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%	TITLE PAGE
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\title[MDP Constellations]{Modeling decisions in operating satellite constellations} 

\author{Will King} % Your name
\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{
Washington State University \\ % Your institution for the title page
\medskip
\textit{william.f.king@wsu.edu} % Your email address
}
\date{\today} % Date, can be changed to a custom date





\begin{document}
\begin{frame}
\titlepage % Print the title page as the first slide
\end{frame}


\begin{frame}[allowframebreaks] %Allow frame breaks
\frametitle{Overview} % Table of contents slide, comment this out to remove it
\tableofcontents 
%Planned TOC
% See ../outline2.txt
\end{frame}

%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}
% Why should we care?
% Uses of space
% Pollution in space and it's impacts
%   Kessler Syndrome 
% What is different now
% 
% 
% 
%-------------------------------------------------------------------------------------

%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Literature}
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Past Literature}
    \begin{enumerate}
        \item Kessler and Cour-Palais: Raised issue of orbital pollution
        \item Adilov et al: Described 2 period salop model of interactions
        \item Adilov et al: Described an infinite period model with symmetric competitive interactions.
        \item Rao and Rondina: Describe a symetric infinite period model (first to do so).
    \end{enumerate}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{}
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model}
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Overview}

    \begin{itemize}
        \item Mathematical Notation
        \item Law of motion for debris
        \item Law of motion for satellite stocks
        \item Kessler Syndrome
        \item Markov Decision Problems
    \end{itemize}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Mathematical Notation}
     \begin{itemize}
         \item $\{s^j_t\}$: The set of constellation satellites stocks
         \item $D_t$: The level of debris
         \item $x^j_t$: The launches from each constellation
         \item 
     \end{itemize}
\end{frame}
%-------------------------------
%------------------------------------
\subsection{Laws of Motion}
%------------------------------------
\begin{frame}
    \frametitle{Debris}
    Law of motion for debris
    \begin{align}
         D_{t+1} =& (1-\delta)D_t  \tag{Debris decay.} \\
                  &+ g\cdot D_t    \tag{Debris produced by collision with debris.} \\
                  &+ \gamma \sum^N_{i=1} \left(1-R^i(\{s^j_t\},D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\
                  &+ \Gamma \sum^n_{j=1} \{x^j_t\} \tag{Debris produced by launches.}
     \end{align}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Satellite Stocks}
\end{frame}
%-------------------------------
%------------------------------------
\subsection{Kessler Syndrome}
%------------------------------------
\begin{frame}
    \frametitle{Explanation of Kessler Syndrome}
    \begin{block}{Kessler Syndrome}
        The situation in which collisions between
        objects in orbit produced debris and this debris begins collisions
        with other objects, leading to a runaway growth in debris.
        As debris can persist for millenia, this may make some orbits unusable.
 
        \autocite{Kessler1978}
    \end{block}
    Often described as a condition with an exponential growth of debris.

    Similar to what is expected with Climate Change
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Past approaches to Kessler Syndrome}
        \begin{itemize}
        \item \cite{Adilov2018}:\\
            Develops an analog of kessler syndrome where the condition is met when satellites 
            are destroyed immediately after launch by debris.
                    \begin{align}
                        \left\{ (\{s^j_t\},D_t) : R^i(\{s^j_t\},D_t) = 0 \forall i\right\}
                    \end{align}
        \item \cite{RaoRondina2020}:\\
            A working paper in which the authors develop a dynamic model and a definition of 
            kessler syndrome that captures all increasing debris levels.
                    \begin{align}
                        \left\{ (\{s^j_t\},D_t) : 
                                \lim_{t\rightarrow \infty} D_{t+1}(\{s^j_t\},D_t) = \infty 
                        \right\}
                    \end{align}
    \end{itemize}
\end{frame}
%-------------------------------
\begin{frame}
\frametitle{My contributions}
	I propose two refinements of these definitions to
	simplify analyzing kessler syndrome in computational models.
	\begin{itemize}
		\item $\epsilon$-Kessler Region
		\item Proto Kessler Region
	\end{itemize}
\end{frame}
%-------------------------------------------------------
\begin{frame}
\frametitle{$\epsilon$-Kessler Region}
    \begin{block}{$\epsilon$-Kessler Region}
        \begin{align}
            \kappa = \left\{ \left(\{s^j_t\}, D_t \right) : \forall k \geq 0, 
                            ~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
        \end{align}
    \end{block}
    
    Notable Features
    \begin{itemize}
        \item $\epsilon$ can be calibrated to capture only economically significant growth.
        \item Requires an explicit description of what is considered economically significant.
	\item Guarantees divergent behavior.
	\item Simulated transition paths can identify the region.
    \end{itemize}
\end{frame}
%-------------------------------------------------------
\begin{frame}
    \frametitle{Proto Kessler Region}

    \begin{block}{Proto Kessler Region}
    \begin{align}
        \left\{ 
	    \left(\{s^j_t\},D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto}
        \right\}
    \end{align}
    \end{block}

