LaSalle for time varying systems I am looking for an explanation, why LaSalles theorem is in general not applicable to time varying systems. Can someone provide an example system with
$$
\dot{x}(t) = A(t)x(t) \tag{1}
$$
I.e., why can't LaSalles theorem be used if I have a Lyapunov function $V(t, x)$ for the system $(1)$ with $\dot{V}(t, x) \leq 0$?

Consider the system 
$$
\left\{\begin{array}{lll}\tag{1}
\dot x&=&0\\
\dot y&=&0\\
\end{array}\right.
$$
and the function $V(t,x,y)=e^{-t}(x^2+y^2)$. The directional derivative is negative definite:
$$
\dot V= -e^{-t}(x^2+y^2)+e^{-t}(2x\dot x+2y\dot y)=-e^{-t}(x^2+y^2),
$$
but the solutions of (1) do not approach the set
$$
E=\{ (x,y):\; \dot V= 0\}=\{(0,0)\}.
$$
This example is possible because $V(t,x)$ may decrease due to the explicit dependence on $t$, regardless of the approaching of the solution to the set $E$.
