I have an exercise to find a Lyapunov-function for the following system:
$\dot{x}=-x+y\\ \dot{y}=-16x-8y+y^2$
I know which conditions such a Lyapunov-function has to have, but I dont have any idea how to find it...
Finding a Lyapunov-function for a nonlinear system
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$\begingroup$
ordinary-differential-equations
nonlinear-system
1 Answers
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Hint: A good way to construct a Lyapunov-function is to construct a quadratic form $[x,y]P[x,y]^T$, where $P$ is a two by two symmetric matrix. You need to play around with the entries in $P$ to find a "good" Lyapunov function.
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0Ok, thats what I also did already, but you need the equilibrium points first to get such a $P$, right? Is it correct to find a $P$ which solves the equation $PA^T+AP=-Q$, where $A$ is the Hessian-Matrix for an equilibrium point and $Q$ the 2x2 Identity matrix? – 2017-02-28