    Notable Features
    \begin{itemize}
	\item $\epsilon_\text{proto}$ can be calibrated to capture only economically significant growth.
        \item Requires an explicit description of what is considered economically significant.
	\item Does not guarantee divergent behavior.
        \item Easily computable kessler regions.
    \end{itemize}
\end{frame}
%-------------------------------------------------------
\begin{frame}
    \frametitle{Proto Kessler Region}
    With the given law of motion for debris, the proto-kessler region is:
    \begin{align}
        \left\{
            \left(S_t,D_t \right) : 
            (g-\delta) D_t   
                + \gamma \sum^n_{i=1} 1-R^i(S_t,D_t) 
                + \Gamma \sum^n_{i=1} x^i_t(S_t,D_t)
	    \geq \epsilon_\text{proto}
        \right\}
    \end{align}
\end{frame}
%------------------------------------
\subsection{Markov Decision Problem Formulation}
%------------------------------------
\begin{frame}
    \frametitle{Operator's Problem}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Benefit Functions}

    Possible profit functions
    \begin{itemize}
        \item Linear (Currently working on this one)
        \item Cournot Profits
        \item Profits under Partial substitutability
        \item Military capabilities (Keeping up with the Jones')
    \end{itemize}

\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Planner's Problem}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Planned model expansions}
    Multiple interacting orbital shells and debris terms.
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis}
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Issues}
    The following issues characterize the Operators' or Planner's problem
    \begin{itemize}
        \item Curse of Dimensionality
        \item Unsure of state space, no idea what equilibrium (if it exists) would look like.
        \item Strategic Interaction (Computability issues)
    \end{itemize}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Possible approaches}

    Possible approaches
    \begin{itemize}
        \item Standard VFI/Howards algorithm.
        \item VFI with sparse state space and other computational benefits.
        \item Maliar et al's approachs to supervised learning.
        \item Reinforcement Learning.
    \end{itemize}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Chosen approach}
    % maliar et al - Bellman Residuals
    % present basic approach (using my notation)
    % Discuss basic training loop (use sutton and bartos term Generalized Policy Iteration)

    Bellman Residual minimization \cref{MALIAR2018}

    Use NN to approximate $V(S_t,D_t|\theta_1)$ and $X(X_t,D_t|\theta_2)$. 

    The loss function is:
    \begin{align}
        %0 =& \left[
        %    V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
        %    \right]^2
        %    - v \left[
        %        F(S_t, D_t, X_t) + \beta V(S_{t+1}, D_{t+1})
        %    \right] \\
        0 =& \left[
                V(S_t, D_t) - F(S_t, D_t, X_t)- \beta V(S_{t+1}, D_{t+1})
                - \frac{v}{2}
            \right]^2
            - v \left[
                V(S_{t}, D_{t}) + \frac{v}{4}
            \right]
    \end{align}

    Iterate between Value and Policy function estimation.
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Training Loop: Planner}
    
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Training Loop: Operators}
    Using a branched policy function and individual value funcitons

    For each training epoch
    Draw random data
    For each operator
        select operator's policy parameters.
        train policy function
        train value function
\end{frame}
%-------------------------------
%------------------------------------
\subsection{Analysis so far}
%------------------------------------
\begin{frame}
    \frametitle{State of the Code}
    Currently functioning
    %Planner training
    %Operator training
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Results}
    %simulated debris paths (increase the ammount of debris a couple orders of magnitude)
    %Protokessler region plots?
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%-------------------------------------------------------------------------------------
%-------------------------------
\begin{frame}
    \frametitle{Summary}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Future Work}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Questions?}
\end{frame}
%\begin{frame}
%    \frametitle{MarginalRevenue}
%    \begin{figure}
%        \tikzfig{../Assets/owned/ch8_MarginalRevenue}
%        \includegraphics[height=\textheight]{../Assets/copyrighted/KrugmanObsterfeldMeliz_fig8-7.jpg}
%        \label{FIG:costs}
%        \caption{Average Cost Curve as firms enter.}
%    \end{figure}
%\end{frame}
\end{document} 
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%     \frametitle{Columns}
%      \begin{columns}
%         \begin{column}{0.5\textwidth}
%         \end{column}
%         \begin{column}{0.5\textwidth}
%                \begin{figure}
%                    \tikzfig{../Assets/owned/ch7_EstablishedAdvantageExample2}
%                    \label{FIG:costs}
%                    \caption{Setting the Stage}
%                \end{figure}
%         \end{column}
%      \end{columns}
% \end{frame}
% %---------------------------------------------------------------