$6\mathrm{.}34$

$$Consider the linear system

x

k

$+$

$1$

$=$

a

x

k

$+$

b

u

k

$+$

$1$

$,$

k

$>=$

$0$

$,$

$$where

x

i

in

R

$,$

u

i

inR, and the initial condition is

x

$0$

$=$

$0$

$.$

$$Find the values of the

control inputs

u

$1$

$,$

dots,

u

n

$$to minimize

$-$

q

x

n

$+$

r

$$

i

$=$

$1$

n

u

i

$2$

$,$

$$

where

q

$,$

r

$>$

$0$

$$are given constants. This can be interpreted as desir

$-$

ing to make

x

n

$$as large as possible but at the same time desiring to

make the total input energy

$\backslash$Sigma

i

$=$

$1$

n

u

i

$2$

$$as small as possible. The con

$-$

stants

q

$$and

r

$$reflect the relative weights of these two objectives.