140a_2009_midterm

140a_2009_midterm - 1 CALIFORNIA INSTITUTE OF TECHNOLOGY...

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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
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2 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.
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.

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140a_2009_midterm - 1 CALIFORNIA INSTITUTE OF TECHNOLOGY...

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