\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 119, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2018/119\hfil
Optimal design of minimum mass structures]
{Optimal design of minimum mass structures for a generalized Sturm-Liouville
problem on an interval and a metric graph}
\author[B. P. Belinskiy, D. H. Kotval \hfil EJDE-2018/119\hfilneg]
{Boris P. Belinskiy, David H. Kotval}
\address{Boris P. Belinskiy \newline
University of Tennessee at Chattanooga,
Department of Mathematics,
Dept 6956, 615 McCallie Ave.,
Chattanooga TN 37403-2598, USA}
\email{boris-belinskiy@utc.edu}
\address{David H. Kotval \newline
Middle Tennessee State University,
Department of Mathematical Sciences,
MTSU BOX 34, 1301 East Main Street,
Murfreesboro TN 37132-0001, USA}
\email{dhk2e@mtmail.mtsu.edu}
\dedicatory{Communicated by Suzanne M. Lenhart}
\thanks{Submitted December 4, 2017. Published May 17, 2018.}
\subjclass[2010]{34L15, 74P05, 49K15, 49S05, 49R05}
\keywords{Sturm-Liouville Problem; vibrating rod; calculus of variations;
\hfill\break\indent optimal design; boundary conditions with spectral parameter;
complete bipartite graph}
\begin{abstract}
We derive an optimal design of a structure that is described by a
Sturm-Liouville problem with boundary conditions that contain the
spectral parameter linearly. In terms of Mechanics, we determine necessary
conditions for a minimum-mass design with the specified natural frequency
for a rod of non-constant cross-section and density subject to the boundary
conditions in which the frequency (squared) occurs linearly. By virtue of
the generality in which the problem is considered other applications are
possible. We also consider a similar optimization problem on a complete
bipartite metric graph including the limiting case when the number of
leafs is increasing indefinitely.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{problem}[theorem]{Problem}
\allowdisplaybreaks
\section{Introduction} \label{Sect1}
The optimal design of an axially vibrating rod supporting a non-structural
point mass was considered by Turner \cite{Turner}. He determined an optimal
cross-sectional mass distribution $m(x)$ such that a rod of given principal
eigenvalue is designed with the least possible mass. Such an optimization
allows for greater economy in a design that must meet certain minimum
requirements for natural frequency. Due to a duality principle,
Turner's technique can also be used to determine the optimal distribution
$m(x)$ such that a rod of given total mass is made with the largest
principal eigenvalue. Such an optimization would give the greatest
resistance to resonance. Taylor \cite{Taylor} considered the same problem
and proved that the design of Turner was indeed optimal.
Taylor also clearly articulated the duality principle employed by Turner
in a form that assists in generalizing the method.
We begin with a brief review of \cite{Turner}. The axial displacement of a
rod can be modeled by the wave equation
\begin{equation}\label{e-1}
m\frac{\partial^2 u}{\partial t^2}
- \frac{E}{\rho}\frac{\partial }{\partial x}
\Big( m\frac{\partial u}{\partial x}\Big)
= 0,\quad 00.
\end{gather}
Here $\alpha \in [0,\pi)$, $\beta_k$, and $\beta_k'$, $k=1,2$ and
$r(x)>0$ are the (known) parameters and function and the assumption that
$\delta > 0$ is required for the problem to be self-adjoint \cite{Hinton},
and therefore for all eigenvalues to be real and bounded below.
It is known (see \cite{Belinskiy1,Belinskiy2} and the references therein)
that problems of this type arise in the study of many diverse physical models
including oscillations of a rotating string, a Timoshenko-Mindlin
beam with a tip mass, a rotating beam with a tip mass (which models a propeller),
and a beam of non-uniform cross section with one end elastically restrained
and the other end carrying a guided mass.
The consideration of the more general model was also motivated by the results
of Hinton and McCarthy \cite{HM} where the authors consider oscillations of a
string fixed at one end with a mass connected to a spring at the other end.
This study also considered minimizing the principal eigenvalue subject to
a fixed total mass constraint.
We also consider optimization problem on a graph. Our consideration of the
differential equations on a metric graph was motivated by the known
extensive study of the mechanical and electrical networks, such as circuit
equations with distributed parameters, string equations with the tip masses,
and systems of beam equations that model the structural constructions
(see \cite{XuMastorakis}). To our best knowledge, only the direct problem has
been studied so far, but we consider optimization. Though we consider a simple
graph, we believe that our research represents just the first step in this
promising direction.
The plan of the paper is as follows. In Section~\ref{Subsection2.1}
we formulate the problem. In Section 2.2 we formulate our main result.
The proof of it occupies Sections 2.3, 3 and 4. In Section 2.3
we use the methods of the Calculus of Variations to find critical points of
the ``mass'' functional, i.e. functions $p(x)$ and also $y(x)$.
These functions contain several arbitrary constants. In Section 3,
we find some conditions on the parameters that guarantee that the function
$y(x)$ satisfies the boundary conditions. In particular, we discover some zones
of existence and non-existence of the parameters. We find an explicit
formula for every critical point $p(x)$. In Section 4,
we derive an explicit expression for the ``mass'' at each critical
point and compare them. We also show that the result by \cite{Turner}
appears as a particular case of our general formulas.
