For a dynamical system, set $A$ is an invariant set with a function $V_1$, whose derivative is semi negative definite on $A$, and the region where the derivative is $0$ is the set $B$, which is also an invariant set. According to the theory of invariant sets, the trajectory in $A$ can only approach $B$ as it cannot go outside of $A$. Then there is also a function $V_2$ in $B$, whose derivative is semi negative definite on $B$. Similarly, the trajectory in $B$ must tend towards a certain set $C$. The current question is whether the trajectory in $A$ necessarily tends towards $C$.
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