Homework 3, due Wednesday March 2.

Due Wednesday, March 2, 2005.
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goodwine
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Homework 3, due Wednesday March 2.

Post by goodwine »

  1. Consider the motion of a particle of mass m attached to a spring with a nonlinear stiffness
    • Image
    where
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    and x is the displacement. The differential equation governing the system is
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    1. Using the total energy as a Lyapunov function, what can you conclude about the stability of the system?
    2. If damping is added to the system in the form
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      what can you conclude about the stability of the system?
  2. Using the Lyapunov function
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    obtain conditions on the coefficients
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    sufficient for global asymptotic stability of the origin of the Lorenz equations
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    where
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  3. Investigate the stability of the origin of
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    using a Lyapunov function of the form
    • Image
  4. Consider the linear first order system
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    1. Show that
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      is the solution of the differential equation.
    2. Is the origin a stable fixed point?
    3. Is the origin an asymptotically stable fixed point?
    4. Is the origin a uniformly asymptotically stable fixed point?
    In each case, justify your answer.
  5. Investigate the stability of the origin of
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    using a Lyapunov function of the form
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  6. Show that the zero solution of
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    is stable if
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    in a neighborhood of the origin.
  7. Investigate the stability of the origin of
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  8. Using the simple example of
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    work through the proof of case 3 in the table of Lyapunov theorems in Sastry's text.
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