In Section 5 we consider the similar optimization problem on a complete
bipartite metric graph (star). In Section 6 we derive the design and
``mass'' for a star with identical leafs and discuss the limiting case
when the number of leafs is increasing indefinitely. Section 7
contains a discussion of the results.
\section{Calculations} \label{Sect2}
\subsection{Statement of the problem} \label{Subsection2.1}
We reduce our consideration to the particular case $q(x)\equiv 0$.
The reason for this is twofold. First, in many applications of
problem \eqref{e-7}-\eqref{e-7b}, there is no term containing
the function $q(x)$ (see \cite{Taylor,Turner,B,BMH}). Second, the calculations
of the optimal form for $q(x)\not \equiv 0$ seem to be intractable in the
frame of an analytic approach. We briefly outline our plans for this case
in Section 7.
Hence, we consider the Sturm-Liouville problem
\begin{gather}\label{e-8}
(p(x)y'(x))' + \lambda p(x)r(x)y(x) = 0,\\
\label{BC(0)OrigProb}
\cos \alpha\,y(0)+ \sin \alpha\,p(0)y'(0) = 0, \\
\label{BC(1)OrigProb}
-\beta_1 y(1) + \beta_2 p(1)y'(1) = \lambda[\beta_1' y(1) - \beta_2' p(1)y'(1)] .
\end{gather}
Here and everywhere below \eqref{inequality} is implicitly assumed.
Though we consider an abstract optimization problem, we prefer to use
the physical terminology below, by interpreting the variables as
in Table \ref{table2}.
\begin{table}[htbp]
\caption{Interpretation in the Notation in \eqref{e-8} - \eqref{BC(1)OrigProb}}
\label{table2}
\begin{center}
\begin{tabular}{|c|c|} \hline
Quantity & Interpretation \\\hline
$p(x)$ & Cross-Sectional Area of Rod\\
$y(x)$ & Axial Displacement \\
$r(x)$ & Density of Rod Material \\
$\lambda$ & $\omega^2/E$\\
$\omega$ & Angular Frequency\\
\hline
\end{tabular}
\end{center}
\end{table}
As usual in the general theory of Sturm-Liouville problems, we will make
the following assumption motivated by the physical restrictions of designing a rod.
\begin{itemize}
\item[(A1)]
The cross-sectional area $p(x)$ is continuous and strictly positive on $[0,1]$.
Only boundary parameters will be considered admissible which yield a positive
$p(x)$.
\end{itemize}
Note the difference between \eqref{e-2} and \eqref{e-8}
due to the loss of the assumption that the density is constant; this is,
setting $\rho = r(x)$ does not reduce \eqref{e-2} to \eqref{e-8}
since $r(x)$ can not be factored out and incorporated into the spectral parameter.
We now formulate our problem.
\begin{problem} \label{prob2} \rm
Minimize the ``mass'' functional,
\begin{equation}\label{e-total-mass}
M[p] := \int_{0}^{1} p(x)r(x)dx
\end{equation}
associated with the Sturm-Liouville problem \eqref{e-8}-\eqref{BC(1)OrigProb}
if the principal eigenvalue, $\lambda_1>0$, of the problem is given. $\Box$
\end{problem}
In view of (A1), the design $p(x)$ must be positive.
Problem \ref{prob2} is a generalization of the problems considered
in \cite{Turner,Taylor,BMH}.
\subsection{Formulation of the main result}
We now formulate our result on minimizing the ``mass''
functional \eqref{e-total-mass}.
\begin{theorem} \label{thm1}
For the Sturm-Liouville problem \eqref{e-8}-\eqref{BC(1)OrigProb}
subject to the condition \eqref{inequality} and {\rm (A1)},
\begin{itemize}
\item[(a)] If $\alpha\ne \pi/2$, then the functional $M[p]$ has the critical point
\begin{equation}\label{p-I}
p_I(x) = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ \tanh^{-1}(\zeta))}{2\sqrt{\lambda_1 r(x)}\cosh^2(\sqrt{\lambda_1}\varrho(x)
+ \frac{1}{2}\tanh^{-1}(\zeta))},
\end{equation}
and if $\alpha\ne 0, \pi/2$, then this functional has a second critical point
\begin{equation}\label{FinalDesignCaseIII}
p_{II}(x) = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ \tanh^{-1}(\zeta))}{2\sqrt{\lambda_1 r(x)}\sinh^2(\sqrt{\lambda_1}\varrho(x)
+ \frac{1}{2}\tanh^{-1}(\zeta))}.
\end{equation}
Here
\begin{gather} \label{rho-th}
\varrho(x) := \int_{0}^{x} \sqrt{r(s)}ds, \\
\label{B}
B := \frac{\beta_1+\lambda_1\beta_1'}{\beta_2+\lambda_1\beta_2'}, \\
\label{zeta}
\zeta := - \frac{\sinh(2\sqrt{\lambda_1}\varrho(1))}
{\frac{\hat{\alpha}}{B}+\cosh(2\sqrt{\lambda_1}\varrho(1))}, \\
\label{alpha}
\hat{\alpha} := \cot \alpha.
\end{gather}
Here we assume that
\begin{equation}\label{zeta1}
\zeta \in (0,1).
