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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}



%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% 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
% 
% 
% 
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Orbital Debris}
    %Story from monday ISS

    Recent News:
    \begin{itemize}
        \item Russion conducts an Anti-Satellite Missle Test generating at least 1,500 items of trackable debris
        \item The Astronauts and Cosmonauts on the ISS entered lockdown, including donning pressure suits.
        \item The situation is still being monitored although the immediate danger appears to have passed.
    \end{itemize}

    \url{https://edition.cnn.com/2021/11/15/politics/russia-anti-satellite-weapon-test-scn/index.html}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Orbital Debris}
    Other recent events highlight the dangers from orbital debris:
    \begin{itemize}
        \item ISS conducts 3 evasive maneuvers to doge debris in 2020- 
            \href{https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809}{Jerusalem Post}
        \item ISS hit by debris, May 2021- \href{https://www.asc-csa.gc.ca/eng/iss/news.asp}{Canadian Space Agency}
        \item ISS dodged debris from 2007 Anti-Sat Missile, Nov 2021- 
            \href{https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809}{Jerusalem Post (Same as above)}
    \end{itemize}

    %This isn't a unique experience
    %list of other issues the ISS has faced
    
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Orbital Debris}
    %Talk about Fregat breakup
    Not just an issue for manned space flight.
    \begin{itemize}
        \item In May of 2020, the Satellite SL-23 Zenit Fregat's 
            tank suffered a second breakup event.
        \item While only 65 large pieces of debris were initially identified, by Feb. 2021
            over 325 had been attributed to the breakup.
        \item Debris was spread in orbits between 500km and 6,000km.
    \end{itemize}

    \href{https://www.orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/odqnv25i1.pdf}{Orbital Debris Quarterly News - NASA}
    
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Why now?}
    %launch costs
    %nano-satellites

    In recent years two major changes have occured
    \begin{enumerate}
        \item SpaceX: Plummeting Launch costs
        \item CubeSates and other Nano-Satellites.
    \end{enumerate}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Goal of the Paper}
    %Model decision making of satellite operators to be able to investigate policies to reduce kessler syndrome.

    \center
    Project Goals:
    \begin{itemize}
        \item Model the choices  
            facing Satellite Constellation Operators
            and optimal policy policy response.
        \item Investigate the effect of various policies on debris pollution
    \end{itemize}
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{TOC}
%-------------------------------------------------------------------------------------
\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{Literature}
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Past Literature}

    Key elements of recent literature.
    \begin{enumerate}
        \item \cite{Kessler1978}: Raised issue of runaway orbital pollution.
        \item \cite{Adilov2015}: Described 2 period salop model of interactions.
        \item \cite{Adilov2018,Adilov2018a}: Described an infinite period model with symmetric competitive interactions.
        \item \cite{RaoRondina2020}: Describe a symetric infinite period model (first to do so).
        \item \cite{Rao2020}: Examine the effect of Orbital-Use fees, find it would quadruple long term value produced of the space industry.
    \end{enumerate}
\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 $S_t$: The set of constellation satellites stocks.
         \item $s^i_t$: The number of satellites (stock) for constellation $i$.
         \item $D_t$: The level of debris.
         \item $X_t$: The set of launches.
         \item $x^i_t$: The launches from constellation $i$.
         \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_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}

    Law of motion for satellite stocks
    \begin{align}
        s^i_{t+1} =& \left(
             R^i(S_t,D_t,X_t) 
             - \eta 
            \right) \cdot s^i_t  
            + x^i_t 
    \end{align}
    \begin{itemize}
        \item $\eta$ is the orbit decay rate of satellites.
    \end{itemize}
\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.

\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_t,D_t) : R^i(S_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_t,D_t) : 
                                \lim_{t \rightarrow \infty} D_{t+1}(S_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_\epsilon = \left\{ \left(S_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}
        \kappa_\text{proto} = 
        \left\{ 
	    \left(S_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}
        \begin{align}
            V^i(S_t, x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(S_t, D_t) -F(x^i_t) 
            + \beta \left[ V^i(S_{t+1}, x^{\sim i}_{t+1}, D_{t+1}) \right]
        \end{align}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Benefit Functions}

    Possible benefit 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}
        \begin{align}
            W(S_t, D_t) =& \max_{X_t} \left[
               \sum^N_{i=1} \left(u^i(S_t, D_t) - F(x^i_t) \right)
            + \beta \left[ W(S_{t+1}, D_{t+1}) \right]\right] \notag \\
             &\text{subject to:} \notag \\
             & s^i_{t+1} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~  \forall i \notag \\
             & D_{t+1} = (1-\delta + g)D_t  
                 + \gamma \sum^N_{i=1} \left(1-R^i(\vec s_t, D_t)\right) s^i_t
                 + \Gamma \sum^N_{i=1} x^i_t
        \end{align}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Planned model expansions}
    \begin{itemize}
        \item Multiple interacting orbital shells and debris terms.
        \item Stochastic laws of motion 
        \item Multiple types of operators
        \item Operators benefit functions include taxation
    \end{itemize}
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis}
%-------------------------------------------------------------------------------------
\begin{frame}
    \frametitle{Issues}
    The following issues characterize the Operators' and Planner's problem
    \begin{itemize}
        \item Curse of Dimensionality
        \item Strategic Interaction (operators only)
    \end{itemize}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Possible approaches}

    Possible approaches
    \begin{itemize}
        \item Standard VFI/Howards algorithm.
        \item VFI with sparse state space (dimensionality reduction).
        \item Reinforcement Learning.
        \item \cite{MALIAR2018} approaches using machine 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 \autocite{MALIAR2018}

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

    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}

\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Training Loop: Planner}
    
    For each training epoch
    \begin{enumerate}
        \item Draw random data
        \item train policy function
        \item train value function
    \end{enumerate}
\end{frame}
%-------------------------------
\begin{frame}
    \frametitle{Training Loop: Operators}

    For each training epoch
    \begin{enumerate}
        \item Draw random data
        \item For each operator
            \begin{enumerate}
                \item train policy function
                \item train value function
            \end{enumerate}
    \end{enumerate}
\end{frame}
%-------------------------------
%------------------------------------
\subsection{Analysis so far}
%------------------------------------
\begin{frame}
    \frametitle{State of the Code}
    Currently functioning
    \begin{itemize}
        \item Nothing
    \end{itemize}
    %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?

    Results are currently waiting on finishing the code.

    \vspace{12pt}

    Some analyses I plan on completing include
    \begin{itemize}
        \item Trajectory Analysis
        \item Free Entry Conditions
        \item Kessler Regions
    \end{itemize}
\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} 
% \begin{frame}
%     \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}
% %---------------------------------------------------------------
