hw4 - Show that the origin of (1) is asymptotically stable...

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Homework 4 Cover Sheet Name: Space below is for the instructor. Problem Score
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CDS 140a: Homework Set 4 Due: Wednesday, October 31, 2007. Nonlinear Differential Equations and Dynamical Systems , Ferdinand Ver- hulst, Second Edition. Chapter 3: 1, 3, 4, 5 Chapter 8: 1, 4, 5 Consider the discrete-time dynamical system: x k +1 = f ( x k ) (1) for f : R n R n and k Z . For a function V : R n R , the rate of change along solutions to the discrete time dynamical system is given by: Δ V ( x ) = V ( f ( x )) - V ( x ) . In the context of discrete-time dynamical systems: Restate the definitions of stability and asymptotic stability.
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Unformatted text preview: Show that the origin of (1) is asymptotically stable if in a neighborhood of the origin, there is a continuous positive denite function V ( x ) so that V ( x ) is negative denite. Consider the linear discrete-time system: x k +1 = Ax k . Show that the following statements are equivalent: * x = 0 is asymptotically stable. * | i | < 1 for all eigenvalues of A . * Given any Q = Q T > 0, there exists P = P T > 0, which is a unique solution to the linear equation A T PA-P =-Q ....
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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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hw4 - Show that the origin of (1) is asymptotically stable...

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