\end{equation}
\item[(b)] For $\alpha\ne 0, \pi/2$, the ``mass'' of the design $p_I$ is less
than the ``mass'' of the design $p_{II}$.
\end{itemize}
\end{theorem}
\subsection{Solution to Problem \ref{prob2}}
The proof of Theorem \ref{thm1} is given in this Section and Sections 3 and 4.
\begin{proof}[Theorem \ref{thm1} Part I]
We follow the development of Turner \cite{Turner} to find the critical
points. Specifically, we formulate an isoperimetric problem in terms of
the ``mass'' functional
\begin{equation} \label{functional-F[y,p]}
\begin{aligned}
F[y,p]
&:= M[p]+\int_{0}^{1}\Lambda_1(x) \Big( (py')'+\lambda_1 pry \Big) dx \\
&\quad + \Lambda_2 \Big(\cos\alpha\,y(0))+ \sin \alpha\,p(0)y'(0) \Big) \\
&\quad + \Lambda_3 \Big( [-\beta_1 y(1) + \beta_2\, p(1)y'(1)]
- \lambda_1[\beta_1'\,y(1)) - \beta_2'\,p(1)y'(1)] \Big).
\end{aligned}
\end{equation}
Here $\Lambda_1(x)$, $\Lambda_2$, $\Lambda_3$ are Lagrange multipliers.
Similarly to \cite{Turner} (see also \cite{GF}, \cite{B}, \cite{BMH})
we compute the first variation of $F[y,p]$:
\begin{equation} \label{variation-of-F}
\begin{aligned}
\delta F &= \big(\Lambda_1 y' \delta p \big) |_{0}^{1}
+ \big(\Lambda_1 p \delta y' \big) |_{0}^{1} - \big(\Lambda_1' p \delta y\big)
|_{0}^{1} \\
&\quad + \Lambda_2 \Big(\cos\alpha\,\delta y(0) + \sin\alpha\,p(0) \delta y'(0)
+ \delta p(0) y' (0) \Big) \\
&\quad + \Lambda_3 \Big(-\beta_1 \delta y(1) + \beta_2 (\delta p(1)y'(1)
+ \delta y'(1) p(1)) \\
&\quad - \lambda_1[\beta_1'\,\delta y(1) - \beta_2'(\delta p(1)y'(1)
+ \delta y'(1)\,p(1))] \Big)\\
&\quad + \int_{0}^{1} \delta y \Big((\Lambda_1' p)'
+ \Lambda_1 \lambda_1 r p \Big) dx\\
&\quad + \int_{0}^{1} \delta p \Big(-\Lambda_1' y' + \Lambda_1 \lambda_1 r y
+ r \Big) dx.
\end{aligned}
\end{equation}
To find the stationary points, we set $\delta F = 0$ and use the fundamental lemma
of the Calculus of Variations to arrive at the following two differential equations
\begin{gather} \label{eq-in-Lambda-1}
(p \Lambda_1')' + \lambda_1 r p \Lambda_1 = 0, \\
\label{eq-in-Lambda-2}
-\Lambda_1' y' + \Lambda_1 \lambda_1 r y + r = 0.
\end{gather}
Furthermore, we determine the following necessary conditions at the
boundaries by considering the terms in which each of the independent
variations ($\delta y(0)$, $\delta y'(0)$, $\delta p(0)$, $\delta y(1)$,
$\delta y'(1)$, and $\delta p(1)$) appears. The boundary conditions
are as follows:
\begin{equation} \label{VariationsAtBoundary(0)}
\begin{gathered}
\delta y(0): \Lambda_2 \cos \alpha - \Lambda_1'(0)p(0) = 0, \\
\delta y'(0): p(0)(\Lambda_2 \sin \alpha + \Lambda_1(0)) =0, \\
\delta p(0): y'(0)(\Lambda_2 \sin \alpha + \Lambda_1(0)) = 0,
\end{gathered}
\end{equation}
\begin{equation}\label{VariationsAtBoundary(1)}
\begin{gathered}
\delta y(1): \Lambda_1'(1)p(1) - \Lambda_3(\beta_1 + \lambda_1 \beta_1') = 0, \\
\delta y'(1): \Lambda_1(1)p(1) -\Lambda_3 p(1)(\beta_2 + \lambda_1 \beta_2') = 0, \\
\delta p(1): \Lambda_1(1)y'(1) - \Lambda_3 y'(1)(\beta_2 + \lambda_1 \beta_2') =0.
\end{gathered}
\end{equation}
From the set of equations \eqref{VariationsAtBoundary(0)}, we can exclude
$\Lambda_2$ to achieve \eqref{boundary-comparison-1} below and from the
set \eqref{VariationsAtBoundary(1)}, we can exclude $\Lambda_3$ to
achieve \eqref{boundary-comparison-2},
\begin{gather} \label{boundary-comparison-1}
\Lambda_1(0) \cos( \alpha ) + \Lambda_1'(0)p(0) \sin \alpha = 0, \\
\label{boundary-comparison-2}
-\beta_1 \Lambda_1(1) + \beta_2 p(1)\Lambda_1'(1)
= \lambda_1[\beta_1'(\Lambda_1(1)) -\beta_2' p(1)\Lambda_1'(1) ].
