User:EGM6341.s11.team1.Chiu/Mtg38

Mtg 38: Wed, 30 Mar 11


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page38-1

$${\color{red}(1)}{\color{blue}p.37-3} \ Z^=h \overset{z} \ {\color{red}(1)}$$

$$z(s)= \sum_{i=1}^{4} {\color{red}\overline}(s) {\color{red}\overline} \ {\color{red}(2)}$$

$$z(t)=\sum_{i=1}^{4} N_{i}(t)d_{i} \ {\color{red}(3)}$$

Recall:

$$Collocation \ at \ t_{i} \ \rightarrow \ {\color{red}(5)}{\color{blue}p.36-4}$$

$$Collocation \ at \ t_{i+1} \ \rightarrow \ {\color{red}(6)}{\color{blue}p.36-4}$$

Now:

$$Collocation \ at \ t_{i+{\color{red}\frac{1}{2}}} \Rightarrow$$

$$ { \overset{z}}_{i+{\color{red}\frac{1}{2}}}=f_{i+{\color{red}\frac{1}{2}}}=f(z_{i+{\color{red}\frac{1}{2}}}, \ t_{i+{\color{red}\frac{1}{2}}}) \ {\color{red}(4)}$$

$$z_{i+{\color{red}\frac{1}{2}}}=z(s={\color{red}\frac{1}{2}})$$

$$\overset{=}\frac{1}{2}(z_{i}+z_{i+1})+\frac{h}{8}(f_{i}-f_{i+1}) \ {\color{red}(5)}$$

page38-2

$${{\color{red}(1)}{\color{blue}p.38-3} \ \overset{z}}_{i+{\color{red}\frac{1}{2}}}= {z}^_{i+{\color{red}\frac{1}{2}}}{\color{blue}\frac{1}{h}} \ {\color{red}(1)}$$

$${\overset{z}}_{i+{\color{red}\frac{1}{2}}}= z^{\color{red}'}(s={\color{red}\frac{1}{2}}){\color{blue} \overset{ HW^{*}6.6\begin{cases} & {\color{red}(1)}{\color{blue}p.37-2} \\ & {\color{red}(1)}{\color{blue}p.37-3} \end{cases}}}-\frac{3}{2}(z_{i}-z_{i+1})-\frac{1}{4}( {z}^_{i}+ {z}^_{i+1}) \ {\color{red}(2)}$$ $${\color{red}(1) \ _ \ (2):}$$ $${\overset{z}}_{i+{\color{red}\frac{1}{2}}}=\frac{-3}{2{\color{blue}h}}(z_{i}-z_{i+1})-\frac{1}{4}(f_{i}+f_{i+1}) \ {\color{red}(3)}$$ $${\overset{z}}_{i+{\color{red}\frac{1}{2}}}{\color{red} \neq f_{i+\frac{1}{2}}} \ in \ general$$

$$Gap \ = \ \Delta \ {\overset{z}}_{i+{\color{red}\frac{1}{2}}}-f_{i+{\color{red}\frac{1}{2}}} \ {\color{red}(4)}$$

$$Collocation \ at \ t_{i+{\color{red}\frac{1}{2}}} \ \Rightarrow \ \Delta \ = \ 0 \ {\color{red}(5)}$$

$${\color{blue}\underline{Goal:}} \ Find \ (z_{i},z_{i+1}) \ st \ \Delta \ = \ 0 \ {\color{red}(6)}$$

page38-3

$$ \Delta =0 \Rightarrow \ z_{i+1} \overset{=} {\color{blue} \underset{Simpson's \ rule \ {\color{red}! \ (2)} \ p.7-4}{ \underbrace}}$$

$$ \overset{z}=f(z,t) \ {\color{red}(2)}$$

$${\color{blue} \underset{ \underbrace}}={\color{blue} \underset{apply \ simpson's \ rule \ \Rightarrow \ {\color{red}(1)}}{ \underbrace}} \ {\color{red}(3)}$$

Opt. control pb.:{zi, i=1,2,...,n} unknownsolved by NLP(nonlin. progr)

$${\color{blue}IVP \ ( \underset{ \underbrace}pb)}: \ Int. \ {\color{blue} \underset{ \underbrace}}$$

page38-4

Hermite-Simpson time-stepping algo

Assume zi</SUB><U> known, find z</U><SUB>i+1</SUB><U> using(1)p.38-3</U></STRONG>

<STRONG>Need:</STRONG>

$${\color{blue}1)} \ f_{i}=f(z_{i},t_{i}) \ {\color{red}can \ comp.}$$

$${\color{blue}2)} \ f_{i+1}=f({\color{red} \underset{unknown}{ \underbrace}},{\color{red} \underset{known}{ \underbrace}}) \ {\color{red}unknown}$$

$${\color{blue}3)} \ f_{i+{\color{red}\frac{1}{2}}}=f({\color{red} \underset{unknown}{ \underbrace}},{\color{red} \underset{known}{ \underbrace}}) \ {\color{red}unknown}$$

$$t_{i+{\color{red}\frac{1}{2}}}=t_{i}+\frac{h}{2}$$

<STRONG>(5) p.38-1:</STRONG>

$$z_{i+{\color{red}\frac{1}{2}}}=g(z_{i},z_{i+1}) \ {\color{red}(1)}$$

<STRONG>(1) p.38-1:</STRONG>

$$z_{i+1}=z_{i}+\frac{h/2}{3}[f_{i}+4f(g(z_{i},z_{i+1}),t_{i+{\color{red}\frac{1}{2}}})+f_{i+1}] \ {\color{red}(2)}$$

$$ \Leftrightarrow \ F(z_{i+1}) \overset{=} \underset{ \underbrace{0}}$$ $${\color{blue}\Rightarrow \ Newton-Raphson-Simpson}$$