}^{2}$ is clearly positive
definite, hence it achieves its minimum $m_{p}$ over the compact constraint
set $\\{y\,|\,\langle y,Q^{T}D^{-1}Qy\rangle=1\\}$. And as the latter set does
not contain zero, this minimum $m_{p}$ is positive.

To summarize, we have shown that for each fixed $p>0$, there exists a positive
$m_{p}$ such that $f^{\prime\prime}_{y}(\tau)\geq m_{p}$, for all $\tau$ in
$[0,p]$, and for all $y$ in $\\{y\,|\,\langle y,Q^{T}D^{-1}Qy\rangle=1\\}$.
Theorem 6 in the Appendix implies that for every $y$ in $\\{y\,|\,\langle
y,Q^{T}D^{-1}Qy\rangle=1\\}$, the function $f_{y}(\tau)$ is strongly convex
with parameter $m_{p}$, for $\tau$ in $(0,p)$ and thus also for $\tau$ in
$[0,p]$. Theorem 7 in the Appendix then implies that the map $\tau\to
r(D\operatorname{e}^{\tau A})$ is strongly convex with parameter $m_{p}$, for
$\tau$ in $[0,p]$.

∎

Remark: We have seen that $\tau\to r(D\operatorname{e}^{\tau A})$ is convex
for $\tau\in[0,\infty)$ whenever $D$ is a diagonal matrix with positive
diagonal entries, and $A$ is diagonally symmetrizable. As mentioned earlier,
the latter condition on $A$ implies that $A$ must have real eigenvalues. Here
we give an example of a case where $A$ does not have real eigenvalues and
which is such that $\tau\to r(D\operatorname{e}^{\tau A})$ is not convex for
$\tau\in[0,\infty)$.

Let

$D=\begin{pmatrix}1&0\\\ 0&d\end{pmatrix}\textrm{ and
}A=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$

for some $d>0$. Then

$D\operatorname{e}^{\tau A}=\begin{pmatrix}\cos\tau&-\sin\tau\\\
d\sin\tau&d\cos\tau\end{pmatrix},$

and the eigenvalues of $D\operatorname{e}^{\tau A}$ are

$\lambda_{1,2}(D\operatorname{e}^{\tau
A})=\begin{cases}\frac{1+d}{2}\left(1\pm\sqrt{\cos^{2}\tau-4d/(d+1)^{2}}\right),\textrm{
if }\cos^{2}\tau\geq\frac{4d}{(d+1)^{2}}\\\ \frac{1+d}{2}\left(1\pm
i\sqrt{4d/(d+1)^{2}-\cos^{2}\tau}\right),\textrm{ if
}\cos^{2}\tau<\frac{4d}{(d+1)^{2}}\end{cases},$

which implies that

$r(D\operatorname{e}^{\tau
A})=\begin{cases}\frac{1+d}{2}\left(1+\sqrt{\cos^{2}\tau-4d/(d+1)^{2}}\right),\textrm{
if }\cos^{2}\tau\geq\frac{4d}{(d+1)^{2}}\\\
\frac{1+d}{2}\sqrt{\sin^{2}\tau+4d/(1+d)^{2}},\textrm{ if
}\cos^{2}\tau<\frac{4d}{(d+1)^{2}}\\\ \end{cases}$

Claim: Assume that $d\neq 1$, and fix some $p\in(0,\pi/4)$ such that
$p<\arccos(2\sqrt{d}/(d+1))$. Then $r(D\operatorname{e}^{\tau A})$ is strongly
concave for $\tau$ in $[0,p]$ (a function $f$ is strongly concave if $-f$ is
strongly convex).

Indeed, note that for $\tau\in[0,p]$, $r(D\operatorname{e}^{\tau
A})=f(g(\tau))$, where

$g(\tau)=\cos^{2}\tau-\frac{4d}{(d+1)^{2}}\textrm{ and
}f(x)=\frac{1+d}{2}\left(1+\sqrt{x}\right).$

Also note that $g$ is positive, decreasing and strongly concave on $[0,p]$
because $g(\tau)\geq g(p)>0$, $g^{\prime}(\tau)=-\sin(2\tau)<0$, and
$g^{\prime\prime}(\tau)=-2\cos(2\tau)\leq-2\cos(2p)<0$ for $\tau\in[0,p]$.
Furthermore, for $x\in[g(p),g(0)]$, we have that
$f^{\prime}(x)=(1+d)/(4\sqrt{x})\geq(1+d)/(4g(p))>0$ and
$f^{\prime\prime}(x)=-(1+d)/(8x^{3/2})\leq-(1+d)/(8(g(0))^{3/2})<0$.

Since for any twice continuously differentiable functions $f$ and $g$ holds
that

$\frac{d^{2}}{d\tau^{2}}\left(f(g(\tau))\right)=f^{\prime\prime}(g(\tau))(g^{\prime}(\tau))^{2}+f^{\prime}(g(\tau))g^{\prime\prime}(\tau),$

the above estimates imply that there is some $m>0$ such that

$\frac{d^{2}}{d\tau^{2}}\left(r(D\operatorname{e}^{\tau
A})\right)=\frac{d^{2}}{d\tau^{2}}\left(f(g(\tau))\right)\leq-m,\textrm{ for
all }\tau\in(0,p),$

and thus Theorem 6 in the Appendix implies that $r(D\operatorname{e}^{\tau
A})$ is strongly concave on $[0,p]$, as claimed.

##  4 Implications

Our main convexity result provides useful information on the behavior of the
map $\tau\to r(D\operatorname{e}^{\tau A})$ and sets us up to find an upper
bound and optimal $\tau$ that ensures 0 is asymptotically stable for
$(\ref{model})$. The convexity result along with an understanding of the
boundary behavior of this map allows us to classify the behavior of the map
$\tau\to r(D\operatorname{e}^{\tau A})$ based on straight forward properties
of $A$ (Figure 1).

The following Lemma 1 and 2 give a description of the boundary behavior.

######  Lemma 1.

Assume that

  1. 1.

$D$ is diagonal with positive diagonal entries and that there is a unique $k$
such that $D_{kk}>D_{ii}$ for all $i\neq k$, and

  2. 2.

$A$ is a real $N\times N$ matrix.

