%          golden-path-ICD.tex (2007.5.7�t��)��
%        Editorial Assistant Flora Galindo
%       "Elaydi, Saber" <selaydi@trinity.edu>
%        ��
%       �W�œ��e�ł�email-submit�����B
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%          golden-path-ICD.tex 2006.10.3 ==> see golden-path.tex 2006.5.28
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%                    in Discrete-Time Dynamic Optimization Processes
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%     (����ߒ��E�z���ߒ� ==> ���U���ԓ��I�œK)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          golden-path.tex 2006.5.28 ==> see Golden-opti-path-abs.tex     
%          Golden Optimal Path 
%                    in Discrete-Time Dynamic Optimization Processes
%          �i���U���ԓ��I�œK�ߒ��ɂ����鉩���œK�o�H
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%     (����ߒ��E�z���ߒ� ==> ���U���ԓ��I�œK)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% 2006.5.28�t���ŕ\�����L�̂悤�ɕύX����  %%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          golden-opti-path-abs.tex 2006.4.24 ==> golden-path.tex 2006.3.2
%          This is Abstract of (page 1 only)
%     \item %\bibitem{tk51} 
%     S.~Iwamoto and M.~Yasuda, 
%     Golden Optimal Path in Discrete-Time Dynamic Optimization Processes, 
%     The 15-th International Conference on Difference Equations and Application%     s (ICDEA 2006), Kyoto University, Kyoto, Japan, July 24-28, 2006.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          golden-path.tex 2006.3.2 ==> golden-traj.tex 2006.2.6 ==>
%                                      see also cost-rate.tex as for "graphs"
%          golden-path.tex <===> �����œK�o�H
%       
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%                      in control and allocation processes
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%%  1.  �O��  <== �O��
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%  1. The Golden Inequalities (Org)
%  2. Cross Dual on The Golden Inequalities (Org)
%  3. A Class of The Golden Inequalities and Their Dual (Org) 
%
%  The Golden Rule is  1 a principle which states that you should treat others  %                        as you want them to treat you.
%                      2 a very important principle, way of behaiving etc. that %                        should be remembered. 
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%  3. The Golden Optimum (Solutions)
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%  3. The Golden Optimum  Solutions
%  4. The Golden Inequalities
%
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%  2. The Golden Triplet of Optimul Solutions
%  3. The Golden Optimization (Problems)   �����œK��
%  4. The Golden Inequalities
%  5. The Golden Inequalities on Triplet of Quadratic Forms
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%                ==> see golden-ququ.tex   2004.8.31  ==> 
%
%  1. The Golden Optimum Solution
%  2. A Ceratain Class of Problems Which Have the Golden Optimum Solution 
%  3. A Duality on Optimum Value and Optimum Point 
%  4. A Duality on Optimum Value and Optimum Line 
%  5. The Golden Cross $ X $ 
%  6. The Golden Crossing $ X $ 
%  7. A Duality on The Golden Optimum Solution 
%  8. A Duality on The Golden Optimality 
%  9. A Crossing of The Golden Optimality 
% 10. The Golden-Optimum-Solution Crossing 
% 11. The Dual Golden Optimum Solution (System)
% 12. On Dual Golden Optimum Solution 
% 13. (On) Dual Golden Optimality 
% 14. (On) Dual Golden Optimum Points
% 15. Duality of Golden Optimum Solution 
% 16. On Crossing Dual Golden Optimum Solutions 
% 17. Crossing Duality of Golden Optimum Solution 
% 18. Dual Golden Extremum Points
% 19. A Road Map to (of) Dual Golden Extremum Points
% 20. A Road to Dual Golden Extremum Points
% 21. The Gold
% 22. A Road to Dual Golden Optimum Points
% 23. Road to Dual Golden Optimum
% 24. Road to Golden Optimum
% 25. (Return) Road from Golden Optimum
% 26. Road from Golden Optimum
% 27. Crossing Dual and Crossing two Sum on the Golden Optimum Solutions
% 28. Crossing Dual and Two Sum on the Golden Optimum
%
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%
%  1. Quadraric and Square-Root Criteria 
%                       in Nondeterministic Dynamic Programming
%
%  2. Contolled Integral Equations: Quadratic and Square-Root Criteria
%  3. Quadratic and Square-Root Criteria 
%                        in Contolled Integral Equations 
%  1.   Linear-Quadraric Criteria in Nondeterministic Dynamic Programming
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%          bridge.tex   2004.1.4 ==>
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%
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%
%  1.   Non-deterministic Dynamic Programming; continuous state model
%  2.   Multi-stage Decision Problem For Mlultiple Integrals
%  3.   Multi-stage Variational Problems
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\begin{document}%
  \baselineskip = 15pt
\title{Golden optimal path
in discrete-time dynamic optimization processes
%
%�i�O�\��)
%Golden optimal trajectories\\ in \\ control and allocation processes \\[10mm]
% ����ߒ��E�z���ߒ��ɂ�����\\[2mm]
% �����œK�O��  \\[8mm]
% (tentative) \\
%: golden-traj.tex
%
}
 \author{Seiichi IWAMOTO and Masami YASUDA\\
~~ \\
Department of Economic Engineering \\
Graduate School of Economics, Kyushu University \\
Fukuoka 812-8581, Japan \\
{\small tel\&fax. +81(92)642-2488,~ email: iwamoto@en.kyushu-u.ac.jp} \\
and \\
Department of Mathematical Science \\
Faculty of Science, Chiba University \\
Chiba 263-8522, Japan \\
{\small tel\&fax. +81(43)290-3662,~ email: yasuda@math.s.chiba-u.ac.jp}
}
%\date{\today}
\date{}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%          golden-traj.tex <===> �����œK�O��
%          Golden optimal trajectories  
%                      in control and allocation processes
%          �i����ߒ��E�z���ߒ��ɂ����鉩���œK�O���j
%          golden-traj.tex 2006.2.6 ==>
%                ==> see golden-opt-poli.tex 2006.1.4  and   golden.tex
%          golden-traj.tex �� 
%           (1) ����ߒ��E�z���ߒ��̉����O�����œK�ł��邱�Ƃ������B
%           (2) ���I�v��̉������􂪍œK�ł��邱�Ƃ𓯎����s�I�Ɏ����B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
%\tableofcontents
%\listoftables 
%\listoffigures 
%

