Example.

Consider the mass-spring system introduced in §2.1. If the system is undamped, it will continue to vibrate forever once started. The question we ask is, Can the user grab and move one of the masses, say the $k$th one, in such a way as to eventually bring all the masses to rest? The question is about controllability, and the answer depends both on the value of $k$ and on the number and values of the spring constants. The calculation involves computing the smallest reducing subspace of $[Y, A - \lambda I]$, where $A$ is defined in (2.4) and $Y$ depends on $k$.

only discuss a few special nonlinear eigenproblems, we cannot say much about its general theory.