\end{gather}
We note that the boundary-value problem
\eqref{eq-in-Lambda-1}, \eqref{boundary-comparison-1}, \eqref{boundary-comparison-2}
is the same as \eqref{e-8}-\eqref{BC(1)OrigProb}.
For this problem, it is well-known that the eigenspace is one dimensional.
Therefore the multiplicity of the principal eigenvalue $\lambda_1$ is one,
and we may conclude that $\Lambda_1(x) = k y(x)$ or $\Lambda_1(x) = -k y(x)$
(for a constant $k \in \mathbb{R} \setminus \{0\}$). Our necessary conditions
\eqref{eq-in-Lambda-1} and \eqref{eq-in-Lambda-2} then become the original
ODE \eqref{e-8}:
\begin{equation}\label{e-9}
(p y')' + \lambda_1 pry = 0
\end{equation}
and one of the following non-linear differential equations:
\begin{equation} \label{e-10}
-k(y')^2 + k\lambda_1 r y^2 + r = 0
\end{equation}
or
\begin{equation} \label{e-11}
k(y')^2 - k \lambda_1 r y^2 + r = 0.
\end{equation}
We observe that the sign of $k$ is not important and assume further that $k>0$.
The solution of the equations \eqref{e-10} and \eqref{e-11} leads to valid
critical points of the functional \eqref{functional-F[y,p]}.
We find respectively,
\begin{equation} \label{e-addin}
y_{1}(x) = \frac{1}{\sqrt{\lambda_1 k}} \sinh(\sqrt {\lambda_1} \varrho(x) +C_1)
\end{equation}
and
\begin{equation} \label{e-13}
y_{2}(x) = \frac{1}{\sqrt{\lambda_1 k}} \cosh(\sqrt {\lambda_1} \varrho(x) +C_2),
\end{equation}
where $\varrho(x)$ is defined by \eqref{rho-th}.
Note that due to the non-linear nature of \eqref{e-10} and \eqref{e-11},
linear combinations of these solutions are not necessarily solutions
to \eqref{e-10} and \eqref{e-11}.
The original differential equation \eqref{e-9} now becomes a first order
linear differential equation for the unknown design $p(x)$.
It may be rewritten in two ways depending on what function
$y_j(x)$, $j=1,2$ is used,
\begin{gather}\label{e-A}
(p y_1')' + \lambda_1 pry_1 = 0, \\
\label{e-B}
(p y_2')' + \lambda_1 pry_2 = 0.
\end{gather}
Solving the differential equation \eqref{e-A} gives the design,
\begin{equation}\label{solution-A}
p_1(x) = C_3 \frac{ \sqrt{r(0)} \cosh^2(C_1) }{ \sqrt{r(x)}
\cosh^2( \sqrt{\lambda_1} \varrho(x) +C_1) }
\end{equation}
with the arbitrary constants $C_3$ and $C_1$. We note that by (A1) $C_3 > 0$.
Solving \eqref{e-B} gives the design
\begin{equation}\label{solution-B}
p_2(x)= C_4 \frac{ \sqrt{r(0)} \sinh^2(C_2) }{ \sqrt{r(x)}
\sinh^2( \sqrt{\lambda_1} \varrho(x) +C_2) }
\end{equation}
with the arbitrary constants $C_4$ and $C_2$. We note that by (A1)
the design should be continuous and strictly positive. This requires that
$C_4>0$ and $C_2 \in (-\infty, -\sqrt{\lambda_1}\varrho(1)) \cup (0, \infty)$.
The condition on $C_2$ can be derived by enforcing that the arguments of the
$\sinh^2$ functions in both the numerator and the denominator not be equal
to zero. This derivation is as follows:
Observe that if $C_2>0$, (A1) is obviously satisfied
(see the definition \eqref{rho-th} of $\varrho(x)$).
Similarly, if $C_2=0$, the denominator is equal to zero at $x=0$.
Further, if $C_2<-\sqrt{\lambda_1} \varrho(1)$, the arguments of both $\sinh^2$
functions are negative and the design is strictly positive.
If $0>C_2>-\sqrt{\lambda_1}\varrho(1)$,
the argument has a unique zero at the point $x_0\in (0,1)$ where
\begin{equation}\label{not-allowed}
\sqrt{\lambda_1} \int_0^{x_0}\,\sqrt{r(s)} ds=-C_2.
\end{equation}
Therefore (A1) is satisfied when $C_2 \in (-\infty, -\sqrt{\lambda_1}\varrho(1))
\cup (0, \infty)$.
Thus, we have two distinct stationary points of our variational problem.