Then $\tau\to r(D\operatorname{e}^{\tau A})$ is continuously differentiable
for all sufficiently small $\tau$. Furthermore,

$\frac{d}{d\tau}\left(r(D\operatorname{e}^{\tau
A}\right)|_{\tau=0}=D_{kk}A_{kk}.$

Consequently, if $A_{kk}<0$ (respectively $A_{kk}>0$), then
$r(D\operatorname{e}^{\tau A})$ is strictly decreasing (respectively strictly
increasing) for all sufficiently small $\tau$.

###### Proof.

Note that the map $\tau\to B(\tau):=D\operatorname{e}^{\tau A}$ is
continuously differentiable for all $\tau$ in $\mathbb{R}$. Furthermore,
$B(0)=D$ has a simple eigenvalue $D_{kk}=r(B(0))$ which is strictly larger
than all other eigenvalues of $D$. This dominant eigenvalue has corresponding
left and right eigenvectors $u(0)=e_{k}$ and $v(0)=e_{k}$ (where $e_{k}$ is
the $k$th standard basis vector in $\mathbb{R}^{N}$) respectively, i.e.
$B(0)u(0)=r(B(0))u(0)$ and $B^{T}(0)v(0)=r(B(0))v(0)$. It follows from the
Implicit Function Theorem that there exists some $\epsilon>0$ such that for
all $\tau$ with $|\tau|<\epsilon$, $r(B(\tau))$ is continuously
differentiable, and that there exist differentiable vectors $u(\tau)$ and
$v(\tau)$ such that

$B(\tau)u(\tau)=r(B(\tau))u(\tau)\textrm{ and
}B^{T}(\tau)v(\tau)=r(B(\tau))v(\tau),$

which are normalized as follows

$\langle v(\tau),u(\tau)\rangle=1.$

Note that $r(B(\tau))=\langle v(\tau),B(\tau)u(\tau)\rangle$ for all
$|\tau|<\epsilon$, and differentiating this identity yields

$\displaystyle\frac{d}{d\tau}\left(r(B(\tau))\right)$ $\displaystyle=$
$\displaystyle
r(\tau)\left(\langle\frac{d}{d\tau}\left(v(\tau)\right),u(\tau)\rangle+\langle
v(\tau),\frac{d}{d\tau}\left(u(\tau)\right)\rangle\right)+$
$\displaystyle\langle
v(\tau),\frac{d}{d\tau}\left(B(\tau)\right)u(\tau)\rangle$ $\displaystyle=$
$\displaystyle\langle v(\tau),DA\operatorname{e}^{\tau A}u(\tau)\rangle,$

where the normalization identity was used to show that the first term above is
zero. Evaluating at $\tau=0$ yields

$\frac{d}{d\tau}\left(r(B(\tau))\right)|_{\tau=0}=\langle
e_{k},DAe_{k}\rangle=D_{kk}A_{kk},$

from which the conclusion follows. ∎

######  Lemma 2.

Assume that

  1. 1.

$D$ is diagonal with positive diagonal entries, and

  2. 2.

$A$ is a diagonally symmetrizable and let $\lambda_{1}$ be its largest
eigenvalue.

Then $\lim_{\tau\to+\infty}r(D\operatorname{e}^{\tau
A})=\begin{cases}+\infty,\textrm{ if }\lambda_{1}>0\\\ 0,\textrm{ if
}\lambda_{1}<0\end{cases}$.

###### Proof.

Recall from the proof of Theorem 1 that for all $\tau\geq 0$,

$r(D\operatorname{e}^{\tau A})=\max_{y:\langle
y,Q^{T}D^{-1}Qy\rangle=1}\langle y,\operatorname{e}^{\tau\Lambda}y\rangle,$

where ${\tilde{A}}=T^{-1}AT$ for some real diagonal and invertible matrix $T$,
and $Q$ is an orthogonal matrix such that $Q^{T}{\tilde{A}}Q=\Lambda$, for
some diagonal matrix $\Lambda$ whose diagonal entries $\lambda_{i}$ are the
eigenvalues of ${\tilde{A}}$ (and of $A$). Note that the ordering of the
diagonal entries of $\Lambda$ is such that the entry in the top left corner of
$\Lambda$ is $\lambda_{1}$, which is the largest eigenvalue of $A$.

  * •

Assume that $\lambda_{1}>0$, and thus $A$ is unstable. Let $\alpha$ be such
that for $y_{\alpha}:=\alpha e_{1}$ holds that $\langle
y_{\alpha},Q^{T}D^{-1}Qy_{\alpha}\rangle=1$. Such a non-zero $\alpha$ exists
because $Q^{T}D^{-1}Q$ is positive definite. Then it follows that for all
$\tau\geq 0$,

$r(D\operatorname{e}^{\tau A})\geq\langle
y_{\alpha},\operatorname{e}^{\tau\Lambda}y_{\alpha}\rangle=\alpha^{2}\operatorname{e}^{\lambda_{1}\tau},$

and since $\lambda_{1}>0$, the conclusion follows from taking the limit in the
above inequality as $\tau\to+\infty$.

  * •

Assume that $\lambda_{1}<0$ and thus $A$ is stable. Then
$\operatorname{e}^{\lambda_{j}\tau}\leq\operatorname{e}^{\lambda_{1}\tau}$ for
all $\tau\geq 0$ and for all $j$, because $\lambda_{1}$ is the largest
eigenvalue of $A$. Then for all $\tau\geq 0$,

$r(D\operatorname{e}^{\tau A})=\max_{y:\langle
y,Q^{T}D^{-1}Qy\rangle=1}\langle
y,\operatorname{e}^{\tau\Lambda}y\rangle\leq\operatorname{e}^{\lambda_{1}\tau}\max_{y:\langle
y,Q^{T}D^{-1}Qy\rangle=1}\langle y,y\rangle,$

and taking the limit as $\tau\to+\infty$ yields the desired result.

∎

The following is our main result concerning the effectiveness of periodic
impulsive control of an unstable linear system ${\dot{x}}=Ax$. It provides a
sufficient condition on $A$ guaranteeing that this control methodology can
stabilize the system, as long as the period between successive impulsive
control interventions remains below a unique threshold. It also shows that
there is a unique period $\tau$ for which $r(D\operatorname{e}^{\tau A})$ is
minimized. This is important for applications where the control objective is
to stabilize the system as efficiently as possible.

######  Theorem 2.

Assume that

  1. 1.

