\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\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 0<x<L.
 \end{equation}

 Here and below we use the notation given in Table \ref{table1}.
  After separating variables and removing the harmonic (in time) term,
we come up with the following Sturm-Liouville optimization problem for a
rod supporting a non-structural mass $M_1$.

\begin{table}[hbt]
\caption{Physical Interpretation of Parameters} \label{table1}
 \begin{center}
 \begin{tabular}{|c|c|} \hline
 Quantity & Interpretation \\\hline
 $E$ & Young's Modulus \\
 $u(x,t)$ & Axial Displacement \\
 $\rho$ & Density of Rod Material \\
 $A(x)$ & Cross-sectional Area\\
 $m(x)$ & Mass per Unit Length (=$\rho A(x)$)\\
 $\gamma^2$ & $\omega^2\rho/E$ \\
 $\omega$ & Angular Frequency \\
 $M_1$ & Non-Structural Mass Supported at the End of The Rod \\
\hline
\end{tabular}
\end{center}
 \end{table}


\begin{problem} \label{prob1} \rm
Let $u(x)$ be a nontrivial solution of the differential equation
 \begin{equation}\label{e-2}
 \frac{\mathrm{d} }{\mathrm{d} x}\big( m\frac{\mathrm{d} u}{\mathrm{d} x} \big) 
+ \gamma_1^2mu = 0,\quad 0<x<L,
 \end{equation}
 for specified natural frequency $\omega_1(=\gamma_1\sqrt{E/\rho}$), 
subject to the boundary conditions
 \begin{equation}\label{boundary-1}
 u(0)=0,\quad m u'(L)=\gamma_1^2 M_1 u(L).
 \end{equation}
Find the mass distribution $m(x) = m_{opt}(x)$ such that the total mass functional,
 \begin{equation}
 M_{0}[m]:= \int _{0}^{L} m dx
 \end{equation}
 attains its minimum value. $\Box$
\end{problem}

Since the problem is homogeneous, we may normalize the solution as follows
 \begin{equation}\label{boundary-2}
 u(L)= 1\quad\text{so that }mu'(L)=\gamma_1^2 M_1.
 \end{equation}
Note that the spectral parameter $\gamma_1^2$ appears linearly in the boundary 
condition. To determine a solution to this problem, Turner seeks to minimize 
the following mass functional in which the equations of motion and the 
boundary conditions are introduced as isoperimetric constraints 
\cite{Turner,Taylor}:
 \begin{align}\label{e-3}
 \Phi[m,u] &:= M_{0}[m] + \int_{0}^{L} \lambda(x)[ (mu')' 
+ \gamma_{1}^{2}mu] dx + \lambda_{1}[\gamma_{1}^{2} M_{1} - m(L)u'(L)].
 \end{align}
Here the $\lambda$'s are Lagrange multipliers. Turner carries out an analysis 
using the techniques of the Calculus of Variations \cite{GF} to find that 
the optimal mass distribution $m_{opt}(x)$ is given by
 \begin{equation}\label{e-5}
 m_{opt}(x)=m(L) \cosh^2(\gamma_{1}L)/\cosh^2(\gamma_{1}x)
 \end{equation}
 where
 \begin{equation}\label{e-6}
 m(L) = \gamma_{1} M_{1} \tanh \gamma_{1}.
 \end{equation}
The total mass for this design is then
 \begin{equation}\label{totalmassturner}
 M_0[m_{opt}]=M_1\sinh^2(\gamma_1L).
 \end{equation}
Formulas \eqref{e-5} and \eqref{e-6} represent the complete
solution of Problem \label{prob1}\ref{prob1}.

 In this article, rather than working with
Problem \ref{prob1} which models the axial vibrations of a rod, 
we consider a general Sturm-Liouville problem with the spectral parameter
 that appears linearly in the boundary conditions. For the general theory 
of this problem see Hinton \cite{Hinton}, Fulton \cite{Fulton1,Fulton2}, 
and Walter \cite{Walter}. This generalization results in some new phenomena, 
such as the occurrence of an additional critical point and some conditions of 
solvability, that did not occur in the models 
\cite{Turner,Taylor,BMH}.

