University of Florida/Egm6341/s11.team1.Gong/Mtg38
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University of Florida
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Egm6341
< s11.team1.Gong
Mtg 38: Wed, 30 Mar 11
page38-1
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{\displaystyle {\color {red}(1)}{\color {blue}p.37-3}\ Z^{\color {red}'}=h{\overset {\color {red}\bullet }{z}}\ {\color {red}(1)}}
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{\displaystyle z(s)=\sum _{i=1}^{4}{\color {red}{\overline {\color {black}N_{i}}}}(s){\color {red}{\overline {\color {black}d_{i}}}}\ {\color {red}(2)}}
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{\displaystyle z(t)=\sum _{i=1}^{4}N_{i}(t)d_{i}\ {\color {red}(3)}}
Recall:
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{\displaystyle Collocation\ at\ t_{i}\ \rightarrow \ {\color {red}(5)}{\color {blue}p.36-4}}
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{\displaystyle Collocation\ at\ t_{i+1}\ \rightarrow \ {\color {red}(6)}{\color {blue}p.36-4}}
Now:
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{\displaystyle Collocation\ at\ t_{i+{\color {red}{\frac {1}{2}}}}\Rightarrow }
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{\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}=f_{i+{\color {red}{\frac {1}{2}}}}=f(z_{i+{\color {red}{\frac {1}{2}}}},\ t_{i+{\color {red}{\frac {1}{2}}}})\ {\color {red}(4)}}
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{\displaystyle z_{i+{\color {red}{\frac {1}{2}}}}=z(s={\color {red}{\frac {1}{2}}})}
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6.6
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{\displaystyle {\overset {\color {blue}HW^{*}6.6}{=}}{\frac {1}{2}}(z_{i}+z_{i+1})+{\frac {h}{8}}(f_{i}-f_{i+1})\ {\color {red}(5)}}
page38-2
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{\displaystyle {{\color {red}(1)}{\color {blue}p.38-3}\ {\overset {\color {red}\bullet }{z}}}_{i+{\color {red}{\frac {1}{2}}}}={z}_{i+{\color {red}{\frac {1}{2}}}}^{\color {red}'}{\color {blue}{\frac {1}{h}}}\ {\color {red}(1)}}
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∗
6.6
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{\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}=z^{\color {red}'}(s={\color {red}{\frac {1}{2}}}){\color {blue}{\overset {HW^{*}6.6{\begin{cases}&{\color {red}(1)}{\color {blue}p.37-2}\\&{\color {red}(1)}{\color {blue}p.37-3}\end{cases}}}{\color {black}=}}}-{\frac {3}{2}}(z_{i}-z_{i+1})-{\frac {1}{4}}({z}_{i}^{\color {red}'}+{z}_{i+1}^{\color {red}'})\ {\color {red}(2)}}
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:
{\displaystyle {\color {red}(1)\ _{\ }(2):}}
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{\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}={\frac {-3}{2{\color {blue}h}}}(z_{i}-z_{i+1})-{\frac {1}{4}}(f_{i}+f_{i+1})\ {\color {red}(3)}}
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{\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}{\color {red}\neq f_{i+{\frac {1}{2}}}}\ in\ general}
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{\displaystyle Gap\ =\ \Delta \ {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}-f_{i+{\color {red}{\frac {1}{2}}}}\ {\color {red}(4)}}
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{\displaystyle Collocation\ at\ t_{i+{\color {red}{\frac {1}{2}}}}\ \Rightarrow \ \Delta \ =\ 0\ {\color {red}(5)}}
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{\displaystyle {\color {blue}{\underline {Goal:}}}\ Find\ (z_{i},z_{i+1})\ st\ \Delta \ =\ 0\ {\color {red}(6)}}
page38-3
Δ
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!
