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?

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