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67 lines
2.9 KiB
TeX
67 lines
2.9 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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The Social (Fleet) Planner's problem can be written in the bellman form as:
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\begin{align}
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W(\vec s_t, D_t) =& \max_{\vec x_t} \left[
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\left(\sum^N_{i=1} u^i(\vec s_t, D_t) - F(x^i_t) \right)
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+ \beta \left[ W(\vec s_{t+1}, D_{t+1}) \right]\right] \notag \\
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&\text{subject to:} \notag \\
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& s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~ \forall i \notag \\
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& D_{t+1} = (1-\delta)D_t + g(D_t)
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+ \gamma \sum^N_{i=1} l^i(\vec s_t, D_t)
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+ \Gamma \sum^N_{i=1} x^i_t
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\end{align}
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Some particular features of the model include:
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\begin{itemize}
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\item The single period welfare function consists only of constellation operators.
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Although satellites do deorbit and occasionally pose a risk to humans living on the
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earth's surface\footnote{Skylab fell in Australia, with some pieces landing near towns.}
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modeling this risk properly would require adding a deorbit decisions,
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including uncontrolled deorbits.
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\item Although the social planner controls each constellation, they do not reap additional
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collision avoidance efficiencies.
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One justification is that no social planner could concieve of every use of orbit
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at any single point in time, and thus constellations are added sequentially.
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This allows only the intra-constellation benefits to be achived.
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\end{itemize}
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\subsubsection{Euler Equation}
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In accordance with Appendix \cref{APX:Derivations:EulerEquations},
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we find the $N$ optimality conditions:
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\begin{align}
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0 =& -\der{F(x^i_t)}{x^i_t}{}
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+ \beta \left[
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\nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
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\cdot
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\parder{}{x^I_t}{}[\vec s_{t+1} ~ D_{t+1}]
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\right]
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~~\forall~~i
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\end{align}
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Which in vector form is:
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\begin{align}
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0 =& -\vec f_x +\beta \left[B\cdot \nabla W_{t+1} \right]
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\end{align}
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Similarly, the $N+1$ envelope conditions are:
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\begin{align}
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% \nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =&
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% \sum^N_{i=1} \nabla_{\vec s_{t}, D_{t}} u^i(\vec s_t, D_t)
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% %- \der{}{x^i_t}{}F(x^i_t) \nabla_{\vec s_{t}, D_{t}}x^i_t %This equals zero due to the envelope theorem
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% \notag \\
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% &+ \beta \left[ \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
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% \cdot \nabla_{\vec s_{t}, D_{t}} [\vec s_{t+1} ~ D_{t+1}]
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% \right] \\
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\nabla W_t =& \vec U + \beta \left[C \cdot \nabla W_{t+1} \right]
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\end{align}
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Which gives us the iteration format
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\begin{align}
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\nabla W_{t+1} =& (\beta C)^{-1} \cdot \left(\nabla W_t - \vec U \right)
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\end{align}
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Thus two iterations of the optimality condition are needed, but only to provide $N+1$ binding conditions.
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This lets us discard $N-1$ of the conditions from the second iteration of the optimality condition.
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% NEed to explain better. Not quite true.
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\end{document}
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