 We adopt the notation from 
\cite{Hinton,Fulton1,Fulton2,Walter}, for dealing with this problem, 
that is, we consider
 \begin{gather}\label{e-7}
 (p(x)y'(x))' -q(x)y(x) +\lambda p(x)r(x)y(x) =0,\quad x\in (0,1), \\
 \label{e-7a}
 \cos\alpha\, y(0)+ \sin \alpha\, (p(0)y'(0)) = 0, \\
\label{e-7b}
 -\beta_1 y(1) + \beta_2 p(1)y'(1) = \lambda[\beta_1' y(1) - \beta_2' p(1)y'(1)], \\
\label{inequality}
 \delta :=\beta_1'\beta_2-\beta_1\beta_2'>0.
 \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 0<x_j<a_j,\;j\in J; 
\end{equation}
\begin{equation}\label{BC-1-g}
 y_j(0)=y_k(0)\quad\text{for all } j,\,k\in J,\; j\ne k; 
\end{equation}
\begin{equation}\label{BC-2-g}
 \cos\alpha\,y_j(0)+ \sin\alpha \sum_J (p_j(0)y_j'(0)) = 0, 
\end{equation}
\begin{equation}\label{BC-3-g}
 -\beta_{1,j} y_j(a_j) + \beta_{2,j} p_j(a_j)y_j'(a_j)
 = \lambda_1[\beta_{1,j}' y_j(a_j) - \beta_{2,j}' p_j(a_j)y_j'(a_j)],\quad j\in J.
\end{equation}
% \end{gather*}
 Here and below we use the abbreviation
 $\sum _J:=\sum_{j\in J}$.

The boundary condition %\eqref{BC-1-g} 
(\ref{BC-1-g}) allows us to let $y_j(0):=1$, $j\in J$.
We note that the condition \eqref{BC-2-g} has the meaning of Hook's law, 
and that allows us to view the graph $K_{1,n}$ as a mechanical construction. 
Hence, it is natural to introduce the following simplifying assumption
\begin{equation} \label{eass-2}
 p_j(0)=p_k(0):=p(0), \quad r_j(0)=r_k(0):=r(0)\quad \forall j,k\in J,
\end{equation}
which means that  the cross-sectional area of the rods and their densities
are continuous at the common knot $x=0$.

 Our goal is to optimize the ``mass'' functional
 \begin{equation}\label{mass-g}
 M:=\int_{K_{1,n}}\,rp\,dx.
 \end{equation}
We introduce the functional similar to \eqref{functional-F[y,p]},
 \begin{equation} \label{F-g}
\begin{aligned}
F[p_1,\dots ,p_n,y_1,\dots ,y_n]
&:=\int_{K_{1,n}}
\Big(rp+\Lambda_1\Big(((py')'+\lambda_1 rpy\Big)\Big) dx \\
&\quad +\sum_J \Lambda_2\Big(-\beta_1y+\beta_2py'-\lambda_1 [\beta'_1y-\beta'_2py']\Big).
 \end{aligned}
\end{equation}
 Here and below we use the abbreviations
 $$
\int_{K_{1,n}}\,f dx:=\sum_J\,\int_0^{a_j}\,f_j dx,\quad
\sum_J\,f:=\sum_J\,f_j(a_j).
$$
 As in  Section 2, we use the optimality condition $\delta F=0$.
 We skip cumbersome calculations that are philosophically similar to once
in  Section 2 and allow us to find two types of critical point on each of
the leafs $e_j$,
 \begin{equation} \label{y-p-1}
\begin{gathered}
 y_j^+(x)=\frac{\cosh(\sqrt{\lambda_1}\varrho_j(x)+C_j^+)}{\cosh(C_j^+)},\quad
 p_j^+(x)=\frac{C\sqrt{r(0)}\sinh^2(C_j^+)}{\sqrt{r_j(x)}
 \sinh^2(\sqrt{\lambda_1}\varrho_j(x)+C_j^+)}, \\
 y_j^-(x)=\frac{\sinh(\sqrt{\lambda_1}\varrho_j(x)+C_j^-)}{\sinh(C_j^-)},\quad
 p_j^-(x)=\frac{C\sqrt{r(0)}\cosh^2(C_j^-)}{\sqrt{r_j(x)}
 \cosh^2(\sqrt{\lambda_1}\varrho_j(x)+C_j^-)},
 \end{gathered}
\end{equation}
where
 \begin{equation}\label{rho-g}
 \varrho_j(x):=\int_0^x \sqrt{r_j(s)} ds
 \end{equation}
and $C_j^{\pm},\,C$ are arbitrary constants, so that $C=p_j^{\pm}(0)$.
We note that the constant $C$ is indeed the same for all $j$ in view of
\eqref{eass-2}.

