Outlined presentation and began work on it.
youainti
4 years ago
parent
56a89717ab
commit
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Beamer Presentation
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% LaTeX Template
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% Version 1.0 (10/11/12)
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%
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% This template has been downloaded from:
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% http://www.LaTeXTemplates.com
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%
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% License:
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% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
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%
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% Changed theme to WSU by William King
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%----------------------------------------------------------------------------------------
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% PACKAGES AND THEMES
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%----------------------------------------------------------------------------------------
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\documentclass[xcolor=dvipsnames,aspectratio=169]{beamer}
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%Import Preamble bits
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\input{../Assets/preambles/FormattingPreamble.tex}
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\input{../Assets/preambles/TikzitPreamble.tex}
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\input{../Assets/preambles/MathPreamble.tex}
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\input{../Assets/preambles/BibPreamble.tex}
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\input{../Assets/preambles/GeneralPreamble.tex}
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%----------------------------------------------------------------------------------------
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% TITLE PAGE
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%----------------------------------------------------------------------------------------
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\title[MDP Constellations]{Modeling decisions in operating satellite constellations}
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\author{Will King} % Your name
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\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
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{
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Washington State University \\ % Your institution for the title page
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\medskip
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\textit{william.f.king@wsu.edu} % Your email address
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}
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\date{\today} % Date, can be changed to a custom date
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\begin{document}
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\begin{frame}
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\titlepage % Print the title page as the first slide
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\end{frame}
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\begin{frame}[allowframebreaks] %Allow frame breaks
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\frametitle{Overview} % Table of contents slide, comment this out to remove it
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\tableofcontents
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%Planned TOC
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% See ../outline2.txt
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\end{frame}
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Background}
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% Why should we care?
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% Uses of space
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% Pollution in space and it's impacts
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% Kessler Syndrome
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% What is different now
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%
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%
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%
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%-------------------------------------------------------------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Literature}
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%-------------------------------------------------------------------------------------
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\begin{frame}
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\frametitle{Past Literature}
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\begin{enumerate}
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\item Kessler and Cour-Palais: Raised issue of orbital pollution
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\item Adilov et al: Described 2 period salop model of interactions
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\item Adilov et al: Described an infinite period model with symmetric competitive interactions.
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\item Rao and Rondina: Describe a symetric infinite period model (first to do so).
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\end{enumerate}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{}
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\end{frame}
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%-------------------------------
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%-------------------------------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Model}
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%-------------------------------------------------------------------------------------
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\begin{frame}
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\frametitle{Overview}
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\begin{itemize}
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\item Mathematical Notation
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\item Law of motion for debris
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\item Law of motion for satellite stocks
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\item Kessler Syndrome
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\item Markov Decision Problems
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Mathematical Notation}
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\begin{itemize}
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\item $\{s^j_t\}$: The set of constellation satellites stocks
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\item $D_t$: The level of debris
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\item $x^j_t$: The launches from each constellation
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\item
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\end{itemize}
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\end{frame}
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%-------------------------------
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%------------------------------------
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\subsection{Laws of Motion}
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%------------------------------------
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\begin{frame}
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\frametitle{Debris}
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Law of motion for debris
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\begin{align}
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D_{t+1} =& (1-\delta)D_t \tag{Debris decay.} \\
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&+ g\cdot D_t \tag{Debris produced by collision with debris.} \\
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&+ \gamma \sum^N_{i=1} \left(1-R^i(\{s^j_t\},D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\
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&+ \Gamma \sum^n_{j=1} \{x^j_t\} \tag{Debris produced by launches.}
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\end{align}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Satellite Stocks}
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\end{frame}
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%-------------------------------
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%------------------------------------
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\subsection{Kessler Syndrome}
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%------------------------------------
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\begin{frame}
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\frametitle{Explanation of Kessler Syndrome}
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\begin{block}{Kessler Syndrome}
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The situation in which collisions between
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objects in orbit produced debris and this debris begins collisions
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with other objects, leading to a runaway growth in debris.
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As debris can persist for millenia, this may make some orbits unusable.
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\autocite{Kessler1978}
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\end{block}
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Often described as a condition with an exponential growth of debris.
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Similar to what is expected with Climate Change
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Past approaches to Kessler Syndrome}
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\begin{itemize}
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\item \cite{Adilov2018}:\\
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Develops an analog of kessler syndrome where the condition is met when satellites
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are destroyed immediately after launch by debris.
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\begin{align}
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\left\{ (\{s^j_t\},D_t) : R^i(\{s^j_t\},D_t) = 0 \forall i\right\}
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\end{align}
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\item \cite{RaoRondina2020}:\\
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A working paper in which the authors develop a dynamic model and a definition of
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kessler syndrome that captures all increasing debris levels.
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\begin{align}
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\left\{ (\{s^j_t\},D_t) :
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\lim_{t\rightarrow \infty} D_{t+1}(\{s^j_t\},D_t) = \infty
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\right\}
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\end{align}
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\end{itemize}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{My contributions}
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I propose two refinements of these definitions to
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simplify analyzing kessler syndrome in computational models.
