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I wanted to find somewhere (a book or a set of online notes would be ideal) that had the exact theorem about lyapunov functions and how they determine stability but in the case of non-autonomous systems

$$ \dot{x} = f(x,t) $$

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  • $\begingroup$ What kind of stability do you want to show using a Lyapunov function: Lyapunov, asymptotic or exponential? $\endgroup$ Apr 8, 2020 at 9:43
  • $\begingroup$ So I believe there's a theorem for an autonomous ode that says Let W be an open subset of $\mathbb{R}^n$ and $x^* \in W$. Suppose there exists a function $V:W \rightarrow \mathbb{R}, V \in C^1(W)$, satisfying $V(x^*) = 0$ and $V(x)>0$ for all $x \in W \setminus \{x^*\}$. Then If: i) $\dot{V}(x(t)) \leq 0$ for all $x \in W \setminus \{x^*\}$ then $x^*$ is stable ii) $\dot{V}(x(t)) < 0$ for all $x \in W \setminus \{x^*\}$ then $x^*$ is asymptotically stable I want something of a similar form but also taking about a time dependent ode $\endgroup$ Apr 8, 2020 at 12:31
  • $\begingroup$ I'm looking to show $$ f(x(t))-f(x^*) \leq \frac{\|x(0)-x^*\|^2}{t^2} $$ for a particular ode, I've got the function but I want the theorem to show it satsifies the conditions. $\endgroup$ Apr 8, 2020 at 12:48

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