Section $11\mathrm{.}2$ shows how to derive the minimum principle for t$1$ fixed and x$($t$1)$ freely varying,

which is stated as Theorem $11\mathrm{.}1,$ by using Lagrange multipliers. Theorem $11\mathrm{.}4$ states a version

of the minimum principle for t$1$ freely varying and some of the indices of x$($t$)$ specified. In this

problem you are to explain how to derive Theorem $11\mathrm{.}41$ by explaining how the derivation in

$1$Typos: p$.218,$ the equations in part $($b$)$: xi$($ti$)$ should be xi$($t$1).$ In the next line it should be for j in Ic not for

i in Ic

Section $11\mathrm{.}2($starting in the middle of page $208)$ should be modified. Steps $1-4$ are identical

except the function m has t as a second argument: m$($x$($t$1),$ t$).$

$($a$)$ What modifications are needed in Step $5$ to complete the derivation of Theorem $11\mathrm{.}4?$

$($b$)$ Consider Theorem $11\mathrm{.}4$ in the special case I $=$; both t$1$ and x$($t$1)$ are freely varying.

The optimal terminal time t$1$ could equal t$0$ if running the system for a nonzero amount

of time is more expensive than the decrease it brings in the terminal cost. Explain how

the theorem can be extended to cover such case by finding a variation of $(11\mathrm{.}15)$ for

the case t$1=$ t$0.($Hint: Let $$ V $($u$,$ t$1)$ represent the cost as a function of t$1$ if a constant

control $$u is used. Then for t$1=$ t$0$ to be optimal it is necessary that $$ V

$$t$1$

$($u$,$ t$0)0$ for

any $$u$.)$