Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
TakeHome
Test
2
Solutions
1
Problem
T2.1
System
of
ODE
equations
x
˙(
t
)
=
Ax
(
t
)
+
Bψ
(
Cx
(
t
)
+
cos
(
t
))
,
(1.1)
where
A,
B,
C
are
constant
matrices
such
that
CB
=
0
,
and
ψ
:
R
k
is
∈�
R
q
continuously
differentiable,
is
known
to
have
a
locally
asymptotically
stable
nonequilibrium
periodic
solution
x
=
x
(
t
)
.
What
can
be
said
about
trace(
A
)
?
In
other
words,
find
the
set
�
of
all
real
numbers
�
such
that
�
=
trace(
A
)
for
some
A,
B,
C,
ψ
such
that
(1.1)
has
a
locally
asymptotically
stable
nonequilibrium
periodic
solution
x
=
x
(
t
)
.
Answer:
trace(
A
)
<
0.
Let
x
0
(
t
)
be
the
periodic
solution.
Linearization
of
(1.1)
around
x
0
(
·
)
yields
α
˙
(
t
)
=
Aα
(
t
)
+
Bh
(
t
)
Cα
(
t
)
,
where
h
(
t
)
is
the
Jacobian
of
ψ
at
x
0
(
t
),
and
x
(
t
)
=
x
0
(
t
)
+
α
(
t
)
+
o
(
α
(
t
) )
.


Partial
information
about
local
stability
of
x
0
(
·
)
is
given
by
the
evolution
matrix
M
(
T
),
where
T
>
0
is
the
period
of
x
0
(
·
):
if
the
periodic
solution
is
asymptotically
stable
then
all
eigenvalues
of
M
(
T
)
have
absolute
value
not
larger
than
one.
Here
M
˙
(
t
)
=
(
A
+
Bh
(
t
)
C
)
M
(
t
)
,
M
(0)
=
I,
1
Version
of
November
25,
2003
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�
�
�
�
2
and
hence
⎝
⎛�
T
det
M
(
T
)
=
exp
trace(
A
+
Bh
(
t
)
C
)
dt
.
0
Since
trace(
A
+
Bh
(
t
)
C
)
=
trace(
A
+
CBh
(
t
))
=
trace(
A
)
,
det(
M
(
T
))
>
1
whenever
trace(
A
)
>
0.
Hence
trace(
A
)
≈
0
is
a
necessary
condition
for
local
asymptotic
stability
of
x
0
(
·
).
Since
system
(1.1)
with
k
=
q
=
1,
ψ
(
y
)
≥
y
,
⎡
�
⎡
�
A
=
−
a
0
1
�
�
, B
=
,
C
=
0
1
0
a
0
−
has
periodic
stable
steady
state
solution
⎡
�
(1
+
a
2
)
−
1
cos(
t
)
+
a
(1
+
a
2
)
−
1
sin(
t
))
x
0
(
t
)
=
0
for
all
a
>
0,
the
trace
of
A
can
take
every
negative
value.
Thus,
to
complete
the
solution,
one
has
to
figure
out
whether
trace
of
A
can
take
the
zero
value.
It
appears
that
the
volume
contraction
techniques
are
better
suited
for
solving
the
question
completely.
Indeed,
consider
the
autonomous
ODE
⎞
⎨
z
˙
1
(
t
)
=
z
2
(
t
)
,
⎨
⎠
z
˙
2
(
t
)
=
z
1
(
t
)
,
⎛
−
⎝
(1.2)
⎨
z
1
(
t
)
⎨
⎧
z
˙
3
(
t
)
=
Az
3
(
t
)
+
Bψ
Cz
3
(
t
)
+
�
z
1
(
t
)
2
+
z
2
(
t
)
2
,
2
defined
for
z
1
+
z
2
2
∞
=
0.
If
(1.1)
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 Spring '09
 AlexandreMegretski
 Dynamics, Derivative, continuously differentiable function, periodic solution, state feedback linearizability, differentiable function g2

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