\vskip 6mm 

\begin{abstract}%
We are concerned with dynamic optimization processes from a viewpoint of Golden optimality. 
A path is called Golden if any state moves to the next state repeating the same Golden section in each transition. 
A policy is called Golden if it, together with a relevant dynamics, yields a Golden path. 
The probelm is whether an optimal path/policy is Golden or not. 
This paper minimizes a quadratic criterion and maximizes a square-root criterion over an infinite horizon. We show that a Golden path is optimal in both optimizations. The Golden optimal path is obtained by solving a corresponding Bellman equation for dynamic programming. This in turn admits a Golden optimal policy.
\end{abstract}

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%\longdiv{1234}{5}\longdiv{333.33}{2.915}
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%
%*******************************************************************
%
\section{Introduction}
%
%*******************************************************************
Recently the Golden optimal solution, its duality, and its equivalence have been discussed in static optimization problems \cite{e78,e80,e85}. In this paper we consider the Golden optimal solution in dynamic optimization problems. 

We consider two typical types of criterion --- quadratic and square-root --- in a deterministic optimization. 
We minimize quadratic criteria 
\begin{eqnarray*}
I(x) \dql \dsum_{n=0}^{\infty}\left[x_{n}^{2} + (x_{n} - x_{n+1})^{2} \right],~~~J(x) \dql \dsum_{n=0}^{\infty}\left[(x_{n} - x_{n+1})^{2} + x_{n+1}^{2} \right]
\end{eqnarray*}
and maximize square-root criteria 
\begin{eqnarray*}
K(x) \dql \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n}\,} +  \sqrt{x_{n} - x_{n+1}\,} \right),~~~ 
L(x) \dql \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n} - x_{n+1}\,} + \sqrt{x_{n+1}\,} \right) 
\end{eqnarray*}
, respectively. Here $ \beta $ is $ 0 < \beta < 1. $ 
%
%
The differences between $ I $ and $ J $ and between $ K $ and $ L $ are  
\begin{eqnarray*}
J(x) \eql I(x) - x_{0}^{2} \nonumber \\[1.56mm]
L(x) \eql K(x) + \dsum_{n=0}^{\infty} \left(\beta^{n-1} - \beta^{n}\right) \sqrt{x_{n}\,}~~~~~(\beta^{-1} = 0).
\end{eqnarray*}
We show that a Golden path is optimal in these four optimization problems. The Golden optimal path is obtained by solving Bellman equation for dynamic programming \cite{iwae,iwya}. 