 Since any of the critical points may be chosen on the $j-$th leaf, 
the total number of critical points for the optimization problem on the 
graph $K_{1,n}$ is $2^n$. We denote the set of leafs where the point 
$(y_j^{\pm},p_j^{\pm})$ is chosen on the $j-$th leaf as $J^{\pm}$, 
so that $J^+\cup J^-=J$. We do not exclude that one of the sets $J^{\pm}$ is empty. 
The boundary condition \eqref{BC-2-g} implies
 \begin{equation}\label{BC-2-g-1}
 \cos \alpha +\sin \alpha C\sqrt{\lambda_1 r(0)} \Big(\sum_{J^+}
\tanh (C_j^+)+\sum_{J^-}\,\coth (C_j^-)\Big) = 0,
 \end{equation}
so that
 \begin{equation}\label{C-1}
 C=-\frac{\hat{\alpha}}{\sqrt{\lambda_1 r(0)} \Big(\sum_{J^+}\tanh (C_j^+)
+\sum_{J^-}\,\coth (C_j^-)\Big)},
 \end{equation}
where $\hat\alpha$ is defined by \eqref{alpha}. The boundary conditions at
 the vertices $x=a_j,\,j\in J$ of the leafs lead to the equations

 (a) If $j\in J^+$, then
\begin{equation} \label{bc-g-1}
 \begin{aligned}
&-\beta_{1,j} \cosh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^+\big)
 + \beta_{2,j}C\sqrt{\lambda_1 r(0)}\frac{\sinh^2(C_j^+)}
 {\sinh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^+\big)} \\
&=  \lambda_1\Big[\beta'_{1,j} \cosh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^+\big)
  - \beta'_{2,j}C\sqrt{\lambda_1 r(0)}
 \frac{\sinh^2(C_j^+)}{\sinh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^+\big)}\Big].
 \end{aligned}
\end{equation}

 (b) If $j\in J^-$, then
\begin{equation} \label{bc-g-2}
 \begin{aligned}
&-\beta_{1,j} \sinh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^-\big) +
 \beta_{2,j}C\sqrt{\lambda_1 r(0)}\frac{\cosh^2(C_j^-)}
 {\cosh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^-\big)} \\
&=  \lambda_1\Big[\beta'_{1,j} \sin\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^+\big) -
 \beta'_{2,j}C\sqrt{\lambda_1 r(0)}\frac{\cosh^2(C_j^-)}
 {\cosh\big(\sqrt{\lambda_1}\varrho_j(a_i)+C_j^-\big)}\Big].
 \end{aligned}
\end{equation}
 Using notation similar to \eqref{B},
 \begin{equation}\label{B-g}
 B_j:=\frac{\beta_{1,j}+\lambda_1 \beta'_{1,j}}{\beta_{2,j}
+\lambda_1 \beta'_{2,j}},\quad j\in J
 \end{equation}
in equations \eqref{bc-g-1} and \eqref{bc-g-2} we find two expressions for $C$:
 \begin{equation} \label{C}
\begin{gathered}
 C=\frac{B_j}{2\sqrt{\lambda_1 r(0)}}
 \frac{\sinh(2\sqrt{\lambda_1}\varrho_j(a_j)+2C_j^+)}{\sinh^2(C_j^+)}\quad
\text{if }j\in J^+; \\
 C=\frac{B_j}{2\sqrt{\lambda_1 r(0)}}
 \frac{\sinh(2\sqrt{\lambda_1}\varrho_j(a_j)+2C_j^-)}{\sinh^2(C_j^-)}\quad
\text{if }j\in J^+ .
 \end{gathered}
\end{equation}
 Combining the formulas \eqref{C-1}, \eqref{C} we get the system of $n$
equations for $n$ constants $C_j^{\pm},\,j\in J$ similar to \eqref{eq-n-for-C-1}
and \eqref{aux-1},
\begin{equation} \label{system-1}
\begin{aligned}
&\frac{B_j\sinh(2\sqrt{\lambda_1}\varrho_j(a_j)+2C_j^+)}{\sinh^2(C_j^+)} \\
&= -\frac{\hat \alpha}{\sum_{J^+}\,\tanh (C_j^+)+\sum_{J^-}\,\coth (C_j^-)}
\quad \text{if }j\in J^+; \\
&\frac{B_j\sinh(2\sqrt{\lambda_1}\varrho_j(a_j)+2C_j^-)}{\cosh^2(C_j^-)} \\
&= -\frac{\hat \alpha}{\sum_{J^+}\,\tanh (C_j^+)+\sum_{J^-}\,\coth (C_j^-)}
\quad \text{if }j\in J^-.
 \end{aligned}
\end{equation}
 It may be shown that if $J^+=\emptyset,\,J^-=\{1\}$ or
$J^-=\emptyset$, $J^+=\{1\}$, the system \eqref{system-1} results
in \eqref{eq-n-for-C-1} or \eqref{aux-1}. We are not optimistic though
about possibility to solve the system \eqref{system-1} in the general case.