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\begin{itemize}
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\item $\epsilon$-Kessler Region
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\item Proto Kessler Region
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{$\epsilon$-Kessler Region}
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\begin{block}{$\epsilon$-Kessler Region}
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\begin{align}
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\kappa = \left\{ \left(\{s^j_t\}, D_t \right) : \forall k \geq 0,
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~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
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\end{align}
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\end{block}
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Notable Features
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\begin{itemize}
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\item $\epsilon$ can be calibrated to capture only economically significant growth.
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\item Requires an explicit description of what is considered economically significant.
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\item Guarantees divergent behavior.
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\item Simulated transition paths can identify the region.
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{Proto Kessler Region}
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\begin{block}{Proto Kessler Region}
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\begin{align}
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\left\{
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\left(\{s^j_t\},D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto}
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\right\}
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\end{align}
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\end{block}
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Notable Features
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\begin{itemize}
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\item $\epsilon_\text{proto}$ can be calibrated to capture only economically significant growth.
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\item Requires an explicit description of what is considered economically significant.
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\item Does not guarantee divergent behavior.
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\item Easily computable kessler regions.
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------
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\begin{frame}
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\frametitle{Proto Kessler Region}
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With the given law of motion for debris, the proto-kessler region is:
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\begin{align}
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\left\{
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\left(S_t,D_t \right) :
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(g-\delta) D_t
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+ \gamma \sum^n_{i=1} 1-R^i(S_t,D_t)
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+ \Gamma \sum^n_{i=1} x^i_t(S_t,D_t)
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\geq \epsilon_\text{proto}
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\right\}
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\end{align}
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\end{frame}
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%------------------------------------
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\subsection{Markov Decision Problem Formulation}
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%------------------------------------
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\begin{frame}
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\frametitle{Operator's Problem}
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\end{frame}
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%-------------------------------
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\begin{frame}
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\frametitle{Benefit Functions}
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Possible profit functions
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\begin{itemize}
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\item Linear (Currently working on this one)
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\item Cournot Profits
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||||
\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}
|
||||
% \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}
|
||||
% %---------------------------------------------------------------
|
||||
@ -0,0 +1,42 @@
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{backgrounds}
|
||||
\usetikzlibrary{arrows}
|
||||
\usetikzlibrary{shapes,shapes.geometric,shapes.misc}
|
||||
|
||||
% this style is applied by default to any tikzpicture included via \tikzfig
|
||||
\tikzstyle{tikzfig}=[baseline=-0.25em,scale=0.5]
|
||||
|
||||
% these are dummy properties used by TikZiT, but ignored by LaTex
|
||||
\pgfkeys{/tikz/tikzit fill/.initial=0}
|
||||
\pgfkeys{/tikz/tikzit draw/.initial=0}
|
||||
\pgfkeys{/tikz/tikzit shape/.initial=0}
|
||||
\pgfkeys{/tikz/tikzit category/.initial=0}
|
||||
|
||||
% standard layers used in .tikz files
|
||||
\pgfdeclarelayer{edgelayer}
|
||||
\pgfdeclarelayer{nodelayer}
|
||||
\pgfsetlayers{background,edgelayer,nodelayer,main}
|
||||
|
||||
% style for blank nodes
|
||||
\tikzstyle{none}=[inner sep=0mm]
|
||||
|
||||
% include a .tikz file
|
||||
\newcommand{\tikzfig}[1]{%
|
||||
{\tikzstyle{every picture}=[tikzfig]
|
||||
\IfFileExists{#1.tikz}
|
||||
{\input{#1.tikz}}
|
||||
{%
|
||||
\IfFileExists{./figures/#1.tikz}
|
||||
{\input{./figures/#1.tikz}}
|
||||
{\tikz[baseline=-0.5em]{\node[draw=red,font=\color{red},fill=red!10!white] {\textit{#1}};}}%
|
||||
}}%
|
||||
}
|
||||
|
||||
% the same as \tikzfig, but in a {center} environment
|
||||
\newcommand{\ctikzfig}[1]{%
|
||||
\begin{center}\rm
|
||||
\tikzfig{#1}
|
||||
\end{center}}
|
||||
|
||||
% fix strange self-loops, which are PGF/TikZ default
|
||||
\tikzstyle{every loop}=[]
|
||||
@ -0,0 +1,27 @@
|
||||
|
||||
utline
|
||||
|
||||
Tell story
|
||||
- Why do we care? Kessler syndrome.
|
||||
- Why is space access needed? (satellite internet, GPS, military, hobbies, science, R&D, etc.)
|
||||
- Why now? (cubesats, lower launch costs, massive expansion in who launches, etc.)
|
||||
|
||||
|
||||
Present past models
|
||||
-
|
||||
-
|
||||
-
|
||||
|
||||
Present Current model
|
||||
-
|
||||
-
|
||||
-
|
||||
|
||||
|
||||
Present work on solving models.
