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Let an autonomous dynamical system (system of ODEs) be given by

$$\frac{dx}{dt}=f(x)$$

where $f(x): \mathbb{R}^n \to \mathbb{R}^n$ and also $f$ is $C^2(\mathbb{R}^n,\mathbb{R}^n)$. We also assume that there exists a function $V(x(t)) \geq 0$ s.t $\dot V \leq 0$ and also $\dot V=0$ only at fixed points. Under what assumptions we have convergence to a fixed point? Thanks for the help

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If in addition $V$ is radially unbounded, then you have $x(t)$ converging to a fixed point.

There exists systems where $V>0$ and $\dot V \le 0$ globally but there are trajectories that go to infinity.

A weaker condition would require all level sets of $V$ be bounded.

Then there is the LaSalle's theorem.

There is no general theory other than specific cases.

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  • $\begingroup$ Hi and thanks for the reply. Can you give me some references for your claims or a formal proof (for radially unbounded?). I am worried about when for an initial condition I can have two or more accumulation points which are fixed points. Can this be the case? $\endgroup$
    – user61581
    Feb 18, 2014 at 4:27
  • $\begingroup$ Nonlinear Systems by Hassan K. Khalil is a good source. I like the first edition which is not as comprehensive as the thrid, but covers the basics better. $\endgroup$
    – user44197
    Feb 18, 2014 at 13:50

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