 \section{Optimization problem on a star graph with identical leafs. 
The limiting case} \label{Sect6}

 Instead of the general star, we consider a particular case when all leafs are 
of the same length and the $p_j, r_j, B_j$ are the same on all leafs, so 
that we may skip the index $j$. We also let $a_j:=1$ as in  Sections 2-4. 
We assume $J^+=\emptyset$.
 This choice is based on the observation that the design $p_{1;3}$ in 
Theorem \ref{thm1} (Section 4) results in the minimal ``mass''.
 Correspondingly, we assume $\alpha\ne \pi/2$. Following these Sections, 
we denote $C_j^-:=C_1$.

 \begin{theorem} \label{thm2}
 For a star graph with identical leafs, the following statements hold.
\begin{itemize}
\item[(a)] The ``mass'' \eqref{mass-g} has the form
 \begin{equation}\label{mass-g-n-1}
 M=\frac{n B \sinh(\sqrt{\lambda_1}\varrho(1)+C_1)
\sinh(\sqrt{\lambda_1}\varrho(1))}{\lambda_1 \cosh(C_1)}.
 \end{equation}

\item[(b)] If the parameter $\kappa:=n B$ is large and
 $\alpha\in (0,\pi/2)$, then the asymptotic representation holds
 \begin{equation}\label{mass-g-n-11}
 M=\frac{\hat\alpha}{\lambda_1}\,\sinh^2(\sqrt{\lambda_1}
\varrho(1))+O\Big(\frac{1}{\kappa}\Big).
 \end{equation}
\end{itemize}
\end{theorem}

\begin{proof} 
Instead of the system \eqref{system-1} we have
 \begin{equation}\label{C-1-n-1}
 \frac{B \sinh(2\sqrt{\lambda_1}\varrho(1)+2C_1)}{2\cosh^2(C_1)}
=-\frac{\hat\alpha}{n \coth C_1},
 \end{equation}
so that similarly to \eqref{C1} and \eqref{zeta},
 \begin{equation}\label{C-1-n-2}
 C_1=\frac{1}{2}\tanh^{-1}(\zeta_n)
 \end{equation}
 where
 \begin{equation}\label{zeta-n}
 \zeta_n:=-\frac{\sinh(2\sqrt{\lambda_1}\varrho(1))}{\frac{\hat \alpha}{B n}+
 \cosh(2\sqrt{\lambda_1}\varrho(1))}.
 \end{equation}
Here $\varrho(x)$ is defined as in \eqref{rho-th} and \eqref{rho-g}.

The design \eqref{y-p-1} implies
 \begin{equation}\label{p-n}
 p(x)=\frac{B\sinh(2\sqrt{\lambda_1}\varrho(1)+\tanh^{-1}(\zeta_n))}
 {2\sqrt{\lambda_1 r(x)}\cosh^2(\sqrt{\lambda_1}\varrho(x)
+\frac{1}{2}\tanh^{-1}(\zeta_n))},
 \end{equation}
 which is similar to \eqref{FinalDesignCaseII}.

We finally evaluate the ``mass'' \eqref{mass-g},
 \begin{equation}\label{mass-g-n}
 M=\int_{K_{1,n}}\,rp dx =n\,\int_0^1\,rp dx 
 =\frac{n B \sinh(\sqrt{\lambda_1}\varrho(1)+C_1)
\sinh(\sqrt{\lambda_1}\varrho(1))}{\lambda_1 \cosh(C_1)}
 \end{equation}
 and this completes the proof of Theorem \ref{thm2} (a). 
We note that the representation \eqref{mass-g-n} is similar 
to \eqref{mass-p-1-3}.

