Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Problem
Set
6
Solutions
1
Problem
6.1
For
the
following
statement,
verify
whether
it
is
true
or
false
(give
a
proof
if
true,
give
a
counterexample
if
false):
Assume
that
(a)
n, m
are
positive
integers;
(b)
f
:
R
n
×
R
m
and
g
:
R
n
×
R
m
are
continuously
differ
∈
R
n
∈
R
m
entiable
functions;
(c)
the
ODE
y
˙(
t
)
=
g
(¯
x, y
(
t
))
has
a
globally
asymptotically
stable
equilibrium
for
every
x
¯
→
R
n
;
(d)
functions
x
0
:
[0
,
1]
∈
R
n
and
y
0
:
[0
,
1]
∈
R
m
are
continuously
differentiable
and
satisfy
x
˙
0
(
t
)
=
f
(
x
0
(
t
)
, y
0
(
t
))
,
g
(
x
0
(
t
)
, y
0
(
t
))
=
0
±
t
→
[0
,
1]
.
Then
there
exists
0
>
0
such
that
for
every
→
(0
,
0
)
the
differential
equation
x
˙(
t
)
=
f
(
x
(
t
)
, y
(
t
))
,
y
˙(
t
)
=
−
1
g
(
x
(
t
)
, y
(
t
))
,
has
a
solution
x
:
[0
,
1]
∈
R
n
,
y
:
[0
,
1]
∈
R
m
such
that
x
e
(0)
=
x
0
(0)
,
y
(0)
=
y
0
(0)
,
and
x
(
t
)
converges
to
x
0
(
t
)
as
0
for
all
t
→
[0
,
1]
.
1
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 Spring '09
 AlexandreMegretski
 Dynamics, Derivative, Trigraph, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, DYNAMICS OF NONLINEAR SYSTEMS

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