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29 lines
1.4 KiB
TeX
29 lines
1.4 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(S_t, D_t) =& \max_{X_t} \left[
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\sum^N_{i=1} \left( u^i(S_t, D_t) - F(x^i_t) \right)
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+ \beta \left[ W(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} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~ \forall i \notag \\
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& D_{t+1} = (1-\delta + g)D_t
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+ \gamma \sum^N_{i=1} \left(1-R^i(\vec s_t, D_t)\right) s^i_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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%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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Although the social planner controls each constellation, note that 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 future use of an orbit
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and consequentally constellations will be designed sequentially.
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This prevents intra-constellation benefits to be achieved across the entire fleet.
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\end{document}
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