�
�
�
�
Spr
2008
16.323
4–3
•
Assuming
J
�
(
x
(
t
+ Δ
t
)
, t
+ Δ
t
)
has
bounded
second
derivatives
in
both
arguments,
can
expand
this
cost
as
a
Taylor
series
about
x
(
t
)
, t
∂J
�
J
�
(
x
(
t
+ Δ
t
)
, t
+ Δ
t
)
≈
J
�
(
x
(
t
)
, t
) +
(
x
(
t
)
, t
)
Δ
t
∂t
∂J
�
+
(
x
(
t
)
, t
)
(
x
(
t
+ Δ
t
)
−
x
(
t
))
∂
x
–
Which
for
small
Δ
t
can
be
compactly
written
as:
J
�
(
x
(
t
+ Δ
t
)
, t
+ Δ
t
)
≈
J
�
(
x
(
t
)
, t
) +
J
�
(
x
(
t
)
, t
)Δ
t
t
+
J
x
�
(
x
(
t
)
, t
)
a
(
x
(
t
)
,
u
(
t
)
, t
)Δ
t
•
Substitute
this
into
the
cost
calculation
with
a
small
Δ
t
to
get
J
�
(
x
(
t
)
, t
) =
u
(
t
)
∈U
{
g
(
x
(
t
)
,
u
(
t
)
, t
)Δ
t
+
J
�
(
x
(
t
)
, t
)
min
+
J
�
(
x
(
t
)
, t
)Δ
t
+
J
�
(
x
(
t
)
, t
)
a
(
x
(
t
)
,
u
(
t
)
, t
)Δ
t
}
t
x
•
Extract
the
terms
that
are
independent
of
u
(
t
)
and
cancel
0 =
J
�
(
x
(
t
)
, t
)+ min
(
x
(
t
)
, t
)
a
(
x
(
t
)
,
u
(
t
)
, t
)
}
u
(
t
)
∈U
{
g
(
x
(
t
)
,
u
(
t
)
, t
) +
J
�
t
x
–
This
is
a
partial
differential
equation
in
J
�
(
x
(
t
)
, t
)
that
is
solved
backwards
in
time
with
an
initial
condition
that
J
�
(
x
(
t
f
)
, t
f
) =
h
(
x
(
t
f
))
for
x
(
t
f
)