$D$ is diagonal with $D_{ii}\in(0,1]$ for all $i=1,\dots,N$; furthermore,
there is a unique $k$ such that $D_{kk}>D_{ii}$ for all $i\neq k$.

  2. 2.

$A$ is a non-singular, diagonally symmetrizable and unstable matrix (i.e., the
largest eigenvalue $\lambda_{1}$ of $A$ is positive), and $A_{kk}<0$.

Then the function $\tau\to r(D\operatorname{e}^{\tau A})$, where
$\tau\in[0,\infty)$, has the following properties:

  1. 1.

There is a unique $\tau_{s}>0$ such that

$r(D\operatorname{e}^{\tau A})\begin{cases}<1,\textrm{ if
}\tau\in(0,\tau_{s})\\\ =1,\textrm{ if }\tau=\tau_{s}\\\ >1,\textrm{ if
}\tau>\tau_{s}\end{cases}$

  2. 2.

There is a unique $\tau_{m}\in(0,\tau_{s})$ such that

$\inf_{\tau\geq 0}r(D\operatorname{e}^{\tau
A})=r(D\operatorname{e}^{\tau_{m}A}).$

###### Proof.

The map $\tau\to r(\tau):=r(D\operatorname{e}^{\tau A})$ is continuous for
$\tau\geq 0$, and Lemma 1 implies that it is strictly decreasing for all
sufficiently small $\tau$ near zero. Thus $r(\tau)<r(0)=D_{kk}\leq 1$ for all
sufficiently small $\tau>0$. Then by Lemma 2 and the Intermediate Value
Theorem, there exists some $\tau_{s}>0$ such that $r(\tau_{s})=1$ and
$r(\tau)>1$ for all $\tau>\tau_{s}$. Furthermore, Theorem 1 implies that
$r(\tau)$ is strongly convex with some parameter $m_{\tau_{s}}>0$ for
$\tau\in[0,\tau_{s}]$. Hence $r(\tau)$ is strictly convex for
$\tau\in[0,\tau_{s}]$, and this implies that $r(\tau)<1$ for all
$\tau\in(0,\tau_{s})$. This establishes item $(1)$.

Since $r(\tau)$ is continuous, it achieves its minimum on the compact set
$[0,\tau_{s}]$, say for $\tau=\tau_{m}$, and clearly,
$\tau_{m}\in(0,\tau_{s})$ and $r(\tau_{m})<1$. Theorem 8 in the Appendix
implies that $r(\tau)$ has a unique minimum at $\tau=\tau_{m}$ on
$[0,\tau_{s}]$. But $r(\tau)>1$ for all $\tau>\tau_{s}$, and thus $r(\tau)$
also has a unique minimum at $\tau=\tau_{m}$ on $[0,\infty)$. This establishes
item $(2)$, and concludes the proof. ∎

An example of the map $\tau\to r(\tau):=r(D\operatorname{e}^{\tau A})$ for
given matrices $A$ and $D$ which illustrates the previous theorem is shown in
the bottom left panel of Figure 1. Next, we consider the other cases.

Remark If all conditions of Theorem 2 hold, except that $A_{kk}>0$ instead of
assuming that $A_{kk}<0$, then the function $\tau\to
r(\tau):=r(D\operatorname{e}^{\tau A})$ behaves differently. We claim that in
this case, $r(\tau)$ is strictly increasing for $\tau\geq 0$, and
$\lim_{\tau\to+\infty}r(\tau)=\infty$. The latter limit is immediate from
Lemma 2. To see why $r(\tau)$ is strictly increasing, we argue by
contradiction. If it were not, there would be some $0\leq\tau_{1}<\tau_{2}$
such that $r(\tau_{2})\leq r(\tau_{1})$. Since by Lemma 1, $r(\tau)$ is
strictly increasing near zero, we may assume that $\tau_{1}>0$. But then
$\tau_{1}$ belongs to the interval $(0,\tau_{2})$ and since $r(\tau)$ is
strictly convex on $[0,\tau_{2}]$ (by Theorem 1), there exists some
$\lambda\in(0,1)$ such that

$r(\tau_{1})<\lambda r(0)+(1-\lambda)r(\tau_{2}).$

But since $r(\tau_{2})\leq r(\tau_{1})$, this implies that $r(\tau_{1})<r(0)$.
Now consider the interval $[0,\tau_{1}]$. Since $r(\tau)$ is strictly
increasing near $\tau=0$, there exists some $\tau_{3}\in(0,\tau_{1})$ such
that $r(0)<r(\tau_{3})$. Then strict convexity of $r(\tau)$ on $[0,\tau_{1}]$
(by Theorem 1) implies that there is some $\mu\in(0,1)$ such that
$r(\tau_{3})<\mu r(0)+(1-\mu)r(\tau_{1})<r(0)$ (because $r(\tau_{1})<r(0)$),
which contradicts that $r(0)<r(\tau_{3})$.

Consequently, in this case, and assuming in addition that $r(0)=D_{kk}<1$,
there will be a unique $\tau_{s}$ such that

$r(D\operatorname{e}^{\tau A})\begin{cases}<1,\textrm{ if
}\tau\in[0,\tau_{s})\\\ =1,\textrm{ if }\tau=\tau_{s}\\\ >1,\textrm{ if
}\tau>\tau_{s}\end{cases},$

and $r(\tau)$ has its unique miminizer over $\tau\geq 0$ at $\tau_{m}=0$. An
example illustrating this case is given in the top left of Figure 1.

Remark Theorem 2 and the subsequent Remark consider the case where $A$ has a
positive eigenvalue, and therefore is unstable. It is natural to ask what
would happen if instead $A$ is assumed to be stable, although this scenario is
perhaps less interesting from a control perspective. After all, why would one
want to stabilize an already stable system with impulsive periodic control? We
will show that in some sense, it is indeed best to not apply impulsive control
in this case. We start with the following

######  Lemma 3.

Assume that $A$ is a stable, diagonally symmetrizable matrix. Then $A_{ii}<0$
for all $i$.

