Lecture XVIII
A Reintroduction to the Control Problem and Derivation of the Euler
Conditions
I.
Another Overview of Dynamic Optimization
A.
The basic calculus problem is to choose the levels of variable x in a way that
max ( )
x
Fx
B.
We are interested in expanding this problem into the time dimension.
1.
One way of incorporating the time dimension is simply
max
( , ( ))
()
xt
T
Ftxt d
t
st
x t
0
0
∫
≥
However, this problem is separable in that it can be broken down into
several separate problems.
Remember x
t
in the discrete optimization
problem.
2.
Another specification that is truly dynamic is
max
( , ( ), ( ))
’( )
( , ( ), ( ))
, ()
xt ut
T
Ftxt ut d
t
xt gtx
tu
t
0
∫
=
In this approach we are trying to derive the path or a function u(t) that
describes the control process which adjusts x(t) in a way so as to
maximize “profit”.
The resulting conditions are:
(29
∂
∂
λ
∂
∂
λλ
λ
∂
∂λ
H
u
fg
H
x
tt
f
g
H
t
uu
xx
=⇒
+=
=
⇒
=

+
=⇒=
00
’
’
(, ()
)
3.
To arrive at this rule, we will first examine the calculus of variations
formulation:
max
( , ( ), ’( ))
T
Ftxt x t d
t
st x t
x
x
0
0
∫
≥=
II.
Example Solved
A.
A firm has received an order for B units of product to be delivered by time T.
It seeks a production function schedule for filling this order at the specified
delivery data at minimum cost, bearing in mind
that unit production cost rises
linearly with the production rate and that the unit cost of holding inventory per
unit time is constant.
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Let x(t) denote the inventory accumulated by time t.
Then we have
x(0)=0 and must achieve x(T)=B.
2.
The inventory level, at any moment, is the cummulated past production;
the rate of change of inventory is the production rate dx/dt=x’(t).
3.
Thus, the firm’s total cost at any moment, t, is
[]
cx t x t c xt c x t
cxt
12
1
2
2
’
() ’
()
’
+=
+
a.
If the cost of producing rises linearly, the pc=c
1
x’(t).
Thus, the
total cost of production is pc x’(t)=c
1
[x’(t)]
2
pc
x
b.
The second term is the unit cost of holding inventory.
4.
The firm’s objective is to determine the production plan x’(t) that
minimizes the total cost to the firm.
Thus,
(29
m
in
’
() , ( )
,’
cx
td
t
st x
x T
B x t
T
1
2
2
0
00
0
+
==≥
∫
5.
One possible plan is to produce at a uniform rate x’(t)=B/T.
Then,
xt
B
T
dt
Bt
T
T
==
∫
0
and the cost incurred will be
c
B
T
c
Bt
T
dt
c
B
T
c
BT
T
1
2
2
0
1
2
2
2
+
=+
∫
while this plan is feasible, it may not minimize costs.
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 Fall '08
 Moss
 Calculus, Derivative, Yi, dt

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