%\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Golden Paths}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A real number 
$$ \phi = \dfrac{1 + \sqrt{5}}{2} \, \thickapprox \, 1.618 $$
is called {\it Golden number} \cite{abbp,dunl,wals}. It is the larger of the two solutions to quadratic equation  
\begin{equation}  
    x^{2} - x - 1 = 0.    \label{eqn:fi1} 
\end{equation}
Sometimes (\ref{eqn:fi1}) is called {\it Fibonacci quadratic equation}. 
The Fibonacci quadratic equation has two real solutions: $ \phi $ and its {\it conjugate}  $ \overline{\phi} := 1 - \phi $. We note that 
$$    \phi + \overline{\phi} \dql 1,~~~~~\phi \ndot \overline{\phi} \dql -1. $$
Further we have 
\begin{eqnarray*}
  & \phi^{2} \dql 1+ \phi, ~~~~~\overline{\phi}^{2} \dql 2 - \phi&  \\
  &\phi^{2} + \overline{\phi}^{2} \dql 3,~~~~~\phi^{2} \ndot \overline{\phi}^{2} \dql 1. &
\end{eqnarray*}

A point $ (2 - \phi)x $ splits an interval $[0, x] $ into two intervals $[0, (2 - \phi)x] $ and $[(2 - \phi)x, x] $. A point $ (\phi -1)x $ splits the interval into $[0, (\phi -1)x] $ and $[(\phi -1)x, x] $. In either case, the length constitutes the Golden ratio $ (2 - \phi) : (\phi -1) = 1 : \phi. $ Thus both divisions are the {\it Golden section}. 

\begin{df} A sequence $ x: \{0, 1, \ldots \} \to R^{1} $ is called {\it Golden} if and only if either
\begin{equation}   
\dfrac{\,x_{t+1}\,}{\,x_{t} \,} \dql \phi - 1~~~~~\mbox{or}~~~~~\dfrac{\,x_{t+1}\,}{\,x_{t} \,} \dql 2 - \phi.  \nonumber %  \label{eqn:dg1}
\end{equation}
\end{df}


\begin{lem} A Golden sequence $ x$ is either 
\begin{equation}   
 x_{t} \dql x_{0}(\phi - 1)^{t}~~~~~\mbox{or}~~~~~x_{t} \dql x_{0}(2 - \phi)^{t}.  \nonumber % \label{eqn:dg3}
\end{equation}
\end{lem}

\clearpage
%%%%%%%%%%%%%%%%%%%%%%    �����o�H�̃O���t   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  �ɒl���@Extremize == Locally maximize and/or locally minimize
%%%%%%%%%%%%%%%%%%%%%%  �����o�H�̃O���t�i�����j gol-traj-1.tex  %%%%%%
\input{gl-traje-dis46.tex}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
Fig.~1~~Golden paths $ x = c(\phi - 1)^{t}~~~c = 1, 2, 3 $ 
\end{center}
%%%%%%%%%%%%%%%%%%%%%%  �����o�H�̃O���t gol-traj-1-c.tex  %%%%%%
\input{gl-traje-dis64.tex}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
Fig.~2~~Golden paths (c)  $ x = c(2 - \phi)^{t}~~~c = 1, 2, 3 $ 
\end{center}

\clearpage


We remark that 
\begin{equation}
(\phi - 1)^{t} = \phi^{-t},~~~~~~(2 - \phi)^{t} = (1 + \phi)^{-t} 
 \nonumber % \label{eqn:a1}
\end{equation} 
where
$$ \phi - 1 \dql \phi^{-1} \,\thickapprox \, 0.618,~~~~~2 - \phi  \dql (1 + \phi)^{-1} \,\thickapprox \, 0.382  $$ 

