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 $E$.