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How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

\begin{equation} \left.\begin{aligned} \dot{x_1} &= x_2 \\ \dot{x_2} &= -\frac{x_1}{1 + x_2^2} \label{eq:q2} \end{aligned}\qquad\right\} \end{equation}

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  • $\begingroup$ Are you familiar with Lyapunov functions? And what have you tried yourself? $\endgroup$ Nov 12, 2018 at 16:58
  • $\begingroup$ I have tried general quadratic functions, such as (x1^2 + x2^2), but a term xy remains and I can't conclude properly. $\endgroup$
    – Scholar
    Nov 12, 2018 at 17:25

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You could try to get the Lyapunov function in the form $$ V=\frac12x_1^2+g(x_2) $$ so that $$ \dot V=x_1x_2+g'(x_2)\frac{-x_1}{1+x_2^2} $$ so that one would get a usable result with $g'(x_2)=x_2(1+x_2^2)$, integrating to,for example, $g(x_2)=\frac14(1+x_2^2)^2$.

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  • $\begingroup$ Thank you so much. I was taking something similar to g(x2) but it was constant and I was stuck. Thanks for your help :) $\endgroup$
    – Scholar
    Nov 12, 2018 at 18:55
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    $\begingroup$ @SampleTime : Then consider $V-V(0)$ to get the normalized value. I prefer the completed square for its simplicity. $\endgroup$ Nov 12, 2018 at 21:18
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    $\begingroup$ @SampleTime : Could you elaborate? How to get a negative value from $V-V(0)=\frac12x_1^2+\frac12x_2^2+\frac14x_2^4$? $\endgroup$ Nov 12, 2018 at 21:27
  • $\begingroup$ @LutzL You are correct, nice! I made an error when writing down the equation, sorry for the confusion. $\endgroup$
    – SampleTime
    Nov 12, 2018 at 21:40

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