Lecture XIX
Euler Equations and a Basic Example
I.
Review
A.
Concepts borrowed from calculus
1.
Maximum/Minimum
a.
The variable x
*
maximizes f(x) if f(x
*
)f(x) > 0 for all x in the
neighborhood of x
*
.
b.
The function y
0
(x) maximizes I[y] if I[y
0
]I[y] > 0 for all
functions in the neighborhood of y
0
.
2.
Continuity
a.
Univariate:
lim
( )
lim
( )
(
)
xx
fx
fx fx
→→
+
==
00
0
b.
Continuity of functional I[y]:
If
ε
>0 there is a
δ
>0 such that
y
0
(x)y(x) <
δ
implies that I[y
0
(x)]I[y(x)]<
ε
, then I[y] is
continuous at y
0
(x).
c.
Why are we concerned?
If the functional is continuous around
y
0
(x), the optimal path, then we can use concepts like
differential calculus to determine the optimal path.
3.
Norms
a.
The norm of a function is the largest absolute value that a
function obtains over a stated range.
b.
The difference between two functions is then defined as the
largest absolute difference between the two functions over a
stated range.
c.
Extending the concept of the norm y(x)
0
is the largest
absolute value that the function y(x) obtains
y(x)
1
is the
largest absolute value of the function plus the largest absolute
value of the first derivative.
B.
Derivation of the Necessary Conditions
1.
In the Lecture XVIII we developed the general calculus of variation
problem as:
max [ ]
( , ( ), ’( ))
()
,
Iy
f xyx y x d
x
yx
y yx
y
x
x
=
∫
0
1
11
We assumed that there existed an optimal path y(x).
Further, we
assumed z(x) was in the neighborhood of y(x).
z(x) was constructed as
a deviation of y(x):
zx
=+
=
δ
δ
Note that if z(x) is near to y(x), z(x) approaches y(x), and z’(x)
approaches y’(x).
Thus, both
δ
y(x) and
∆
(
δ
y(x))
become very small.
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Again changing notation
δε
φ
yx
x
()
=
where
φ
(x
0
)=
φ
(x
1
)=0.
Further,
ε
is a scalar that we can let approach
zero.
Thus, the alternative feasible path can be expressed as
zx
x
=+
ε
φ
The functional value as a function of the alternative feasible path can
then be expressed as:
If
x
y
x
x
y
x
x
d
x
x
x
o
+
∫
(,()
,’
’
)
εφ
εφ
1
3.
To derive the maximum, we will differentiate the functional expressed
with respect to the alternative path with respect to
ε
.
To be technically
correct, we must first define two new functions
Φ(ε29
and
Φ
’
(ε29
based
on the z(x) and z’(x) functions.
Specifically,
x
x
’( )
⇒=
+
+
Φ
Φ
εε
φ
φ
The above functional then becomes
x
d
x
x
x
=
∫
(, ()
, ’
)
ΦΦ
0
1
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 Fall '08
 Moss
 Calculus, Derivative, Euler equation, c1 III, c2 dt c2

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