Hi!

Let's say that we have a dynamical system described by

$\dot{x} = f(x)$,

where f is some nonlinear function, which has several equilibria. Assume that we have found a continously differentiable Liapunov function V such that

$\dot{V} = 0 \Rightarrow \dot{x} = 0$.

Then, assuming that V is radially unbounded, by the LaSalle invariance principle we should be able to say that the system always converges to an equilibrium point. However, in some works I have seen the additional requirement that in order to show convergence, all equilibrium points must be isolated, otherwise the system could move indefinitely inside a connected set of equilibrium points. Can that really be the case for the situation described above? Doesn't $\dot{x} = 0$ mean that the system has "stopped" (assuming that $x$ completely describes the state of the system)? It seems to me that in my case, the assumption about isolated equilibria is unnecessary.

Kind regards

Olav