###### Proof.

As $A$ is diagonally symmetrizable, there exists a real, diagonal and
invertible matrix $T$ such that

${\tilde{A}}=T^{-1}AT$ (4)

is a symmetric matrix. The eigenvalues of ${\tilde{A}}$ and $A$ coincide, and
since $A$ is stable, this implies that the largest eigenvalue of ${\tilde{A}}$
is negative. Thus, ${\tilde{A}}$ is a negative semi-definite matrix. And since
${\tilde{A}}$ is symmetric, this implies that ${\tilde{A}}_{ii}<0$ for all
$i$. Indeed, the largest eigenvalue $\lambda_{1}$ of ${\tilde{A}}$ is
negative, and then Raleigh quotient

$\lambda_{1}=\max_{x:\langle x,x\rangle=1}\langle x,{\tilde{A}}x\rangle,$

implies that for all $i$,

$\langle e_{i},{\tilde{A}}e_{i}\rangle={\tilde{A}}_{ii}\leq\lambda_{1}<0,$

where $e_{i}$ denotes the $i$th standard basis vector of $\mathbb{R}^{N}$. But
note that since $T$ in $(\ref{simil})$ is a diagonal matrix, it follows that
$A_{ii}={\tilde{A}}_{ii}$ for all $i$. Consequently, $A_{ii}<0$ for all $i$,
which concludes the proof. ∎

The above Lemma is illustrated in the top right of Figure 1. Equipped with
this Lemma, we can now describe what happens when stabilizing a stable system
with periodic impulsive control.

######  Theorem 3.

Assume that

  1. 1.

$D$ is diagonal with $D_{ii}\in(0,1]$ for all $i=1,\dots,N$; furthermore,
there is a unique $k$ such that $D_{kk}>D_{ii}$ for all $i\neq k$.

  2. 2.

$A$ is a diagonally symmetrizable and stable matrix (i.e., the largest
eigenvalue $\lambda_{1}$ of $A$ is negative).

Then the function $\tau\to r(D\operatorname{e}^{\tau A})$ is strictly
decreasing for $\tau\in[0,\infty)$, and
$\lim_{\tau\to+\infty}r(D\operatorname{e}^{\tau A})=0$.

###### Proof.

For $\tau\geq 0$, we set $r(\tau):=r(D\operatorname{e}^{\tau A})$. By the
variational description of $r(\tau)$, see for instance the second line of the
proof of Lemma 2, we see that $r(\tau)>0$ for all $\tau\geq 0$. Since
$A_{ii}<0$ for all $i$ by Lemma 3, Lemma 1 implies that $r(\tau)$ is strictly
decreasing for all sufficiently small $\tau$, and Lemma 2 implies that
$\lim_{\tau\to\infty}r(\tau)=0$. It remains to be shown that $r(\tau)$ is
strictly decreasing for all $\tau\geq 0$. If this were not the case, then
there would exist $0<\tau_{1}<\tau_{2}$ such that $0<r(\tau_{1})\leq
r(\tau_{2})$. Since $\lim_{\tau\to\infty}r(\tau)=0$, there exists
$\tau_{3}>\tau_{2}$ such that $r(\tau_{3})<r(\tau_{1})$. Note that
$\tau_{2}\in(\tau_{1},\tau_{3})$, and then the strong convexity of $r(\tau)$
on $[\tau_{1},\tau_{3}]$ (by Theorem 1) implies that there exists some
$\lambda\in(0,1)$ such that

$r(\tau_{2})<\lambda r(\tau_{1})+(1-\lambda)r(\tau_{3})<r(\tau_{1}),$

which contradicts that $r(\tau_{1})\leq r(\tau_{2})$. ∎

Since $\tau\to r(D\operatorname{e}^{\tau A})$ is strictly decreasing on
$[0,\infty)$ when the assumptions in Theorem 3 hold, this function has no
minimizer. The limiting behavior of this function suggest that the minimizer
“occurs as $\tau$ approaches infinity”, a scenario which corresponds to not
applying any impulsive periodic control at all. An example is given in the
bottom right of Figure 1.

Figure 1: Spectral radius as a function of $\tau$ in example matrices $A$ and
$D$. All examples have a common matrix $D=\begin{bmatrix}0.5&0\\\
0&0.25\end{bmatrix}$ but a different $A$. In the top left, bottom left, and
bottom right, the matrices are $A=\begin{bmatrix}0.2&1\\\
1&-0.2\end{bmatrix},A=\begin{bmatrix}-2&1\\\
1&1\end{bmatrix},A=\begin{bmatrix}-2&1\\\ 1&-2\end{bmatrix}$, respectively.
Here, $k=1$ since $D_{11}>D_{22}$, i.e., control for class 1 is weaker than
for class 2.

To summarize, if we assume that the largest diagonal entry of $D$, $D_{kk}$,
is less than $1$ (so control involves a multiplicative reduction in all
variables), we can categorize the behavior into 3 distinct cases. First, if
$\lambda_{\max}(A)<0$, the optimal strategy is to never control (bottom right
in Figure 1). If $\lambda_{\max}(A)>0$ and the variable associated with
weakest control (or largest diagonal entry $D_{kk}$ of $D$) is self-promoting
(i.e., $A_{kk}>0$), then the optimal strategy is to treat as often as possible
(top left in Figure 1). Finally, if $\lambda_{\max}(A)>0$ and this variable is
self-limiting (i.e., $A_{kk}<0$), then the optimal strategy is to treat at a
precise frequency (bottom left in Figure 1).

##  5 Examples

In this section, we provide an example of a real-world problem in biological
control. which was previously discussed in the introduction as a motivating
example. For this, we apply our results to a given concrete model and discuss
the implications to the ongoing control efforts.

The model involves the biological control of a structured population, in which
distinct population classes are impacted by the control intervention
differently. Before discussing the concrete example, we begin with a brief
description of the results for an unstructured (1D) population model, for the
purpose of juxtaposition.

###  5.1 Unstructured populations

In an unstructured population, we have the scalar linear impulse differential
equation

$\displaystyle\frac{dx}{dt}$ $\displaystyle=ax$ (5) $\displaystyle
x(n\tau^{+})$ $\displaystyle=dx(n\tau^{-})$ (6)

The spectral radius is $d\operatorname{e}^{a\tau}$. If $a>0$, then the map
$\tau\rightarrow d\operatorname{e}^{a\tau}$ is strictly increasing, and hence,
minimized when $\tau=0$. Conversely, if $a<0$, then the map is strictly
decreasing.

Interpreting this, if the population is exponentially growing in the absence
of interventions, we should intervene as often as possible. Conversely, if the
population is exponentially declining in the absence of interventions, we
should not intervene. As we will see below, in case the populations are
structured into classes and classes are impacted differently by the control
intervention, there is no such simple dichotomy.

