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CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 140a Midterm Examination
Jerry Marsden
Nov. 6, 2008
This is a three hour, closed book exam
While no aids permitted, results from the course may be used, but
they must be quoted.
Turn in your exam on or before 5pm, Wednesday, Nov. 12.
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1. The graph of the function
V
(
x
) =
x
(
x

1)(
x

2)(
x

4) is shown in the
following ﬁgure.
Consider the planar system
˙
x
=
y
˙
y
=

V
0
(
x
)

νy
where
ν
≥
0 is a constant. In what follows, distinguish the cases
ν >
0 and
ν
= 0.
(a) Show that solution curves exist for all positive time.
(b) Let ¯
x
be the local maximum of
V
lying between 1 and 2. Compute the
linearization of the system at the equilibrium point (¯
x,
0) in terms of
V
00
(¯
x
) and determine the nature of the eigenvalues.
(c) Can you conclude anything about stability or instability of the given non
linear system from the Liapunov eigenvalue theorem at the point (¯
x,
0)?
(d) Sketch qualitatively what the phase portrait looks like.
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 Fall '09
 Marsden

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