If the Lyapunov function is $$ V(x) = x^2_1 + x^2_2-1 $$ And its time derivative is $$ \dot{V}(x) = -(x^2_1 + x^2_2) $$ Why the origin is globally asymptotically stable?
2026-03-25 12:55:37.1774443337
Why the origin is globally asymptotically stable?
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Strictly speaking, $V(x) = x^2_1 + x^2_2-1 $ is not a Lyapunov function because it is not positive definite. But one can take $W(x)=V(x)+1=x^2_1 + x^2_2$. It is positive definite, radially unbounded and its derivative $\dot W(x)=\dot V(x)+0=-(x^2_1 + x^2_2)$ is negative definite; thus, the origin is globally asymptotically stable.