|
||||
-
|
||||
-
|
||||
-
|
||||
|
||||
|
||||
Future work
|
||||
@ -0,0 +1,117 @@
|
||||
\documentclass{article}
|
||||
|
||||
%Setup Subfiles
|
||||
\usepackage{subfiles}
|
||||
|
||||
|
||||
|
||||
%Include preambles
|
||||
\input{assets/preambles/MathPreamble}
|
||||
%\input{assets/preambles/TikzitPreamble}
|
||||
\input{assets/preambles/BibPreamble}
|
||||
\input{assets/preambles/GeneralPreamble}
|
||||
|
||||
\title{Summary of my work on Satellite Constellations}
|
||||
\author{William King}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\section{Introduction}
|
||||
\subfile{sections/00_Introduction} %Currently from past semesters.
|
||||
%roughly done 2021-07-15
|
||||
|
||||
%Describe sections
|
||||
The paper is organized as follows.
|
||||
Section \ref{SEC:Models}, describes the laws of motion
|
||||
governing satellites and debris(\ref{SEC:Laws})%,
|
||||
%places limits on various measures of risk (\ref{SEC:Survival}),
|
||||
and reviews various definitions of kessler syndrome in
|
||||
the context of dynamic modeling, their advantages, and their disadvanteges (\ref{SEC:Kessler}).
|
||||
It then describes the dynamic problem faced by constellation operators
|
||||
(\ref{SEC:Operator}) and social planners (\ref{SEC:Planner}).
|
||||
Section \ref{SEC:Computation} describes the computational approach and the
|
||||
results are reported in \cref{SEC:Results}.
|
||||
Section \ref{SEC:Conclusion} concludes with a discussion of limitations, concerns,
|
||||
and remaining policy questions.
|
||||
|
||||
%\section{Modeling the Environment}\label{SEC:Environment}
|
||||
\section{Model}\label{SEC:Models}
|
||||
\subsection{Laws of motion}\label{SEC:Laws}
|
||||
\subfile{sections/01_LawsOfMotion} %Roughly done 2021-07-15
|
||||
|
||||
%\subsection{Marginal survival rates}\label{SEC:Survival}
|
||||
%\subfile{sections/03_SurvivalAnalysis} %roughly done 2021-07-14
|
||||
% Thoughts on removal: This doesn't add much to the actual questions.
|
||||
% It is interesting, but in a paper is just too much.
|
||||
% I'll keep it here to add it back easily.
|
||||
|
||||
\subsection{Kessler Syndrome}\label{SEC:Kessler}
|
||||
% Kessler syndrome follows laws of motion because it is the main
|
||||
% threat of orbital pollution and needs to be included.
|
||||
% Also, there is not really a better place to place it.
|
||||
\subfile{sections/02_KesslerSyndrome} %roughly done before 2021-07-14
|
||||
\subfile{sections/06_KesslerRegion} %roughly done before 2021-07-14
|
||||
|
||||
\subsection{Constellation Operator's Program}\label{SEC:Operator}
|
||||
\subfile{sections/04_ConstellationOperator} %Reasonably done.
|
||||
|
||||
\subsection{Social Planner's Program}\label{SEC:Planner}
|
||||
\subfile{sections/05_SocialPlanner} %Reasonably done?
|
||||
|
||||
\section{Computation}\label{SEC:Computation}
|
||||
\subfile{sections/07_ComputationalApproach} %needs some clarifications.
|
||||
|
||||
\section{Results}\label{SEC:Results}
|
||||
\subfile{sections/09_Results} %TODO
|
||||
|
||||
\section{Conclusion}\label{SEC:Conclusion}
|
||||
\subfile{sections/08_Conclusion} %TODO
|
||||
|
||||
|
||||
\newpage
|
||||
\section{References}
|
||||
\printbibliography
|
||||
\newpage
|
||||
\section{Appedicies}
|
||||
\subsection{Mathematical Notation}
|
||||
Needs completed.
|
||||
%\subsection{Deriving Marginal Survival Rates}\label{APX:Derivations:SurvivalRates}
|
||||
%\subfile{sections/apx_01_MarginalSurvivalRates}
|
||||
|
||||
\subsection{Deriving Euler Equations}\label{APX:Derivations:EulerEquations}
|
||||
\subfile{sections/apx_02_GeneralizedEuEqSteps}
|
||||
|
||||
|
||||
\subsection{Collected Assumptions and Caveats}\label{APX:CollectedAssumptions}
|
||||
I hope to write a section clearly explaining assumptions, caveats, and shortcomings here.
|
||||
These will later get written back into the other sections, but I want to collect them
|
||||
in a single place first.
|
||||
%time periods are long enough for debris to disperse after collisions.
|
||||
%Only a single type of debris
|
||||
%With my current computational idea; each constellation provides the same risk to each other constellation
|
||||
% That can be easily adjusted in the computational models.
|
||||
|
||||
\newpage
|
||||
%Just for simplicity, remove later
|
||||
\tableofcontents
|
||||
\end{document}
|
||||
|
||||
%%% Notes to keep track of
|
||||
% Possible other things to investigate
|
||||
% - Free-entry conditions: which of the following?
|
||||
% - When for every operator, the current stocks imply no more launches(x = 0).
|
||||
% - When for every type of operator, the current stocks plus an own stock of 0 imply no more launches(x = 0).
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
%
|
||||
Loading…
Reference in New Issue