1
CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 140a Midterm Examination
Jerry Marsden
Nov. 5, 2009
This is a three hour, closed book exam
While no aids are permitted, results from the course may be used
as long as they are accurately quoted.
Turn in your exam on or before 5pm, Friday, Nov. 13, 2009
Print Your Name
:
1
2
3
4
5
/100
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
1. The graph of the function
V
(
x
) =
x
4

4
x
3
+ 4
x
2
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 where appropriate.
(a) Show that solution curves exist for all positive time.
(b) Find the equilibrium points of the system
(c) Compute the linearization of the system at the equilibrium points and
determine the nature of the eigenvalues.
(d) What can you conclude about stability or instability of the given nonlin
ear system from the Liapunov eigenvalue theorem at the equilibria?
(e) Sketch the phase portrait.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Marsden
 Fundamental physics concepts, Linear system, Stability theory, Dynamical systems

Click to edit the document details