�
+
�
Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Problem
Set
2
Solutions
1
Problem
2.1
Consider
the
feedback
system
with
external
input
r
=
r
(
t
)
,
a
causal
linear
time
invariant
forward
loop
system
G
with
input
u
=
u
(
t
)
,
output
v
=
v
(
t
)
,
a
)
−
1
/
2
e
−
t
,
where
¯
and
impulse
response
g
(
t
)
=
0
.
1
�
(
t
)
+
(
t
+
¯
a
0
is
a
parameter,
→
and
a
memoryless
nonlinear
feedback
loop
u
(
t
)
=
r
(
t
)
+
π
(
v
(
t
))
,
where
π
(
y
)
=
sin(
y
)
.
It
is
customary
to
require
wellposedness
of
such
feedback
models,
r
±
�
u
±
G
±
v
π
(
y
)
Figure
2.1:
Feedback
setup
for
Problem
2.1
which
will
usually
mean
existence
and
uniqueness
of
solutions
v
=
v
(
t
)
,
u
=
u
(
t
)
of
system
equations
⎬
t
v
(
t
)
=
0
.
1
u
(
t
)
+
h
(
t
−
δ
)
u
(
δ
)
dδ,
u
(
t
)
=
r
(
t
)
+
π
(
v
(
t
))
0
on
the
time
interval
t
≤
[0
,
⊂
)
for
every
bounded
input
signal
r
=
r
(
t
)
.
1
Version
of
October
8,
2003
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�
2
(a)
Show
how
Theorem
3.1
from
the
lecture
notes
can
be
used
to
prove
wellposedness
in
the
case
when
¯
a
>
0
.
In
terms
of
the
new
signal
variable
y
(
t
)
=
v
(
t
)
−
0
.
1
π
(
v
(
t
))
−
0
.
1
r
(
t
)
system
equations
can
be
rewritten
as
⎬
t
y
(
t
)
=
h
(
t
−
δ
)[
r
(
δ
)
+
�
(
y
(
δ
)
+
0
.
1
r
(
δ
))]
dδ,
0
where
−
t
(
t
+
a
)
−
1
/
2
e
,
t
0
h
(
t
)
=
→
0
,
otherwise,
and
�
:
R
∞�
R
is
the
function
which
maps
z
≤
R
into
π
(
q
),
with
q
being
the
solution
of
q
−
0
.
1
π
(
q
)
=
z.
Since
π
is
continuously
differentiable,
and
its
derivative
ranges
in
[
−
1
,
1],
�
is
con
tinuously
differentiable
as
well,
and
its
derivative
ranges
between
1
/
1
.
1
and
1
/
0
.
9.
For
every
constant
T
≤
[0
,
⊂
),
the
equation
for
y
(
t
)
with
t
T
can
be
rewritten
→
as
⎬
t
y
(
t
)
=
y
(
T
)
+
a
T
(
y
(
δ
)
,
δ,
t
)
dδ,
T
where
a
T
(¯
y,
δ,
t
)
=
h
(
t
−
δ
)[
r
(
δ
)
+
�
(
y
(
δ
)
+
0
.
1
r
(
δ
))]
+
h
T
(
t
)
,
⎬
T
h
T
(
t
)
=
h
˙
(
t
−
δ
)[
r
(
δ
)
+
�
(
y
(
δ
)
+
0
.
1
r
(
δ
))]
dδ.
0
When
parameter
a
takes
a
positive
value,
function
a
=
a
T
satisfies
conditions
of
¯
Theorem
3.1
with
X
=
R
n
,
¯
a
)
being
a
x
0
=
y
(
T
),
r
=
1,
and
t
0
=
T
,
with
K
=
K
(¯
function
of
¯
a
=
0,
and
≥
M
=
M
T
=
M
0
(
a
)(1
+
max
y
(
t
) )
.
t
�
[0
,T
]


Hence
a
solution
y
=
y
(
·
)
defined
on
an
interval
t
≤
[0
,
T
]
can
be
extended
in
a
unique
way
to
the
interval
t
≤
[0
,
T
+
],
where
T
+
−
T
=
min
{
1
/M
T
,
1
/
(2
K
)
}
,
and
max
y
(
t
)
√
M
T
(
T
+
−
T
)
+
max
y
(
t
) )
.
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 Spring '09
 AlexandreMegretski
 Dynamics, Derivative, Continuous function, Limit sets

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