\end{proof}
\section{Boundary conditions: zones of existence and non-existence\label{Sect3}}
\begin{proof}[Proof of Theorem \ref{thm1} part II]
We use the boundary conditions of our problem, \eqref{BC(0)OrigProb} and
\eqref{BC(1)OrigProb}, to determine arbitrary constants, as well conditions
for which a solution exists. We discern three cases, shown in Table \ref{table3}.
\begin{table}[hbt]
\caption{Summary of Cases}
\label{table3}
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
$p(x)$ & $ \alpha $ & Case for Constants and Existence & Final Design \\\hline
$p_1(x)$ & 0 & Case(1) & \eqref{FinalDesignCaseI}\\
& $\pi/2$ & Case(2) & Does Not Exist\\
& $\ne 0, \pi/2$ & Case(3) & \eqref{FinalDesignCaseII}\\ \hline
$p_2(x)$ & 0 & Case(4) &Does Not Exist \\
& $\pi/2$ & Case(5) & Does Not Exist \\
& $\ne 0, \pi/2$ & Case(6) & \eqref{FinalDesignCaseIII-c}\\
\hline
\end{tabular}
\end{center}
\end{table}
First, we consider the solutions stemming from $p_1$.
\smallskip
\noindent \textbf{Case (1)}
In this case $y=y_1$ as given by \eqref{e-addin}, $p=p_1$ as given by
\eqref{solution-A}, and $\alpha = 0$.
The boundary condition \eqref{BC(0)OrigProb} immediately implies
\begin{equation}\label{align1}
C_1 = 0.
\end{equation}
The boundary condition \eqref{BC(1)OrigProb}, after the long but simple
algebraic manipulations leads to the following
\begin{equation}\label{align2}
C_3 = \frac{B \sinh(2\sqrt{\lambda_1}\varrho(1))}{2\sqrt{\lambda_1 r(0)}}.
\end{equation}
Since it is required that $p(x)>0$, a solution exists when
\begin{align}
B > 0
\end{align}
or equivalently
\begin{equation}\label{quadineq}
\beta_1'\beta_2'\Big(\lambda_1+\frac{\beta_1}{\beta_1'}\Big)\Big(\lambda_1+\frac{\beta_2}{\beta_2'}\Big)>0.\;\;
\end{equation}
Here the final design $p_1$ is
\begin{equation} \label{FinalDesignCaseI}
p_{1;1}(x) = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1))}{2\sqrt{\lambda_1 r(x)}
\cosh^2(\sqrt{\lambda_1}\varrho(x))}.
\end{equation}
\smallskip
\noindent \textbf{Case (2)}
Note that for $p_1$, the solution does not exist when $\alpha = \pi/2$.
To see this, consider that when $\alpha = \pi/2$, \eqref{BC(0)OrigProb},
together with \eqref{e-addin}, \eqref{rho-th}, and \eqref{solution-A} becomes
\begin{equation}
C_3\sqrt{r(0)}\cosh(C_1) = 0.
\end{equation}
Due to the condition that $C_3>0$ (which follows from (A1)), this
boundary condition cannot be satisfied.
\smallskip
\noindent \textbf{Case (3)}
In this case $y=y_1$ as given by \eqref{e-addin}, $p=p_1$ as given by
\eqref{solution-A}, and $\alpha \not\in \{0,\pi/2\}$.
The boundary condition \eqref{BC(0)OrigProb} immediately implies
\begin{equation}\label{align11}
C_3 = - \frac{\hat \alpha \tanh (C_1)}{\sqrt{\lambda_1\,r(0)}}.
\end{equation}
Isolating $C_3$ from the boundary condition \eqref{BC(1)OrigProb}
(see also \eqref{e-addin} and \eqref{solution-A}) and equating the result
with \eqref{align11} gives the equation
\begin{equation}\label{eq-n-for-C-1}
\frac{B \sinh(2\sqrt{\lambda_1} \varrho(1)+2C_1)}{2\sqrt{\lambda_1r(0)}
\cosh^2 (C_1)}= C_3 =-\frac{\hat \alpha \tanh (C_1)}{\sqrt{\lambda_1 r(0)}}.
\end{equation}
After some algebraic manipulations and utilization of the notation \eqref{zeta}
we arrive at
\begin{equation}
\tanh(2C_1) = \zeta.
\end{equation}
This results in the following formulas
\begin{gather}\label{C1}
C_1 = \frac{1}{2}\tanh^{-1}(\zeta), \\
\label{C3}
C_3 = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ 2 C_1)}{2\cosh^2(C_1)\sqrt{\lambda_1 r(0)}} = p_1(0),
\end{gather}
the first of which is well-defined since $\zeta \in (0,1)$ by
\eqref{zeta1}.
Here a solution exists as long as the resulting design $p(x)$ is positive definite.
The representation \eqref{solution-A} shows that this is equivalent to the
inequality $C_3>0$, or by \eqref{align11}, $\hat \alpha C_1<0$, or
by \eqref{C1} $\hat \alpha \zeta<0$, or by \eqref{zeta1},
\begin{equation}\label{conditionforsolvability}
\hat\alpha<0.
\end{equation}
The final design is given by
\begin{equation}\label{FinalDesignCaseII}
p_{1;3}(x) = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ \tanh^{-1}(\zeta))}{2\sqrt{\lambda_1 r(x)}\cosh^2(\sqrt{\lambda_1}\varrho(x)
+ \frac{1}{2}\tanh^{-1}(\zeta))}.
\end{equation}
We now consider the solution stemming from $p_2$.
\smallskip
\noindent \textbf{Case (4)} We note that for $\alpha = 0$ the solution does
not exist. Indeed, for $\alpha = 0$, \eqref{BC(0)OrigProb},
together with \eqref{e-13} implies
\begin{equation}
\cosh(C_2) = 0
\end{equation}
which is a contradiction.