###  5.2 Spatially-structured: soil-transmitted helminths

Soil-transmitted helminths are harmful parasitic worms that infect the gut of
humans and several livestock species, causing various ailments such as
malnutrition and developmental issues [16]. They have been subjected to
control efforts, including through preventative chemotherapy interventions,
also known as massive drug administration (MDA). This intervention involves
the widespread distribution of oral anthelmintic drugs a few times a year to
populations where the disease is thought to be prevalent.

Here we focus on three human-infecting species: _Ascaris lumbricoides_
(herafter roundworm), _Trichuris trichuria_ (herafter whipworm), and
_Ancylostoma duodenale_ (herafter hookworm). and compare the frequency of
interventions needed for effective control.

####  5.2.1 Model

The life cycle of STH consists of two distinct spatial locations:

  1. 1.

Adult worms live and produce eggs within the host environment.

  2. 2.

Eggs get secreted into an external environment, where they develop into
larvae.

Following [1], we model this through a set of linear differential equations.
Let $x_{1}(t)$ and $x_{2}(t)$ be the adult worms in the host environment and
the larvae in the external environment, respectively. Then, our uncontrolled
model is

$\frac{dx}{dt}=\begin{bmatrix}-\mu_{1}&\beta N\\\ \lambda&-(\mu_{2}+\beta
N)\end{bmatrix}x$ (7)

where $\beta$ is the per-host rate at which larvae are taken up (depends on
their contact with the external environment), $N$ is the population of hosts
(“size of host environment”- assumed to be fixed), $\lambda$ is the per-adult
worm egg production rate, and $\mu_{1}$ and $\mu_{2}$ are the natural per-
capita death rates in the host and external environment, respectively. These
life cycle parameters differ between species.

We assume the only control intervention is the administration of anthelmintic
drugs, and that this is given to a proportion $c$ of hosts in a given
population periodically at intervals of length $\tau$. The drug targets to
kill adult worms within the host environment and has a different efficacy
$\delta$; larvae in the external environment are not impacted. Under these
assumptions, our controlled model is of the form 1, with

$D=\begin{bmatrix}(1-c\delta)&0\\\ 0&1\end{bmatrix}$

####  5.2.2 Parameterizations

We use values as described in [13] and [3] to parameterize the matrix $A$ for
each species. Additionally, using drug efficacies from [14], (assuming egg
count reduction is proportional to adult worm reduction) and assuming a $75\%$
coverage of school-aged children (target based on the WHO [15]) that compose
of $50\%$ of the total population, we express the control intervention matrix
$D$, for each species.

For roundworms, we have

$A_{1}=\begin{bmatrix}-0.0028&1.3\times 10^{-8}\\\
5000&-0.016\end{bmatrix}\quad D_{1}=\begin{bmatrix}0.62875&0\\\
0&1\end{bmatrix}.$

For whipworms, we have

$A_{2}=\begin{bmatrix}-0.0028&2.089\times 10^{-7}\\\
1000&-0.05\end{bmatrix}\quad D_{2}=\begin{bmatrix}0.8125&0\\\
0&1\end{bmatrix}.$

For hookworms, we have

$A_{3}=\begin{bmatrix}-0.0014&1.18\times 10^{-7}\\\
1500&-0.082\end{bmatrix}\quad D_{3}=\begin{bmatrix}0.64375&0\\\
0&1\end{bmatrix}.$

The treatment drug is most effective against roundworms, which also has the
lowest contact rate but the highest fecundity. The treatment drug is least
effective against whipworms, but these also have the lowest fecundity.
Additionally, in all cases, the larvae class is the least controlled class,
since it is not impacted by the intervention, i.e.,
$[D_{i}]_{22}>[D_{i}]_{11}$ for $i=1,2,3$. Hence, we set $k=2$.

####  5.2.3 Implications

Using each parameterization given above, we see that $\lambda_{\max}(A_{i})>0$
for $i=1,2,3$, which predicts each parasite population will grow in the
absence of any intervention. Additionally, $A_{i}$ is diagonally
symmetrizable, which follows from observing that $A_{i}$ is sign symmetric,
that all $2\times 2$ matrices satisfy the cycle condition $(\ref{cycle})$ and,
then invoking Proposition 1. Alternatively, one can use Corollary 1. Finally,
observe that $[A_{i}]_{kk}<0$ for $i=1,2,3$.

Applying Theorem 2, there is a unique period $\tau_{m}$ that minimizes the
spectral radius map $\tau\rightarrow r(D\operatorname{e}^{\tau A})$, for each
species (Figure 2). The value $\tau_{m}$ is the optimal period to treat at.
For both whipworm and hookworm, the optimal period is very low, indicating
that treatment is best administered every few days. For roundworm, the optimal
treatment period is around every 40 days.

Additionally, applying Theorem 2, there is a unique $\tau_{s}$ such that for
all $\tau\in(0,\tau_{s})$, we have $r(D\operatorname{e}^{\tau A})<1$. The
value $\tau_{s}$ is the upper bound to the amount of time in between drug
administration needed in order to stabilize zero, i.e., effectively control
the population. We find that one needs to treat around once a year, more than
twice a year or a little more than once every two years for roundworm,
whipworm, and hookworm, respectively. The differences between the life history
rates and the drug efficacies of these distinct species drive differences in
the frequency of control interventions needed. Currently, the recommendations
from the WHO for mitigating this disease in human populations are agnostic to
which parasitic species are present [15].

Figure 2: Spectral radius of $D\operatorname{e}^{\tau A}$as a function of
$\tau$ (days) in the parameterized linear model for three species of soil-
transmitted helminths. Parameter values are given in the main text.

##  6 Discussion

Our main result establishes the convexity of the map $\tau\rightarrow
r(D\operatorname{e}^{\tau A})$. This map arises from the consideration of the
periodic impulse control of a system otherwise described by a set of linear
differential equations. The main implications of this result are that:

  * •

We find that there exists a unique range for the period between consecutive
control interventions which guarantees that the system can be stabilized, and

  * •

We find a unique optimal period to achieve this.