Let us introduce a controlled linear dynamics with real parameter $ b $ as follows. 
\begin{equation}   
 x_{t+1} \dql b x_{t} + u_{t} ~~~~~t = 0, 1, \ldots  \label{eqn:dg1}
\end{equation}
where $ u : \{0, 1, \ldots \} \to R^{1} $ is called {\it control}. If $ u_{t} = px_{t} $, the control $ u $ is called {\it proportional}, where $ p $ is a real constant, {\it proportional rate}.  A sequence $ x $ satisfying (\ref{eqn:dg1}) is called {\it path}. 

\begin{df} A proportional control $ u $ on dynamics (\ref{eqn:dg1}) is called {\it Golden} if and only if it generates a Golden path $ x $. 
\end{df}
%Thus the definition of Golden path depends on a dynamics. 

\begin{lem} A proportional control $ u_{t} = px_{t} $ on (\ref{eqn:dg1}) is Golden if and only if 
\begin{equation}   
 p \dql -b + \phi - 1~~~~~\mbox{or}~~~~~p \dql -b + 2 - \phi.   \label{eqn:dg5}
\end{equation}
\end{lem}


%\clearpage
%*******************************************************************
%
\section{Control processes}
%
%******************************************************************
This section minimizes two quadratic cost functions 
$$
\dsum_{n=0}^{\infty}\left[x_{n}^{2} + (x_{n} - x_{n+1})^{2} \right] 
~~~\mbox{and}~~~
 \dsum_{n=0}^{\infty}\left[(x_{n} - x_{n+1})^{2} + x_{n+1}^{2} \right].
$$
Both problems are solved as a control process with criterion 
$$
\dsum_{n=0}^{\infty} \left( x_{n}^{2} + u_{n}^{2} \right)
~~~\mbox{and}~~~
 \dsum_{n=0}^{\infty} \left( u_{n}^{2} + x_{n+1}^{2} \right)
$$
under a common additive dynamics with a given initial state  
\begin{eqnarray*}
 & & x_{n+1} = bx_{n} + u_{n},~~~~~x_{0} = c   
\end{eqnarray*}
where $ c \in R^{1}. $ 
% (see \cite{belc,belm}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quadratic in current state} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us now consider a control process with an additive transition $ T(x, u) = bx + u$. Here $ b $ is a constant, which represents a characteristics of the process : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm minimize} \hspace{0.4cm} 
                  \dsum_{n=0}^{\infty} \left( x_{n}^{2} + u_{n}^{2} \right)   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  x_{n+1} = bx_{n} + u_{n}    \nonumber \\[-2.96mm]
{\rm C}(c)& & \hspace*{6.2cm}   ~~~n \ge 0   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   -\infty < u_{n} < \infty \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x_{0} = c.    \nonumber  
\end{eqnarray}
%

Let $ v(c) $ be the minimum value of $ {\rm C}(c) $. Then the value function $ v $ satisfies Bellman equation \cite{iwae}: 
\begin{equation}
 v(x) \dql \min_{-\infty < u < \infty}\left[x^{2} + u^{2} + v(bx + u) \right]. \label{eqn:cp1} 
\end{equation}
Eq. (\ref{eqn:cp1}) has a quadratic form $ v(x) = v x^{2} $, where $ v~ \in R^{1}. $ 

%%%%%%%%%%%%%    Begin of Lemma   %%%%%%%%%
\begin{lem} \label{lem: le0}
The control process $ {\rm C}(c) $ with characteristic value $ b~ (\in R^{1}) $ has a proportional optimal policy $ f^{\infty}, f(x) = px, $  and a quadratic minimum value function $ v(x) = v x^{2} $, where 
\begin{eqnarray}
 v \eql \dfrac{\, b^{2} + \sqrt{b^{4} + 4 \,} \,}{2},~~~~~~
 p \dql -\,\dfrac{\, v \,}{\, 1 + v \,}b.  \nonumber
\end{eqnarray} 
\end{lem}
%%%%%%%%%%%%%%%  End of Lemma   %%%%%%%%% 