\smallskip
\noindent \textbf{Case (5)}
Likewise, for $\alpha = \pi/2$, \eqref{BC(0)OrigProb} implies
\begin{equation}\label{alphapi/2nogo}
C_4\sqrt{r(0)}\sinh(C_2) = 0.
\end{equation}
If $C_2 = 0$, then $p_2(x) = 0$ for all $x \in (0,1)$ which contradicts (A1).
If $C_4 = 0$, then the same contradiction of (A1) is seen;
therefore \eqref{alphapi/2nogo} cannot be satisfied, and the solution does
not exist.
\smallskip
\noindent \textbf{Case (6)} In this case $y=y_2$ as given by \eqref{e-13},
$p=p_2$ as given by \eqref{solution-B}, and $\alpha \not\in \{0,\pi/2\}$.
The boundary condition \eqref{BC(0)OrigProb} immediately implies that
\begin{equation}
\label{C_4}
C_4 = \frac{-\hat{\alpha}\coth(C_2)}{\sqrt{\lambda_1 r(0)}}.
\end{equation}
Isolating $C_4$ from the boundary condition \eqref{BC(1)OrigProb}
(see also \eqref{e-13} and \eqref{solution-B})
and equating the result with \eqref{C_4} gives the equation
\begin{equation}\label{aux-1}
\frac{B \sinh(2\sqrt{\lambda_1} \varrho(1)+2C_2)}{2\sqrt{\lambda_1r(0)}
\sinh^2 (C_2)}=-\frac{\hat \alpha \coth (C_2)}{\sqrt{\lambda_1 r(0)}}.
\end{equation}
After some algebraic manipulations and utilization of the notation \eqref{zeta},
we arrive at
\begin{equation}\label{C_2zetatanh}
\tanh(2C_2) = \zeta.
\end{equation}
This results in the formulas
\begin{gather}
C_2 = \frac{1}{2}\tanh^{-1}(\zeta),\\
C_4 = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ 2 C_2)}{2\sinh^2(C_2)\sqrt{\lambda_1 r(0)}} = p_2(0)
\end{gather}
provided that $\zeta \in (-1,0) \cup (0,1)$. Note that formula for $C_2$
in this case coincides with the formula for $C_1$ in Case(3).
A solution exists in this case as long as the resulting design is positive
definite, again this means that from \eqref{solution-B}, $C_4 > 0$.
By \eqref{C_4} $\hat\alpha C_2<0$ or by \eqref{C_2zetatanh} $\hat{\alpha}\zeta<0$,
or by \eqref{zeta},
\begin{equation}
\frac{\hat\alpha}{\frac{\hat\alpha}{B}+\cosh(2\sqrt{\lambda_1} \varrho(1)}>0.
\end{equation}
Note that this condition is exactly the same as \eqref{conditionforsolvability}.
The final design is given by
\begin{equation}\label{FinalDesignCaseIII-c}
p_{2;6}(x) = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)
+ \tanh^{-1}(\zeta))}{2\sqrt{\lambda_1 r(x)}\sinh^2(\sqrt{\lambda_1}\varrho(x)
+ \frac{1}{2}\tanh^{-1}(\zeta))}.
\end{equation}
So far, the proof does not establish that $\lambda_1>0$ is actually the principal
eigenvalue. We establish this with the help of the zero properties of
the first eigenfunction, see \cite[Theorem 1, p. 445]{Linden}.
According to this theorem, the first (and only first) eigenfunction has no
zeros in $(0,1)$. We now analyze the eigenfunctions \eqref{e-addin}
and \eqref{e-13}. Obviously the eigenfunction $y_2(x)>0$.
The eigenfunction $y_1(x)>0$ in $(0,1)$ if $C_1\ge 0$ which takes place
because either \eqref{align1} for Case (1) or \eqref{C1} and \eqref{zeta1}
for Case (3), and this completes the proof of
Theorem \ref{thm1} part (a).
\end{proof}
\section{``Mass" functional\label{Sect4}}
We now compare the total ``mass'' of each design (critical point),
i.e. \eqref{FinalDesignCaseII} and \eqref{FinalDesignCaseIII-c} for
$\alpha \ne \{0, \pi/2\}$, when both designs exist. Hence, we compare both
\begin{equation}\label{design-1}
M[p_{1;3}] = \frac{C_3\cosh^{2}(C_1)\sqrt{r(0)}}{\sqrt{\lambda_1}}
[\tanh(\sqrt{ \lambda_1 } \varrho(1) + C_1) - \tanh(C_1)],
\end{equation}
and
\begin{equation}
M[p_{2;6}] = \frac{C_4 \sinh^{2}(C_2)\sqrt{r(0)}}{\sqrt{\lambda_1}}[ \coth(C_2) - \coth(\sqrt{ \lambda_1 } \varrho(1) + C_2)],
\end{equation}
where based on previous considerations
\begin{gather*}
C_1 = \frac{1}{2}\tanh^{-1}(\zeta), \quad
C_2 = \frac{1}{2}\tanh^{-1}(\zeta),\\
C_3 = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1) + 2 C_1)}{2\cosh^2(C_1)
\sqrt{\lambda_1 r(0)}}, \\
C_4 = \frac{B\sinh(2\sqrt{\lambda_1}\varrho(1) + 2 C_2)}{2\sinh^2(C_2)
\sqrt{\lambda_1 r(0)}}.
