What is the significance of having eigenvalues that are 0? I think I heard you say "in that case you don't know anything". Does that mean you cannot determine if the system is stable or asymptotically stable?

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In today's lecture we were trying to analyze (rather than design) the stability of a given system, so the eigenvalues are already fixed. In the case that one or more eigenvalues are zero, you can say the following things depending on the type of system:

  • Linear system, can be transformed into diagonal form (4.8):

System is stable (in the sense of Lyapunov) if all other eigenvalues are strictly negative.

  • Linear system, can be transformed into the block diagonal form on page 106:

System is stable (in the sense of Lyapunov) if the real parts all other eigenvalues are strictly negative.

  • Linear system, cannot be transformed into the above two forms:

In general cannot determine the stability.

  • Linearized version of a nonlinear system:

In general cannot determine the stability.

--Shuo