 We further consider the limiting case $n\to \infty$, that may be interpreted 
as optimization problem for a star with infinitely many leafs. 
More specifically, we assume that the parameter $\kappa=B n$ is large, i.e.
 \begin{equation}\label{B-g-n}
 \kappa:=n B \gg 1.
 \end{equation}
Our goal is to find the leading terms of the asymptotic representation for 
the ``mass'' as $\kappa\to \infty$. Firstly, we find from \eqref{zeta-n}
 \begin{equation}\label{zeta-n-i}
 \zeta_n=-\tanh(2\sqrt{\lambda_1}\varrho(1))+
 \frac{\hat\alpha}{\kappa}
\frac{\sinh(2\sqrt{\lambda_1}\varrho(1))}{\cosh^2(2\sqrt{\lambda_1}
\varrho(1))}+O\Big(\frac{1}{\kappa^2}\Big).
 \end{equation}
Further, from \eqref{C-1-n-2} we derive 
 \begin{equation} \label{C-1-n-3}
 2C_1=\tanh^{-1}(\zeta_n)=-2\sqrt{\lambda_1}\varrho(1)
+\frac{\hat\alpha}{\kappa}\cdot \sinh(2\sqrt{\lambda_1}\varrho(1))
 + O\Big(\frac{1}{\kappa^2}\Big),
\end{equation}
so that
\[
\sqrt{\lambda_1}\varrho(1)+C_1=\frac{\hat\alpha}{\kappa}
 \frac{\sinh(2\sqrt{\lambda_1}\varrho(1))}{2}+
 O\Big(\frac{1}{\kappa^2}\Big).
\]
Based on \eqref{y-p-1}, \eqref{C}, \eqref{C-1-n-3}, we now can find the
asymptotic representation for $y(x)$ and $p(x)$. We skip calculations
and only give the results
 \begin{equation} \label{y-p-n-i}
\begin{gathered}
 y(x)=\frac{\sinh(\sqrt{\lambda_1}(\varrho(1)-\varrho(x)))}
{\sinh(\sqrt{\lambda_1}\varrho(1))}
 +O\Big(\frac{1}{\kappa}\Big), \\
p(x)=\frac{\sinh(2\sqrt{\lambda_1}\varrho(1)) \hat\alpha}
{2n\sqrt{\lambda_1 r(x)} \cosh^2(\sqrt{\lambda_1}(\varrho(1)-\varrho(x)))}
 +O\Big(\frac{1}{\kappa^2}\Big).
\end{gathered}
\end{equation}
The asymptotic representation for the ``mass'' \eqref{mass-g-n} appears based
on the asymptotic representation \eqref{C-1-n-2}.
After some algebraic manipulations we find
 \begin{equation}\label{mass-i}
 M=\frac{\hat\alpha}{\lambda_1}\,\sinh^2(\sqrt{\lambda_1}
\varrho(1))+O\Big(\frac{1}{\kappa}\Big).
 \end{equation}
The answer makes sense if $\alpha\in (0,\pi/2)$. This completes the proof
of part (b), and hence the proof of Theorem \ref{thm2} is complete.
\end{proof}

\begin{remark} \label{rmk5}\rm 
 Comparison of the formulas \eqref{zeta-n} and \eqref{zeta} shows 
that the formulas for one interval and the star with identical leafs are 
quite similar except the parameter $\hat\alpha$ is changed for
 $\hat\alpha/n$.
\end{remark}

\begin{remark} \label{rmk6}\rm
(a) The leading terms of the asymptotic representation for $y(x),p(x)$ 
and $M$ do not depend on the parameters $\beta_k,\beta'_k$, $k=1,2$.
 
(b) It is rather easy to check that the leading terms of the asymptotic 
representation \eqref{y-p-n-i} for $y(x)$ and $p(x)$ satisfy the boundary 
conditions at the vertex $x=0$ exactly and the boundary conditions at 
the vertices $x_j=1$ within an error $O\big(\frac{1}{\kappa}\big)$.
\end{remark}

\begin{remark} \label{rmk7}\rm
 We suggest the following interpretation of our asymptotic formulas.

\noindent (a) The boundary condition \eqref{BC-2-g} at the vertex $x=0$ 
of the graph has the form
 \begin{equation}\label{BC-2-g-n}
 \cos\alpha\,y(0)+\sin\alpha\cdot n\,p(0)y'(0)=0.
 \end{equation}
 It may be viewed as ``almost'' Neumann condition for the Sturm-Liouville
 problem on a single interval $(0,1)$,
 \begin{equation}\label{BC-2-g-n-1}
 p(0)y'(0)=-\frac{\hat\alpha}{n}
 \end{equation}
 where we use our usual normalization $y(0)=1$ and the notation 
$\hat\alpha=\cot \alpha$ \eqref{alpha}. Hence, for $n\to \infty$, 
the star is  split into $n$ disconnected leafs with the boundary condition 
at $x=0$ that ``approaches" Neumann condition. It is not too complex to 
check the following. If we take the formula for the optimal ``mass'' $M[p]$ 
for the Sturm-Liouville problem on one interval \eqref{mass-p-1-3} with 
$\hat\alpha$ formally substituted by $\hat\alpha/n$ (see Remark \ref{rmk5}), 
then multiply this ``mass'' by $n$, and  find the first term of 
asymptotic representation as $n\to \infty$, then we get the leading term 
of the formula for the ``mass'' of the star \eqref{mass-i}. 
We skip this calculation since it almost repeats calculation 
\eqref{zeta-n-i}-\eqref{mass-i}.