Essentially, the matrix $D\operatorname{e}^{\tau A}$comes from considering the
discrete time map of the system at the moment of impulse, either right before
or after the control intervention. Hence, it ignores the dynamics in between
interventions, and rescales time to be relative to the period of the pulses.
Because of this, one must be careful with the interpretation. In Theorem 2,
the optimal period (that minimizes the spectral radius
$r(D\operatorname{e}^{\tau A})$) is indicative of the most efficient period,
i.e., how to get the “closest” to zero per pulse. It does not necessarily tell
us about the system in real time and which control period will lead to zero
the quickest.

Our modeling framework assumes linearity of the system dynamics and, that the
times between consecutive interventions are always equal. It is natural to aim
to extend these results to consider nonlinear systems and more nuanced control
strategies. By considering the linearization around an equilibrium point, the
results may be applicable in some nonlinear systems. Additionally, the
approaches may be readily extended to other periodic control strategies. For
example, we could evaluate whether applying interventions multiple times in a
short burst followed by a longer time in between bursts is more efficient than
when they are all evenly spaced in time. Both considering nonlinear systems or
these other strategies will require additional care and thought.

Finally, here we provide sufficient conditions, most crucially that $A$ is
diagonally symmetrizable, for the convexity of the spectral radius of
$D\operatorname{e}^{\tau A}$. The following example demonstrates that this is
not a necessary condition. Consider an $N\times N$ triangular matrix with at
least one nonzero off-diagonal entry. This is clearly not diagonally
symmetrizable, as it does not satisfy the sign-symmetric condition (see
Proposition 1). But the map $\tau\rightarrow r(D\operatorname{e}^{\tau A})$ is
convex. To see this, observe that for a triangular matrix $A$ and a diagonal
matrix $D$, the matrix $D\operatorname{e}^{\tau A}$ remains triangular and the
eigenvalues are of the form $D_{ii}\operatorname{e}^{A_{ii}\tau}$. If $D$ is a
non-negative matrix, then each eigenvalue is also non-negative, real, and
convex with respect to $\tau$. Thus, the spectral radius
$r(D\operatorname{e}^{\tau})$ equals
$\max_{i}\left(D_{ii}\operatorname{e}^{A_{ii}\tau}\right)$, which by Theorem 7
in the Appendix, is convex with respect to $\tau$. Future work will be to
weaken or provide alternate conditions for this convexity.

## Appendix

In this Appendix we review some properties of convex functions. Most of the
material presented here can be found in [11], although for completeness, we
include some proofs of results whose proof was omitted in [11]. We recall the
definitions of three types of convexity. In what follows, $||x||$ denotes the
Euclidean norm of the vector $x$ in $\mathbb{R}^{N}$.

######  Definition 5.

Let $C$ be a convex set in $\mathbb{R}^{N}$. We say that a function
$f:C\to\mathbb{R}$ is

  * •

convex if

$f(\lambda x_{1}+(1-\lambda)x_{2})\leq\lambda
f(x_{1})+(1-\lambda)f(x_{2}),\textrm{ for all }x_{1},x_{2}\textrm{ in
}C\textrm{ and }\lambda\in[0,1].$

  * •

strictly convex if

$f(\lambda x_{1}+(1-\lambda)x_{2})<\lambda
f(x_{1})+(1-\lambda)f(x_{2}),\textrm{ for all }x_{1}\neq x_{2}\textrm{ in
}C\textrm{ and }\lambda\in(0,1).$

  * •

strongly convex 555It is a standard exercise to show that this definition of
strong convexity of $f$ with parameter $m>0$ is equivalent to
$g(x):=f(x)-\frac{m}{2}||x||^{2}$ being convex. with parameter $m>0$ if

$f(\lambda x_{1}+(1-\lambda)x_{2})\leq\lambda
f(x_{1})+(1-\lambda)f(x_{2})-\frac{m}{2}\lambda(1-\lambda)||x_{1}-x_{2}||^{2},$

for all $x_{1}$ and $x_{2}$ in $C$, and all $\lambda$ in $[0,1]$.

For continuously differentiable functions, the next two results provide
characterizations of convexity and strong convexity.

######  Theorem 4.

(Theorem 2.1.2 in [11]) Let $C$ be an open convex set in $\mathbb{R}^{N}$, and
assume that $f:C\to\mathbb{R}$ is continuously differentiable (i.e., the
gradient $\nabla f(x)$ is continuous for all $x$ in $C$). Then $f$ is convex
if and only if

$f(x_{2})\geq f(x_{1})+\langle\nabla f(x_{1}),x_{2}-x_{1}\rangle,\textrm{ for
all }x_{1},x_{2}\textrm{ in }C.$ (8)

######  Theorem 5.

(Theorem 2.1.9 in [11]) Let $C$ be an open convex set in $\mathbb{R}^{N}$, and
assume that $f:C\to\mathbb{R}$ is continuously differentiable. Then the
following statements are equivalent:

  * •

$f$ is strongly convex with parameter $m>0$.

  * •

$f(x_{2})\geq f(x_{1})+\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\frac{m}{2}||x_{2}-x_{1}||^{2},\textrm{ for all
}x_{1}\textrm{ and }x_{2}\textrm{ in }C.$ (9)

  * •

$\langle\nabla f(x_{1})-\nabla f(x_{2}),x_{1}-x_{2}\rangle\geq
m||x_{1}-x_{2}||^{2},\textrm{ for all }x_{1}\textrm{ and }x_{2}\textrm{ in
}C.$ (10)

###### Proof.

  * •

Let’s first show the equivalence of strong convexity of $f$ with parameter
$m>0$ and condition $(\ref{second})$. If $f$ is strongly convex with parameter
$m>0$, then for all $x_{1}$ and $x_{2}$ in $C$, and for all $\lambda$ in
$[0,1)$ (note that we are excluding $\lambda=1$),

$\displaystyle f(x_{2})$ $\displaystyle\geq$ $\displaystyle\frac{f(\lambda
x_{1}+(1-\lambda)x_{2})}{1-\lambda}-\frac{\lambda}{1-\lambda}f(x_{1})+\frac{m}{2}\lambda||x_{1}-x_{2}||^{2}$
$\displaystyle=$ $\displaystyle f(x_{1})+\frac{f(\lambda
x_{1}+(1-\lambda)x_{2})-f(x_{1})}{1-\lambda}+\frac{m}{2}\lambda||x_{1}-x_{2}||^{2}$
$\displaystyle=$ $\displaystyle
f(x_{1})+\frac{f(x_{1}+(1-\lambda)(x_{2}-x_{1}))-f(x_{1})}{1-\lambda}+\frac{m}{2}\lambda||x_{1}-x_{2}||^{2}$

Taking the limit as $\lambda\to 1-$ in the above inequality, the limit of the
last quotient is equal to $\langle\nabla f(x_{1}),x_{2}-x_{1}\rangle$. This
shows that $(\ref{second})$ holds.