The proportional optimal policy $ f^{\infty} $ splits at any time an interval $ [0, \,x] $ into $ [0, \,(b + p)x] = \left[0, \, \dfrac{\, bx \,}{\, 1 + v \,} \right]$ and $ \left[\dfrac{\, bx \,}{\, 1 + v \,}, \,x\right]. $ 
In particular, when $ b = 1 $, the quadratic coefficient $ v $ is reduced to the {\it Golden number} 
$$ 
\phi = \dfrac{1 + \sqrt{5}}{2} \, \thickapprox \, 1.618 
$$
Further the division of $ [0, \,x] $ into  $ \left[0, \, \dfrac{\, x \,}{\, 1 + \phi \,} \right] $ and $ \left[\dfrac{\, x \,}{\, 1 + \phi \,}, \, x \right] $ is Golden. 
%That is, the ratio of length of two intervals constitutes the Golden ratio: 
% $$ 1 \, : \, \phi ~=~ (2 - \phi) \, : \, (\phi - 1). $$ 
A quadratic function $ w(x) = a x^{2} $ is called {\it Golden} if $ a = \phi. $


%%%%%%%%%%%%%%%%%%%%       Begin of Theorem      %%%%%%%%%%%%%%%%
\begin{thm} \label{thm: th0} 
The control process $ {\rm C}(c) $ with characteristic value $ b = 1 $ has a Golden optimal policy $ f^{\infty}, f(x) = (1 - \phi)x, $  and the Golden quadratic minimum value function $ v(x) = \phi x^{2} $. 
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%     End of Theorem     %%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Qquadratic in next state} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we consider the cost function $ r : X \stimes U \to  R^{1} $ which is quadratic in current control and next state : 
\begin{eqnarray*}
  r(x,u) \eql u^{2} + (bx + u)^{2}.   
\end{eqnarray*}
Then a control process is represented by the following sequential minimization problem : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm minimize} \hspace{0.4cm} 
                  \dsum_{n=0}^{\infty} \left( u_{n}^{2} + x_{n+1}^{2} \right)   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  x_{n+1} = bx_{n} + u_{n}      \nonumber \\[-2.96mm]
{\rm C}^{\prime}(c)& & \hspace*{6.2cm}   ~~~n \ge 0   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   -\infty < u_{n} < \infty \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x_{0} = c.    \nonumber 
\end{eqnarray}
%
%
The value function $ v $ satisfies Bellman equation \cite{iwae}: 
\begin{equation}
 v(x) \dql \min_{-\infty < u < \infty}\left[u^{2} + (bx + u)^{2} + v(bx + u) \right].  \label{eqn:cq1} 
\end{equation}
Eq. (\ref{eqn:cq1}) has a quadratic solution $ v(x) = v x^{2} $, where $ v \in R^{1}. $ 

%%%%%%%%%%%%%    Begin of Lemma   %%%%%%%%%
\begin{lem} \label{lem: le4}
The control process $ {\rm C}^{\prime}(c) $ with characteristic value $ b $ has a proportional optimal policy $ f^{\infty}, f(x) = px, $  and a quadratic minimum value function $ v(x) = v x^{2} $, where 
\begin{eqnarray}
 v \eql \dfrac{\, b^{2} - 2 + \sqrt{b^{4} + 4 \,} \,}{2},~~~~~~
 p \dql -\,\dfrac{\, 1 + v \,}{\, 2 + v \,}b.  \nonumber
\end{eqnarray} 
\end{lem}
%%%%%%%%%%%%%%%  End of Lemma   %%%%%%%%% 

The policy $ f^{\infty} $ splits an interval $ [0, \,x] $ into $ \left[0, \, \dfrac{\, bx \,}{\, 2 + v \,} \right]$ and $ \left[\dfrac{\, bx \,}{\, 2 + v \,}, \,x\right]. $ 
When $ b = 1 $, the coefficient $ v $ is reduced to the {\it inverse Golden number} 
$$ \phi^{-1} = \phi - 1 = \dfrac{-1 + \sqrt{5}}{2} \, \thickapprox \, 0.618  
$$ 
Further the division of $ [0, \,x] $ into  $ \left[0, \, (2 - \phi)x \right] $ and $ \left[(2 - \phi)x, \, x \right] $ is Golden.
% : 
% $$ (2 - \phi) \, : \, (\phi - 1) ~=~ 1 \, : \, \phi. $$ 
A quadratic function $ w(x) = a x^{2} $ is called {\it inverse Golden} if $ a = \phi^{-1}. $
%%%%%%%%%%%%%%%%%%%%       Begin of Theorem      %%%%%%%%%%%%%%%%
\begin{thm} \label{thm: th5} 
The control process $ {\rm C}^{\prime}(c) $ with characteristic value $ b = 1 $ has a Golden optimal policy $ f^{\infty}, f(x) = (1 - \phi)x, $  and the inverse Golden quadratic minimum value function $ v(x) = (\phi - 1) x^{2} $. 
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%     End of Theorem     %%%%%%%%%%%%%%%%


