�
Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Problem
Set
5
Solutions
1
Problem
5.1
y
(
t
)
≥
a
is
an
equilibrium
solution
of
the
differential
equation
y
(3)
(
t
)
+
¨
y
(
t
)
+
˙
y
(
t
)
+
2
sin(
y
(
t
))
=
2
sin(
a
)
,
where
a
→
R
and
y
(3)
denotes
the
third
derivative
of
y
.
For
which
values
of
a
→
R
is
this
equilibrium
locally
exponentially
stable?
The
linearized
equations
for
small
�
(
t
)
=
y
(
t
)
−
a
are
given
by
�
(3)
(
t
)
+
�
¨
(
t
)
+
�
˙
(
t
)
+
2
cos(
a
)
�
(
t
)
=
0
.
The
linearized
system
is
asymptotically
stable
if
and
only
if
0
<
cos(
a
)
<
0
.
5.
Hence
the
equilibrium
y
(
t
)
≥
a
of
the
original
system
is
locally
exponentially
stable
if
and
only
if
0
<
cos(
a
)
<
0
.
5.
Problem
5.2
In
order
to
solve
a
quadratic
matrix
equation
X
2
+
AX
+
B
= 0
,
where
A,
B
are
given
n
by
n
matrices
and
X
is
an
n
by
n
matrix
to
be
found,
it
is
proposed
to
use
an
iterative
scheme
X
k
+1
=
X
k
2
+
AX
k
+
X
k
+
B.
Assume
that
matrix
X
�
satisfies
X
2
+
AX
�
+
B
=
0
.
What
should
be
required
of
the
eigenvalues
of
X
�
and
A
+
X
�
in
order
to
guarantee
that
X
k
�
X
�
1
Version
of
November
12,
2003
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�
�
�
�
2
exponentially
as
k
� ≈
when
≡
X
0
−
X
�
≡
is
small
enough?
You
are
allowed
to
use
the
fact
that
matrix
equation
ay
+
yb
=
0
,
where
a,
b,
y
are
n
by
n
matrices,
has
a
nonzero
solution
y
if
and
only
if
det(
sI
−
a
)
=
det(
sI
+
b
)
for
some
s
→
C
.
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 Spring '09
 AlexandreMegretski
 Dynamics, x3, Massachusetts Institute of Technology, Stability theory, x�, nbyn matrices, DYNAMICS OF NONLINEAR SYSTEMS

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