For the converse, note that for all $x_{1}$ and $x_{2}$ in $C$ and all
$\lambda$ in $[0,1]$,

$\displaystyle f(x_{2})$ $\displaystyle\geq$ $\displaystyle f(\lambda
x_{1}+(1-\lambda)x_{2})+\langle\nabla f(\lambda
x_{1}+(1-\lambda)x_{2}),\lambda(x_{2}-x_{1})\rangle$ $\displaystyle+$
$\displaystyle\frac{m}{2}||\lambda(x_{2}-x_{1})||^{2}$ $\displaystyle
f(x_{1})$ $\displaystyle\geq$ $\displaystyle f(\lambda
x_{1}+(1-\lambda)x_{2})+\langle\nabla f(\lambda
x_{1}+(1-\lambda)x_{2}),(1-\lambda)(x_{1}-x_{2})\rangle$ $\displaystyle+$
$\displaystyle\frac{m}{2}||(1-\lambda)(x_{2}-x_{1})||^{2}$

Multiplying the first inequality by $(1-\lambda)$, the second by $\lambda$,
and adding the resulting inequalities shows that $f$ is strongly convex with
parameter $m>0$.

  * •

Let’s show next that $(\ref{second})$ and $(\ref{first})$ are equivalent. If
$(\ref{second})$ holds then for all $x_{1}$ and $x_{2}$ in $C$,

$\displaystyle f(x_{2})$ $\displaystyle\geq$ $\displaystyle
f(x_{1})+\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\frac{m}{2}||x_{2}-x_{1}||^{2},\textrm{ and }$
$\displaystyle f(x_{1})$ $\displaystyle\geq$ $\displaystyle
f(x_{2})+\langle\nabla
f(x_{2}),x_{1}-x_{2}\rangle+\frac{m}{2}||x_{2}-x_{1}||^{2},$

Adding both inequalities yields $(\ref{first})$.

For the converse, assume that $(\ref{first})$ holds. Then for all $x_{1}$ and
$x_{2}$ in $C$,

$\displaystyle f(x_{2})-f(x_{1})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{d\tau}\left(f(x_{1}+\tau(x_{2}-x_{1}))\right)d\tau$
$\displaystyle=$ $\displaystyle\int_{0}^{1}\langle\nabla
f(x_{1}+\tau(x_{2}-x_{1})),x_{2}-x_{1}\rangle d\tau$ $\displaystyle=$
$\displaystyle\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\int_{0}^{1}\frac{1}{\tau}\langle\nabla
f(x_{1}+\tau(x_{2}-x_{1}))-\nabla f(x_{1}),\tau(x_{2}-x_{1})\rangle d\tau$
$\displaystyle\geq$ $\displaystyle\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\int_{0}^{1}\frac{1}{\tau}m||\tau(x_{2}-x_{1})||^{2}d\tau$
$\displaystyle=$ $\displaystyle\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\int_{0}^{1}\tau m||\tau(x_{2}-x_{1})||^{2}d\tau$
$\displaystyle=$ $\displaystyle\langle\nabla
f(x_{1}),x_{2}-x_{1}\rangle+\frac{m}{2}||x_{2}-x_{1}||^{2},$

and thus $(\ref{second})$ holds.

∎

Next we review characterizations of convexity and strong convexity for
functions which are twice continuously differentiable. For such functions
$f:C\to\mathbb{R}$, we denote the Hessian of $f$ at any $x$ in $C$ by
$H_{f}(x)$, i.e. $[H_{f}(x)]_{ij}=\partial^{2}f/\partial x_{i}\partial
x_{j}(x)$, for all $i,j$ in $\\{1,\dots,N\\}$ and all $x$ in $C$. For a
symmetric $N\times N$ matrix $B$, the notation $B\succeq 0$ simply means that
$B$ is a positive semi-definite matrix (or equivalently, that all the
eigenvalues of $B$ are non-negative real numbers).

######  Theorem 6.

(Theorems 2.1.4 and 2.1.11 in [11])) Assume that $C$ is an open convex set in
$\mathbb{R}^{N}$, and that $f:C\to\mathbb{R}$ is twice continuously
differentiable. Then

  * •

$f$ is convex if and only if $H_{f}(x)\succeq 0$ for all $x$ in $C$.

  * •

$f$ is strongly convex with parameter $m>0$ if and only if $H_{f}(x)-mI\succeq
0$, for all $x$ in $C$.

###### Proof.

We only provide the proof of the last statement.

Assuming that $f$ is twice continuously differentiable and strongly convex
with parameter $m>0$, it follows from $(\ref{first})$ in Theorem 5 that for
all sufficiently small $\tau>0$ and for every $s$ in $\mathbb{R}^{N}$,

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\frac{1}{\tau}\left(\langle\nabla f(x+\tau s)-\nabla f(x),\tau
s\rangle-m\langle\tau s,\tau s\rangle\right)$ $\displaystyle=$
$\displaystyle\langle\nabla f(x+\tau s)-\nabla f(x),s\rangle-m\tau\langle
s,s\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\frac{d}{d\lambda}\left(\langle\nabla f(x+\lambda
s),s\rangle\right)d\lambda-\int_{0}^{\tau}\langle ms,s\rangle d\lambda$
$\displaystyle=$ $\displaystyle\int_{0}^{\tau}\langle H_{f}(x+\lambda
s)s,s\rangle-\langle ms,s\rangle d\lambda$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\langle\left(H_{f}(x+\lambda
s)-mI\right)s,s\rangle d\lambda$

As the integrand in the last integral is continuous in $\lambda$, it follows
from taking the limit as $\tau\to 0+$ that

$0\leq\langle\left(H_{f}(x)-mI\right)s,s\rangle,\textrm{ for all }s\textrm{ in
}\mathbb{R}^{N}.$

In other words, $H_{f}(x)-mI\succeq 0$, for all $x$ in $C$.