\end{gather*}
Then it follows that
\begin{gather}\label{mass-p-1-3}
M[p_{1;3}] = \frac{B \sinh(\sqrt{\lambda_1}\varrho(1)
+ C_1)\sinh(\sqrt{\lambda_1}\varrho(1))}{\lambda_1 \cosh(C_1)}, \\
M[p_{2;6}] = \frac{-B \cosh(\sqrt{\lambda_1}\varrho(1)
+ C_2)\sinh(\sqrt{\lambda_1}\varrho(1))}{\lambda_1 \sinh(C_2)}.
\end{gather}
At this point, we note that the total ``mass'' for design $p_{2;6}(x)$,
formally speaking, may be negative for some combination of parameters.
Rather than discuss when this ``mass'' is positive, we consider the
following quotient
\begin{equation}
\Big| \frac{M[p_{1;3}]}{M[p_{2;6}]} \Big|
= \Big| \frac{-\sinh(\sqrt{\lambda_1}\varrho(1)+C_1)\sinh(C_2)}
{\cosh(\sqrt{\lambda_1}\varrho(1)+C_2)\cosh(C_1)}\Big|.
\end{equation}
Noting that $C_1 = C_2$, we have
\begin{equation}
\Big| \frac{M[p_{1;3}]}{M[p_{2;6}]} \Big|
= \Big| -\tanh(\sqrt{\lambda_1}\varrho(1)+C_1)\tanh(C_1) \Big| < 1.
\end{equation}
So regardless of when $p_{2;6}(x)$ has a positive ``mass'',
we conclude that the design corresponding to $p_{1;3}(x)$ will always have less
``mass'' than the one corresponding to $p_{2;6}(x)$, and this completes
the proof of part (b), and hence the proof of Theorem \ref{thm1}.
\begin{remark} \label{rmk3} \rm
We analyze the design \eqref{FinalDesignCaseII} as the function of $\alpha$.
It is easy to check that if $\alpha\to 0$, i.e.
$\hat \alpha\to \infty$, then $\zeta\to 0$, and the design
\eqref{FinalDesignCaseII} approaches the design \eqref{FinalDesignCaseI}.
Similarly, if $\alpha\to \pi/2$, i.e. $\hat \alpha\to 0$, then
$\zeta\to -\tanh(2\sqrt{\lambda_1}\varrho(1))$, so that
$2\sqrt{\lambda_1}\varrho(1) + \tanh^{-1}(\zeta)\to 0$, and the design
\eqref{FinalDesignCaseII} is not positive (see Case (2) above).
\end{remark}
\begin{remark} \label{rmk4} \rm
If $\alpha = 0$ then
\begin{equation}
M[p_{1;1}] = \frac{C_3\cosh^{2}(C_1)\sqrt{r(0)}}{\sqrt{\lambda_1}}
[\tanh(\sqrt{ \lambda_1 } \varrho(1) + C_1) - \tanh(C_1)],
\end{equation}
where $C_1 = 0$ as in \eqref{align1} and $C_3$ is given by \eqref{align2}.
Substituting in these values and simplifying gives
\[
M[p_{1;1}] = \frac{B \sinh^2(\sqrt{\lambda_1}\varrho(1))}{\lambda_1}
= \frac{\beta_1+\lambda_1\beta_1'}{\beta_2+\lambda_1\beta_2'}
\frac{ \sinh^2(\sqrt{\lambda_1}\varrho(1))}{\lambda_1}.
\]
In this case, we can recover the result of Turner \cite{Turner}.
To see this, set $\beta_1=\beta_2=\beta_2'=0$, $\beta_1'=M_1$ and $r(x) = \rho$.
This gives
\begin{equation}
M[p_{1;1}] = M_1 \sinh^2(\sqrt{\lambda_1}\int_{0}^{1}\sqrt{\rho} dx)
= M_1 \sinh^2(\sqrt{\lambda_1\rho}).
\end{equation}
Recall from Table \ref{table1} and Table \ref{table2} that
$\lambda = \frac{\omega^2 }{E}$
and
$\gamma^2 = \frac{\omega^2 \rho}{E}$.
From these two equations, it follows that
\begin{equation}\label{sqrtlambda}
\sqrt{\lambda_1} = \frac{\omega_1}{\sqrt{E}}
\end{equation}
and
\begin{equation}\label{gamma}
\gamma_1 = \frac{\omega_1 \sqrt{\rho}}{\sqrt{E}}.
\end{equation}
We see by substituting \eqref{sqrtlambda} into \eqref{gamma} that we have
\begin{equation}
M[p_{1;1}] = M_1 \sinh^2(\gamma_1).
\end{equation}
We see complete agreement with the result of Turner in \eqref{totalmassturner}
since for our problem $L = 1$.
\end{remark}
\section{Optimization problem on a metric graph\label{Sect5}}
We now consider the similar optimization problem on a complete bipartite
metric graph $K_{1,n}$, $n>1$ that we will call the {\sl star} for brevity.