%\clearpage 
%*******************************************************************
%
\section{Allocation processes}
%
%******************************************************************
This section maximizes two discounted square-root reward functions 
$$
\dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n}\,} +  \sqrt{x_{n} - x_{n+1}\,} \right)
~~~\mbox{and}~~~
 \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n} - x_{n+1}\,} + \sqrt{x_{n+1}\,} \right).
$$
Both problems are solved as an allocation process with criterion 
$$
\dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n}\,} + \sqrt{u_{n}\,} \right)
~~~\mbox{and}~~~
 \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{u_{n}\,} + \sqrt{x_{n+1}\,} \right)
$$
under a common subtractive dynamics with a given initial state  
\begin{eqnarray*}
 & & x_{n+1} = x_{n} - u_{n},~~~~~x_{0} = c   
\end{eqnarray*}
where $ c \ge 0. $ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Square-root in current state} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us now consider an allocation process with a subtractive transition $ T(x, u) = x - u$ : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm Maximize} \hspace{0.4cm} 
                  \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{x_{n}\,} + \sqrt{u_{n}\,} \right)   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  x_{n+1} = x_{n} - u_{n}     \nonumber\\[-2.96mm]
{\rm A}(c)& & \hspace*{6.2cm}   ~~~n \ge 0   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   0 \le u_{n} \le x_{n} \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x_{0} = c.    \nonumber 
\end{eqnarray}

Let $ v(c) $ be the maximum value of $ {\rm A}(c).$  Then the maximum value function $ v $ satisfies the following Bellman equation:


\begin{equation}
 v(x) \dql \Max_{0 \le u \le x}\left[\sqrt{x \,} + \sqrt{u \,} + \beta v(x - u) \right].    \label{eqn:dc1} 
\end{equation}
Eq. (\ref{eqn:dc1}) has a square-root form $ v(x) = v \sqrt{x\,} $, where $ v \in R^{1}. $ 


Let us adopt a proportional policy $ f^{\infty}~(f(x) = px)$ with proportional rate $ p~(0 < p < 1)$. Then state $ x $ under the control $ u = px $ goes deterministically to the next state $ T(x,u) = x - u = x - px = (1 - p)x. $ Thus we have $ x = (1 - p)x + px. $ The state transition of control process $ {\rm A}(c)$ governed by the proportional policy $f^{\infty} $ means that the current control $ u = px $ splits the state interval $ [0, \,x] $ into two intervals $ [0, \,(1 - p)x] $ and $ [(1 - p)x, \,x]. $ When the split yields a {\it Golden section}, the proportional policy $ f^{\infty}~(f(x) = px)$ is called {\it Golden}. 


%%%%%%%%%%%%%    Begin of Lemma   %%%%%%%%%
\begin{lem} \label{lem: le6}
The allocation process $ {\rm A}(c) $ has a proportional optimal policy $ f^{\infty}, f(x) = px, $  and a square-root maximum value function $ v(x) = v \sqrt{x\,} $, where 
\begin{eqnarray}
 v \eql \dfrac{\, 2 \,}{\, 1 - \beta^{2} \,}, ~~~~~
 p \dql \dfrac{\, (1 - \beta^{2})^{2} \,}{\, (1 + \beta^{2})^{2}\,}.  \nonumber
\end{eqnarray} 
\end{lem}
%%%%%%%%%%%%%%%  End of Lemma   %%%%%%%%% 

