It is possible to distinguish the properties "time-varying" and "nonautonomous" in dynamical systems regarding Lyapunov stability analysis?
Does it make a difference if the system depends explicitly on $t$ or indirectly on $t$ due to a time-varying parameter?
I want to explain the problem in detail:
Let a dynamical system denoted by $\dot x = f$, with state $x$. We say that a dynamical system is nonautonomous if the dynamics $f$ depend on time $t$, i.e. $$\dot x = f(t,x).$$
For instance the systems $$ \dot x = - t x^2 $$ and $$ \dot x = -a(t)x,$$ are nonautonomous. Let $a(t)$ be a bounded time-varying parameter, i.e. $||a(t)||<a^+$ and strictly positive, i.e. $a(t) > 0$.
Particularly, the second example is more likely denoted as a time-varying linear system, but of course it is nonautonomous.
In Lyapunov stability analysis autonomous and nonautonomous systems must be strongly distinguished to make assertions about stability of the system, and the Lyapunov analysis for nonautonomuos systems is much more difficult.
And here for me some questions arise. When i want to analize stability of the second example must i really use the Lyapunov theory for nonautonomous systems?
It follows for the candidate $V = 1/2 x^2$
$$\dot V = -a(t)x^2,$$
which is negative definite. Is the origin really asymptotically stable, as i suppose, or must i take the nonautomous characteristic into account in this case?
I would suppose it makes a difference if a system depends explicitly on $t$ as in the first example or just indirect due to a time-varying parameter, since $t$ approaches infinity, but a parameter does not.