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{\displaystyle \Delta =0\Rightarrow \ z_{i+1}{\overset {\color {red}(1)}{=}}{\color {blue}{\underset {Simpson's\ rule\ {\color {red}!\ (2)}\ p.7-4}{\underbrace {\color {black}z_{i}+{\frac {h/2}{3}}[f_{i}+4f_{i+{\frac {1}{2}}}+f_{i+1}]} }}}}
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{\displaystyle {\overset {\color {red}\bullet }{z}}=f(z,t)\ {\color {red}(2)}}
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{\displaystyle {\color {blue}{\underset {\color {black}z_{i+1}-z_{i}}{\underbrace {\color {black}\int _{t_{i}}^{t_{i+1}}{\overset {\color {red}\bullet }{z}}dt} }}}={\color {blue}{\underset {apply\ simpson's\ rule\ \Rightarrow \ {\color {red}(1)}}{\underbrace {\color {black}\int _{t_{i}}^{t_{i+1}}dt} }}}\ {\color {red}(3)}}
Opt. control pb.
:
{z
i
, i=1,2,...,n}
unknown
solved by NLP
(nonlin. progr)
I
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{\displaystyle {\color {blue}IVP\ ({\underset {{\color {red}(4)}{\color {black}z(t_{o})=z_{0}}}{\underbrace {\color {blue}Initial\ Value} }}pb)}:\ Int.\ {\color {blue}{\underset {\color {blue}(2)}{\underbrace {\color {black}nonlinear\ ODEs} }}}}
page38-4
Hermite-Simpson time-stepping algo
Assume
z
i
known
, find z
i+1
using
(1)
p.38-3
Need:
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{\displaystyle {\color {blue}1)}\ f_{i}=f(z_{i},t_{i})\ {\color {red}can\ comp.}}
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{\displaystyle {\color {blue}2)}\ f_{i+1}=f({\color {red}{\underset {unknown}{\underbrace {\color {black}z_{i+1}} }}},{\color {red}{\underset {known}{\underbrace {\color {black}t_{i+1}} }}})\ {\color {red}unknown}}
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{\displaystyle {\color {blue}3)}\ f_{i+{\color {red}{\frac {1}{2}}}}=f({\color {red}{\underset {unknown}{\underbrace {\color {black}z_{i+{\color {red}{\frac {1}{2}}}}} }}},{\color {red}{\underset {known}{\underbrace {\color {black}t_{i+{\color {red}{\frac {1}{2}}}}} }}})\ {\color {red}unknown}}
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{\displaystyle t_{i+{\color {red}{\frac {1}{2}}}}=t_{i}+{\frac {h}{2}}}
(5)
p.38-1:
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{\displaystyle z_{i+{\color {red}{\frac {1}{2}}}}=g(z_{i},z_{i+1})\ {\color {red}(1)}}
(1)
p.38-1:
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{\displaystyle z_{i+1}=z_{i}+{\frac {h/2}{3}}[f_{i}+4f(g(z_{i},z_{i+1}),t_{i+{\color {red}{\frac {1}{2}}}})+f_{i+1}]\ {\color {red}(2)}}
⇔
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{\displaystyle \Leftrightarrow \ F(z_{i+1}){\overset {\color {red}(3)}{=}}{\underset {\color {blue}nonlinear\ alg.\ eq.}{\underbrace {0} }}}
⇒
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{\displaystyle {\color {blue}\Rightarrow \ Newton-Raphson-Simpson}}
Recall:
Now:
page38-2
page38-3
![{\displaystyle \Delta =0\Rightarrow \ z_{i+1}{\overset {\color {red}(1)}{=}}{\color {blue}{\underset {Simpson's\ rule\ {\color {red}!\ (2)}\ p.7-4}{\underbrace {\color {black}z_{i}+{\frac {h/2}{3}}[f_{i}+4f_{i+{\frac {1}{2}}}+f_{i+1}]} }}}}](../../../4c532dc2d696d0ba7334f8a0b52fd212605763fd.svg)
Opt. control pb.:{zi, i=1,2,...,n} unknownsolved by NLP(nonlin. progr)
page38-4
Hermite-Simpson time-stepping algo
Assume zi known, find zi+1 using(1)p.38-3
Need:
(5) p.38-1:
(1) p.38-1:
![{\displaystyle z_{i+1}=z_{i}+{\frac {h/2}{3}}[f_{i}+4f(g(z_{i},z_{i+1}),t_{i+{\color {red}{\frac {1}{2}}}})+f_{i+1}]\ {\color {red}(2)}}](../../../71d6e8ffbb7171452f77baec8a5c9986e119c3f4.svg)