We remark that the coefficient $ v $ is the solution to 
$$
  v \dql 1 + \sqrt{ 1 + (\beta v)^{2} \,}, ~~~v \ge 2. 
$$
Let us solve $ 1 - p = \phi - 1~\,\mbox{or}~\, 2 - \phi. $ Then we have the following result. 
%%%%%%%%%%%%%    Begin of Theorem   %%%%%%%%%
\begin{thm} \label{thm: 6le}
When $ \beta = \phi \, (1 - \sqrt{\phi - 1\,}) \thickapprox  0.346~~\mbox{or}~~\beta = \sqrt{\phi \,} - \sqrt{\phi - 1\,} \thickapprox 0.486 $, the proportional policy $ f^{\infty}, f(x) = px, $ is Golden optimal. 
\end{thm}
%%%%%%%%%%%%%%%  End of Theorem   %%%%%%%%%
%\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Square-root in next state} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we consider an allocation process with transition $ T(x, u) = x - u$ : 
\begin{eqnarray}
 & & \hspace*{0.2cm} {\rm Maximize} \hspace{0.4cm} 
                  \dsum_{n=0}^{\infty} \beta^{n} \left( \sqrt{u_{n}\,} + \sqrt{x_{n+1}\,} \right)   \nonumber \\
& & \hspace*{0.2cm}{\rm subject ~ to \hspace{0.4cm}(i)} \hspace{0.4cm} 
                  x_{n+1} = x_{n} - u_{n}     \nonumber \\[-2.96mm]
{\rm A^{\prime}}(c)& & \hspace*{6.2cm}   ~~~n \ge 0   \nonumber \\[-2.96mm]
& & \hspace*{2.3cm} {\rm(ii)}\hspace{0.4cm}  
                   0 \le u_{n} \le x_{n} \nonumber \\
& & \hspace*{2.2cm} {\rm(iii)}\hspace{0.4cm}  
                   x_{0} = c.    \nonumber 
\end{eqnarray}
%
Let $ v(c) $ be the maximum value of $ {\rm A^{\prime}}(c) $. Then the maximum value function $ v $ satisfies an optimality equation:
%
\begin{equation}
 v(x) \dql \Max_{0 \le u \le x}\left[\sqrt{u \,} + \sqrt{x - u \,} + \beta v(x - u) \right].  \label{eqn:dp1} 
\end{equation}
Eq. (\ref{eqn:dp1}) has a square-root solution $ v(x) = v \sqrt{x\,} $, where $ v \in R^{1}. $ 

Let us adopt a proportional policy $ f^{\infty}~(f(x) = px)$ with $ p~(0 < p < 1)$. Then 
%
%state $ x $ under $ u = px $ moves to the nextstate $ x - u = (1 - p)x. $ Thus we have $ x = (1 - p)x + px. $ The state transition of control process $ {\cal A} $ governed by $f^{\infty} $ means that 
%
%
the current control $ u = px $ splits the interval $ [0, \,x] $ into $ [0, \,(1 - p)x] $ and $ [(1 - p)x, \,x]. $ 
%
%When the split yields a {\it Golden section}, the proportional policy $ f^{\infty}~(f(x) = px)$ is called {\it Golden}. 


%%%%%%%%%%%%%    Begin of Lemma   %%%%%%%%%
\begin{lem} \label{lem: le4}
The allocation process $ {\rm A^{\prime}}(c) $ has a proportional optimal policy $ f^{\infty}, f(x) = px, $  and a square-root maximum value function $ v(x) = v \sqrt{x\,} $, where 
\begin{eqnarray}
 v \eql \dfrac{\, \beta + \sqrt{2 - \beta^{2} \,} \,}{\, 1 - \beta^{2} \,}, ~~~~~
 p \dql \dfrac{\, 1 - \beta \sqrt{2 - \beta^{2}\,} \,}{\, 2 \,}.  \nonumber
\end{eqnarray} 
\end{lem}
%%%%%%%%%%%%%%%  End of Lemma   %%%%%%%%% 


Note that the coefficient $ v $ is the positive solution to 
$$
  v \dql \sqrt{ 1 + (1 + \beta v)^{2} \,}. 
$$
By solving $ 1 - p = \phi - 1$, we have the following result.  
%%%%%%%%%%%%%    Begin of Theorem   %%%%%%%%%
\begin{thm} \label{thm: 4le}
When  $ \beta = \sqrt{ 1 - 2 \sqrt{2 \phi - 3 \,} \,} \thickapprox 0.168 $, the proportional policy $ f^{\infty}, f(x) = px, $ is Golden optimal.
\end{thm}
%%%%%%%%%%%%%%%  End of Theorem   %%%%%%%%%

%
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
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