We denote $J:=\{1,\dots ,n\}$ and equip every leaf $e_j$, $j\in J$ of the graph
with the coordinate $x_j\in [0,a_j]$, where $x_j=0$ is the common vertex
of all leafs. The wave type partial differential equations on a metric graph
appear naturally in engineering problems relating to mechanical and electrical
networks \cite{XuMastorakis}. One of the models is a system of strings (or rods)
with the tip masses. After separating variables and removing the harmonic
(in time) factor, we come up with a Sturm-Liouville problem on the system of
strings (see Fig. 1). We assume that the displacements are continuous at
the common point of all string and this point is attached to an elastic string,
so that Hook's law is satisfied. Further, we assume that some masses are
attached to the other end points of the strings (see the boundary condition
\eqref{boundary-1}). Hence, we come to the following problem.
\begin{figure}[hb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1} % Graph.pdf
\end{center}
\caption{Graph $K_{1,n}$}
\end{figure}
We consider the Sturm-Liouville problem on the metric graph
% \begin{gather*}
\begin{equation}\label{SL-1-g}
(p_j(x)y_j'(x))' + \lambda_1 p_j(x)r_j(x)y_j(x) = 0,\quad 00$.
Removing this condition is non-trivial since even to analyze the
Sturm-Liouville problem itself, before solving optimization problem,
it is necessary to work in a space with indefinite metric \cite{AZ}.
\subsection*{Acknowledgments}
B. P. Belinskiy was partially supported by the Tennessee Higher
Education Commission through a CEACSE grant.
D. H. Kotval would like to thank the Honors College, the
Office of the Provost, and the Department of Mathematics at the
University of Tennessee at Chattanooga for supporting this research.
The authors are grateful to the anonymous referees for the numerous
suggestions toward the improvement of this article.
\begin{thebibliography}{99}
\bibitem{Belinskiy2} S. A. Avdonin, B. P. Belinskiy;
On controllability of a rotating string,
\emph{J. of Math. Analysis and Applications}, 321 (1) (2006) 198-212.
\bibitem{B} B. P. Belinskiy, J. V. Matthews, J. W.~Hiestand;
Piecewise uniform optimal design of a bar with an attached mass,
\emph{Electron. J. Diff. Equ.}, 133 (2015) 1-17.
\bibitem{BMH} B. P. Belinskiy, J. W. Hiestand, M. L. McCarthy;
Optimal design of a bar with an attached mass for maximizing the heat transfer,
\emph{Electron. J. Diff. Equ.}, 2012 (181) (2012), 1-13.
\bibitem{Belinskiy1} B. P. Belinskiy, J. P. Dauer;
Eigenoscillations of mechanical systems with boundary conditions containing
the frequency, \emph{Quarterly Appl. Math}, 56 (3) (1998) 521-541.
\bibitem{Cardou} A. Cardou;
Piecewise uniform optimum design for axial vibration requirement,
\emph{AIAA J.}, 11 (1973), 1760--1761.
\bibitem{Fulton2} C. T. Fulton;
Singular eigenvalue problems with eigenvalue parameter contained in the
boundary conditions, \emph{Proc. Roy. Soc. Edinburg}, 87A (1980), 1-34.
\bibitem{Fulton1} C. T. Fulton;
Two-point boundary value problems with eigenvalue parameter contained in
the boundary conditions, \emph{Proc. Roy. Soc. Edinburg}, 77A (1977), 293-308.
\bibitem{GF} I. M. Gelfand, S. V. Fomin;
\emph{Calculus of Variations}, Mineola, New York: Dover, 1963.
\bibitem{HM} D. Hinton, M. McCarthy;
Bounds and optimization of the minimum eigenvalue for a vibrating system,
\emph{Electron. J. Diff. Equ.}, 48 (2013), 1-22.
\bibitem{Hinton} D. Hinton;
An expansion theorem for an eigenvalue problem with eigenvalue parameter
in the boundary condition, \emph{Quart. J. Math. Oxford}, 2 (3) (1979), 33–42.
\bibitem{Linden} H. Linden;
Leighton's bounds for Sturm-Liouville eigenvalues with eigenvalue parameter
in the boundary conditions, \emph{J. of Math. Analysis and Applications}, 156 (1991),
444-456.
\bibitem{Taylor} J. E. Taylor;
Minimum mass bar for axial vibration at specified natural frequency,
\emph{AIAA Journal}, 5 (10) (1967) 1911-1913.
\bibitem{Turner} M. J. Turner;
Design of minimum mass structures with specified natural frequencies,
\emph{AIAA Journal}, 5 (3) (1967), 406-412.
\bibitem{Walter} J. Walter;
Regular eigenvalue problems with eigenvalue parameter in the boundary condition,
\emph{Math. Z.}, 133 (1973), 301-312.
\bibitem{XuMastorakis} G. Q. Xu, N. E. Mastorakis;
\emph{Differential Equation on Metric Graph}, World Scientific
Engineering Academy and Society Press, 2010.
\bibitem{AZ} A. Zettl;
\emph{Sturm-Liouville Theory. Mathematical Surveys and Monographs, v. 121},
Rhode Island: American Mathematical Society, 2005.
\end{thebibliography}
\end{document}