For the converse, assume that $f$ is twice continuously differentiable and
that there is some $m>0$ such that $H_{f}(x)-mI\succeq 0$ for all $x$ in $C$.
Then for all $x$ and $y$ in $C$,

$\displaystyle f(y)$ $\displaystyle=$ $\displaystyle
f(x)+\int_{0}^{1}\frac{d}{d\tau}\left(f(x+\tau(y-x))\right)d\tau$
$\displaystyle=$ $\displaystyle f(x)+\int_{0}^{1}\langle\nabla
f(x+\tau(y-x)),y-x\rangle d\tau$ $\displaystyle=$ $\displaystyle
f(x)+\langle\nabla
f(x),y-x\rangle+\int_{0}^{1}\int_{0}^{\tau}\frac{d}{d\lambda}\left(\langle\nabla
f(x+\lambda(y-x)),y-x\rangle\right)d\lambda d\tau$ $\displaystyle=$
$\displaystyle f(x)+\langle\nabla
f(x),y-x\rangle+\int_{0}^{1}\int_{0}^{\tau}\langle
H_{f}(x+\lambda(y-x))(y-x),y-x\rangle d\lambda d\tau$ $\displaystyle\geq$
$\displaystyle f(x)+\langle\nabla
f(x),y-x\rangle+\int_{0}^{1}\int_{0}^{\tau}\langle m(y-x),y-x\rangle\rangle
d\lambda d\tau$ $\displaystyle=$ $\displaystyle f(x)+\langle\nabla
f(x),y-x\rangle+\frac{m}{2}||y-x||^{2}$

Thus, we have shown that $(\ref{second})$ holds, and therefore Theorem 5
implies that $f$ is strongly convex with parameter $m>0$. ∎

In particular, if $C$ is an open and convex subset of $\mathbb{R}$ and if
$f:C\to\mathbb{R}$ is a twice continuously differentiable function, then $f$
is strongly convex with parameter $m>0$ if and only if
$f^{\prime\prime}(x)\geq m$ for all $x$ in $C$. For example, this shows that
if $f:(-1,1)\to\mathbb{R}$ is defined as $f(x)=x^{4}$, then $f$ is not
strongly convex with respect to any positive parameter $m>0$, because
$f^{\prime\prime}(0)=0$.

Next we show that the point-wise supremum of a collection of convex functions
is also convex, and furthermore, that the point-wise supremum of a collection
of strongly convex functions all having the same parameter, is also strongly
convex with the same parameter.

######  Theorem 7.

Let $C$ be a convex set, and assume that $f_{\alpha}:C\to\mathbb{R}$ is
convex, for all $\alpha$ in some (possibly infinite) index set $I$. Then

$g(x):=\sup_{\alpha\in I}f_{\alpha}(x)$

is convex.

Furthermore, if there is some $m>0$ such that for all $\alpha$ in $I$, the
function $f_{\alpha}$ is strongly convex with parameter $m$, then $g$ is also
strongly convex with parameter $m$.

###### Proof.

Assume first that all $f_{\alpha}$ are convex, for all $\alpha$ in $I$. Then
for all $\alpha$ in $I$, all $x_{1}$ and $x_{2}$ in $C$ and $\lambda$ in
$[0,1]$, holds that

$\displaystyle f_{\alpha}(\lambda x_{1}+(1-\lambda)x_{2})$ $\displaystyle\leq$
$\displaystyle\sup_{\alpha\in I}\left(\lambda
f_{\alpha}(x_{1})+(1-\lambda)f_{\alpha}(x_{2})\right)$ $\displaystyle\leq$
$\displaystyle\lambda\sup_{\alpha\in
I}f_{\alpha}(x_{1})+(1-\lambda)\sup_{\alpha\in I}f_{\alpha}(x_{2})$

Taking the supremum over all $\alpha$ in $I$ yields that $g$ is convex.

Next assume that there is some $m>0$ such that every $f_{\alpha}$ is strongly
convex with parameter $m$, for all $\alpha$ in $I$. Then for all $\alpha$ in
$I$, all $x_{1}$ and $x_{2}$ in $C$ and $\lambda$ in $[0,1]$, holds that

$\displaystyle f_{\alpha}(\lambda x_{1}+(1-\lambda)x_{2})$ $\displaystyle\leq$
$\displaystyle\sup_{\alpha\in I}\left(\lambda
f_{\alpha}(x_{1})+(1-\lambda)f_{\alpha}(x_{2})\right)-\frac{1}{2}m\lambda(1-\lambda)||x_{1}-x_{2}||^{2}$
$\displaystyle\leq$ $\displaystyle\lambda\sup_{\alpha\in
I}f_{\alpha}(x_{1})+(1-\lambda)\sup_{\alpha\in
I}f_{\alpha}(x_{2})-\frac{1}{2}m\lambda(1-\lambda)||x_{1}-x_{2}||^{2}.$

Taking the supremum over all $\alpha$ in $I$ yields that $g$ is strongly
convex with parameter $m$.

∎

It is natural to ask if the point-wise supremum of a collection of strictly
convex functions is strictly convex. While it is easy to see that this is true
if the collection is finite, it is not necessarily true if the collection is
infinite. Example: Let $n\in\mathbb{N}$ and suppose that
$f_{n}:[-1,1]\to\mathbb{R}$ is defined as $f_{n}(x)=(x^{2}-1)/n$. Clearly,
each $f_{n}$ is strictly convex, but $g(x)=\sup_{n}f_{n}(x)=0$ for all $x$ in
$[-1,1]$, and this function is convex but not strictly convex.

Finally, we recall the importance of strict convexity on the uniqueness of
minimizers.

######  Theorem 8.

Let $C$ be a convex set in $\mathbb{R}^{N}$, and assume that
$f:C\to\mathbb{R}$ is strictly convex. Then $f$ has at most one global
minimizer in $C$.

###### Proof.

Assume that $x_{1}$ and $x_{2}$ are two distinct global minimizers of $f$ in
$C$. Then $f(x_{1})=f(x_{2})$. However, since $f$ is strictly convex, it
follows that $f(\lambda x_{1}+(1-\lambda)x_{2})<\lambda
f(x_{1})+(1-\lambda)f(x_{2})=f(x_{1})$, for all $\lambda$ in $(0,1)$. This
contradicts that $x_{1}$ is a global minimizer of $f$. ∎

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