140aMidterm2008

140aMidterm2008 - 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. 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. Print Your Name : 1 2 3 4 5 /100
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2 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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140aMidterm2008 - 1 CALIFORNIA INSTITUTE OF TECHNOLOGY...

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