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