\noindent (b) It is interesting to note that if we consider the limiting case 
of the boundary condition \eqref{BC-2-g-n-1}, i.e. $y'(0)=0$, then, in
 terms of the boundary condition \eqref{e-7a}, we need to let $\alpha=\pi/2$, 
and, as we show in  Section 3, the corresponding critical point does 
not exist. Simultaneously, the leading term of the asymptotic representation
 \eqref{mass-i} for the ``mass'' of the star vanishes. That shows that indeed 
the value of the parameter $\alpha=\pi/2$ should be excluded from consideration.
\end{remark}

 \section{Conclusion and discussion}\label{Sect7}

 We consider the optimal design problem modeled by a Sturm-Liouville problem 
on an interval or a complete bipartite graph and find the explicit formulas 
for the optimal design. We analyze the intervals of the parameters where 
such a design exists. We are motivated by the known publications on 
(a) the Sturm-Liouville problem with the spectral parameter that appears 
in the boundary conditions linearly; 
(b) optimization problem for an  elastic rod with an attached mass; 
(c) differential equations describing  mechanical and electrical networks.
\smallskip

\noindent  1. There are two surprises the authors discovered in this study. 
(a) The existence of a solution corresponding to the design $p_2(x)$,
 not only to $p_1(x)$ as in \cite{Turner}, \cite{Taylor}, \cite{BMH}.
 (b) The existence of the limit of the ``mass'' functional for the 
star as the number of leafs increases indefinitely. 
As for (a), this other solution was not expected, though the fact that 
it does not exist for either $\alpha = 0$ or $\alpha = \pi/2$ explains, 
to some extent, why it was elusive. In this work, it appears unfruitful 
since it does not lead to a minimum ``mass'' design, yet we feel it is 
important to include since this critical point might be of interest 
for other optimization problems. It is also intriguing that this solution 
does not exist for both $\alpha = 0$ and $\alpha = \pi/2$. 
As for (b), we interpret this phenomenon in terms of the split of the star 
into disconnected leafs with ``almost'' Neumann condition at the vertex $x=0$. 
Generally speaking, for a mechanical construction, disconnection of the 
leafs may result in a destruction of this construction. Both phenomena 
(a) and (b) may give an interesting chance for further studies.
\smallskip

\noindent 2. In the case $\alpha=0$, the functional $M[p]$ has only one 
critical point, and based on the duality that was derived in \cite{Taylor}, 
we may expect that the following two problems have the same optimal solution $p(x)$:

 (I) Given $r(x)$, $\beta_1$, $\beta_2$, $\beta_1'$, $\beta_2'$, and $\lambda_1$, 
find $p(x)$ such that $M[p]$ $\to \min$.

 (II) Given $r(x)$, $\beta_1$, $\beta_2$, $\beta_1'$, $\beta_2'$, and $M$, 
find $p(x)$ such that $\lambda_1[p]$ $\to \max$. 
 
 We have solved Problem (I) but may hope that the optimal $p(x)$ from solving 
(I) is the same as the optimal $p(x)$ from solving (II). The validity of this 
duality in the case of multiple critical points should be studied further.
\smallskip

 \noindent 3. We have made some restrictions on the data in the process of the 
construction of the optimal solution. Removing them would represent a 
challenging problem. (a) We assumed $q(x)\equiv 0$. The reason for this 
is twofold. First, in many applications of the problem \eqref{e-7}-\eqref{e-7b}, 
there is no term containing the function $q(x)$. Second, the calculations 
of the optimal form for $q(x)\not \equiv 0$ seem to be intractable in the 
frame of an analytic approach. Yet, the complete analysis here is probably 
possible at the numerical level. For example, an alternative approach for 
a similar but simpler problem based on the discretization is developed 
in \cite{Cardou}, \cite{B}, \cite{BMH}. (b) We assumed $r(x)>0$. 
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. 

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